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9880:. Convergence of the net implies convergence of the eventuality filter. This correspondence allows for any theorem that can be proven with one concept to be proven with the other. For instance, continuity of a function from one topological space to the other can be characterized either by the convergence of a net in the domain implying the convergence of the corresponding net in the codomain, or by the same statement with filter bases.
1544:{\displaystyle {\begin{alignedat}{4}&x_{\bullet }&&\to \;&&x&&\;\;{\text{ in }}X\\&x_{a}&&\to \;&&x&&\;\;{\text{ in }}X\\\lim \;&x_{\bullet }&&\to \;&&x&&\;\;{\text{ in }}X\\\lim _{a\in A}\;&x_{a}&&\to \;&&x&&\;\;{\text{ in }}X\\\lim _{a}\;&x_{a}&&\to \;&&x&&\;\;{\text{ in }}X.\end{alignedat}}}
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necessarily a sequence. Moreso, a subnet of a sequence may be a sequence, but not a subsequence. But, in the specific case of a sequential space, every net induces a corresponding sequence, and this relationship maps subnets to subsequences. Specifically, for a first-countable space, the net
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16586:, which states that the product of any collection of compact topological spaces is compact. But if every compact space is also Hausdorff, then the so called "Tychonoff's theorem for compact Hausdorff spaces" can be used instead, which is equivalent to the
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argues that despite their equivalence, it is useful to have both concepts. He argues that nets are enough like sequences to make natural proofs and definitions in analogy to sequences, especially ones using sequential elements, such as is common in
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334:), there is always some element that is "above" both of them (greater than or equal to each); in this way, directed sets generalize the notion of "a direction" in a mathematically rigorous way. Importantly though, directed sets are
16594:. Nets can be used to give short proofs of both version of Tychonoff's theorem by using the characterization of net convergence given above together with the fact that a space is compact if and only if every net has a convergent
9682:
is a related idea in topology that allows for a general definition for convergence in general topological spaces. The two ideas are equivalent in the sense that they give the same concept of convergence. More specifically, every
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9854:
4771:. Topologies can also be characterized by closed subsets since a set is open if and only if its complement is closed. So the characterizations of "closed set" in terms of nets can also be used to characterize topologies.
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14100:{\displaystyle {\begin{alignedat}{4}\pi _{l}:\;&&{\textstyle \prod }X_{\bullet }&&\;\to \;&X_{l}\\&&\left(x_{i}\right)_{i\in I}&&\;\mapsto \;&x_{l}\\\end{alignedat}}}
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In the context of topology, sequences do not fully encode all information about functions between topological spaces. In particular, the following two conditions are, in general, not equivalent for a map
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around a point. Therefore, while sequences do not encode sufficient information about functions between topological spaces, nets do, because collections of open sets in topological spaces are much like
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Virtually all concepts of topology can be rephrased in the language of nets and limits. This may be useful to guide the intuition since the notion of limit of a net is very similar to that of
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The analogue of "subsequence" for nets is the notion of a "subnet". There are several different non-equivalent definitions of "subnet" and this article will use the definition introduced in
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In general, this statement would not be true if the word "net" was replaced by "sequence"; that is, it is necessary to allow for directed sets other than just the natural numbers if
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With this change, the conditions become equivalent for all maps of topological spaces, including topological spaces that do not necessarily have a countable or linearly ordered
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of a net of real numbers can be defined in a similar manner as for sequences. Some authors work even with more general structures than the real line, like complete lattices.
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to the
Hausdorff condition on the space, and indeed this may be taken as the definition. This result depends on the directedness condition; a set indexed by a general
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17306:, this characterization says that a net converges to a point in a metric space if and only if the distance between the net and the point converges to zero. If
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While condition 1 always guarantees condition 2, the converse is not necessarily true. The spaces for which the two conditions are equivalent are called
4070:). Although Cauchy nets are not needed to describe completeness of normed spaces, they are needed to describe completeness of more general (possibly non-
20162:
11752:
which makes this one instance where (non-sequence) nets must be used because sequences alone can not reach the desired conclusion. Compare elements of
10974:, are sequential spaces, but not all topological spaces are sequential. Nets generalize the notion of a sequence so that condition 2 reads as follows:
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15072:{\displaystyle \pi _{i}\left(f_{\bullet }\right)\;{\stackrel {\scriptscriptstyle {\text{def}}}{=}}\;\left(\pi _{i}\left(f_{a}\right)\right)_{a\in A}}
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14368:{\displaystyle \pi _{i}\left(f_{\bullet }\right)~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~\left(\pi _{i}\left(f_{a}\right)\right)_{a\in A}}
8003:
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10210:. Nets are frequently denoted using notation that is similar to (and inspired by) that used with sequences. For example, the subscript notation
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1824:, every net has at most one limit, and the limit of a convergent net is always unique. Some authors do not distinguish between the notations
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A net has a limit if and only if all of its subnets have limits. In that case, every limit of the net is also a limit of every subnet.
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The learning curve for using nets is typically much less steep than that for filters, which is why many mathematicians, especially
14753:{\displaystyle \pi _{i}\left(f_{\bullet }\right)~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~\pi _{i}\,\circ \,f_{\bullet }.}
9691:
using the filter's pointed sets, and convergence of the filter base implies convergence of the associated net. Similarly, any net
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3477:, every net has some subnet that is an ultranet, but no nontrivial ultranets have ever been constructed explicitly. If
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has the property that every finite subcollection has non-empty intersection. Thus, by the remark above, we have that
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R. G. Bartle, Nets and
Filters In Topology, American Mathematical Monthly, Vol. 62, No. 8 (1955), pp. 551–557.
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1790:{\displaystyle \lim x_{\bullet }=x\;~~{\text{ or }}~~\;\lim x_{a}=x\;~~{\text{ or }}~~\;\lim _{a\in A}x_{a}=x}
89:
that (in general), sequences are unable to characterize (this shortcoming of sequences motivated the study of
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Limit superior of a net of real numbers has many properties analogous to the case of sequences. For example,
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is not empty, then the axiom of choice would (in general) still be needed to conclude that the projections
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16783:{\displaystyle \limsup x_{a}=\lim _{a\in A}\sup _{b\succeq a}x_{b}=\inf _{a\in A}\sup _{b\succeq a}x_{b}.}
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21446:. Mathematics and its Applications 268. Dordrecht: Kluwer Academic Publishers Group. pp. xii, 340.
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Every constant net is a (trivial) ultranet. Every subnet of an ultranet is an ultranet. Assuming the
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By the proof given in the next section, it is equal to the set of limits of convergent subnets of
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19682: – Use of filters to describe and characterize all basic topological notions and results.
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example of a directed set. A sequence is a function on the natural numbers, so every sequence
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Characterizations of the category of topological spaces § Convergent net characterization
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It is in this way that nets are generalizations of sequences: rather than being defined on a
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17586:{\displaystyle \left\|m-m_{\bullet }\right\|:=\left(\left\|m-m_{a}\right\|\right)_{a\in A}.}
16971:
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10662:
9895:. In any case, he shows how the two can be used in combination to prove various theorems in
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14845:{\displaystyle L=\left(L_{i}\right)_{i\in I}\in {\textstyle \prod \limits _{i\in I}}X_{i},}
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4066:) if and only if every Cauchy sequence converges to some point (a property that is called
4062:, which is a special type of topological vector space, is a complete TVS (equivalently, a
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8:
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19679:
19673:
17291:{\displaystyle d\left(m,m_{\bullet }\right):=\left(d\left(m,m_{a}\right)\right)_{a\in A}}
16508:{\displaystyle {\textstyle \prod }X_{\bullet }={\textstyle \prod \limits _{j\in I}}X_{j}}
16397:(because then there is nothing to choose between), which happens for example, when every
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can be interpreted as a limit of a net. Specifically, the net is eventually in a subset
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10529:
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10271:
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10091:
10042:
10022:
9892:
9859:
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9635:
9615:
9591:
9571:
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9392:
9372:
9226:
9206:
9186:
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8361:
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7448:
7363:
7307:
7264:
6899:
6879:
6770:
6687:
6667:
6647:
6627:
6624:; that there is always such a point follows from the fact that no open neighborhood of
6607:
6538:
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6242:
6196:
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5555:
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5266:
5201:
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4143:
4123:
3860:
3727:
3541:
3398:
3307:
3223:
3055:
2818:
2367:
2177:
2005:
1900:
1554:
1258:
1235:
1204:
1134:
1107:
901:
789:
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602:
475:
317:
147:
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21683:
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21607:
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21420:
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13846:
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792:
notation, the filled disk or "bullet" stands in place of the input variable or index
66:
21571:
21060:
20099:
19691:
17689:
17349:
16885:
13912:
11651:{\displaystyle \mathbf {0} \in \operatorname {cl} _{\mathbb {R} ^{\mathbb {R} }}E.}
11316:
10963:
9896:
9883:
8833:
The set of cluster points of a net is equal to the set of limits of its convergent
6451:(since the intersection of every two such neighborhoods is an open neighborhood of
5223:
2936:
347:
90:
42:
31:
30:
This article is about nets in topological spaces. For unfoldings of polyhedra, see
20245:{\displaystyle \left(x_{\bullet }\circ \varphi \right)(r)=x_{\varphi (r)}=0=s_{r}}
9922:
is directed. Therefore, every function on such a set is a net. In particular, the
21659:
21503:
21457:
21430:
21048:
20403:
17881:
as", so that "large enough" with respect to this relation means "close enough to
16965:
16591:
16425:
16265:
13600:
9849:{\displaystyle \left\{\left\{x_{a}:a\in A,a_{0}\leq a\right\}:a_{0}\in A\right\}}
9609:
3775:
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2563:
1821:
351:
114:
21628:
21597:
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21515:
16610:
16606:
16576:
13218:
11440:
everywhere except for at most finitely many points (that is, such that the set
9923:
4767:
It is these characterizations of "open subset" that allow nets to characterize
4071:
118:
13192:
is a directed set, where the direction is given by reverse inclusion, so that
21698:
21665:
21650:
20506:{\displaystyle \left(s_{i}\right)_{i\in \mathbb {N} }:=(1,1,2,2,3,3,\ldots )}
17303:
13791:
10178:
9661:
9410:
7381:
4042:
3779:
3663:{\displaystyle f\circ x_{\bullet }=\left(f\left(x_{a}\right)\right)_{a\in A}}
343:
21499:
21507:
17345:
16937:
16288:
exists; the axiom of choice is not needed in some situations, such as when
12060:
11235:{\displaystyle {\textstyle \prod \limits _{x\in \mathbb {R} }}\mathbb {R} }
11125:
10971:
10207:
6448:
4063:
4059:
4011:
141:
129:
74:
58:
21687:
21541:
18854:
15975:
8172:{\displaystyle \bigcap _{a\in A}\operatorname {cl} E_{a}\neq \varnothing }
7074:{\displaystyle \lim _{}\left(f\left(x_{a}\right)\right)_{a\in A}\to f(x).}
6186:{\displaystyle \lim _{}\left(f\left(x_{a}\right)\right)_{a\in A}\to f(x).}
5954:{\displaystyle \lim _{}\left(f\left(x_{a}\right)\right)_{a\in A}\to f(x),}
21044:
17299:
16889:
10181:
9919:
9907:
9684:
339:
110:
38:
10636:
that is, if and only if infinitely many elements of the sequence are in
6810:
this neighborhood is a member of the directed set whose index we denote
4086:. The following set of theorems and lemmas help cement that similarity:
21397:
Sundström, Manya Raman (2010). "A pedagogical history of compactness".
21214:, Dover Books on Mathematics, Courier Dover Publications, p. 260,
21072:
20586:{\displaystyle \left(x_{i}\right)_{i\in \mathbb {N} }:=(1,2,3,\ldots )}
18860:
20341:{\displaystyle \left(s_{r}\right)_{r\in R}=x_{\bullet }\circ \varphi }
19880:{\displaystyle x_{\bullet }=(0)_{i\in \mathbb {N} }:\mathbb {N} \to X}
16869:{\displaystyle \limsup(x_{a}+y_{a})\leq \limsup x_{a}+\limsup y_{a},}
15318:
However, the converse does not hold in general. For example, suppose
13904:{\displaystyle {\textstyle \prod }X_{\bullet }:=\prod _{i\in I}X_{i}}
21403:
21064:
2266:
is a cluster point if and only if it has a subset that converges to
19685:
17861:
In other words, the relation is "has at least the same distance to
15422:{\displaystyle f_{\bullet }=\left(f_{a}\right)_{a\in \mathbb {N} }}
10085:
9888:
9657:
165:
82:
70:
62:
20682:{\displaystyle h(i):=\left\lfloor {\tfrac {i+1}{2}}\right\rfloor }
14654:{\displaystyle \pi _{i}:{\textstyle \prod }X_{\bullet }\to X_{i};}
13455:
in the net are constrained to lie in decreasing neighbourhoods of
21526:. Vol. 27. New York: Springer Science & Business Media.
16568:{\displaystyle \pi _{i}:{\textstyle \prod }X_{\bullet }\to X_{i}}
16268:
might be need to be assumed in order to conclude that this tuple
14591:{\displaystyle f_{\bullet }:A\to {\textstyle \prod }X_{\bullet }}
13553:
11131:
For an example where sequences do not suffice, interpret the set
190:(unless indicated otherwise), with the property that it is also (
20159:
is an order morphism whose image is cofinal in its codomain and
10685:
is a cluster point of the net if and only if every neighborhood
8117:{\displaystyle \{\operatorname {cl} \left(E_{a}\right):a\in A\}}
4612:
14496:
It is sometimes useful to think of this definition in terms of
9652:
with two distinct limits. Thus the uniqueness of the limit is
9433:
are ordered by inclusion) makes it a directed set, and the net
6444:
2531:{\displaystyle x_{\geq a}:=\left\{x_{b}:b\geq a,b\in A\right\}}
21165:
21163:
9612:, the limit of a net, if it exists, is unique. Conversely, if
4077:
314:
In words, this property means that given any two elements (of
16876:
where equality holds whenever one of the nets is convergent.
14489:{\displaystyle \pi _{i}\left(f_{\bullet }\right):A\to X_{i}.}
10084:
Conversely, any net whose domain is the natural numbers is a
9906:, prefer them over filters. However, filters, and especially
8058:{\displaystyle E_{a}\triangleq \left\{x_{b}:b\geq a\right\}.}
4724:
is open if and only if every net converging to an element of
2895:{\displaystyle s_{i}=x_{h(i)}\quad {\text{ for all }}i\in I.}
2330:{\textstyle \operatorname {cl} _{X}\left(x_{\bullet }\right)}
8309:
has a convergent subnet. For the sake of contradiction, let
1571:
is clear from context, it may be omitted from the notation.
740:
Notation for nets varies, for example using angled brackets
21160:
8643:
This net cannot have a convergent subnet, because for each
2559:
2428:{\textstyle \operatorname {cl} _{X}\left(x_{\geq a}\right)}
21131:
21129:
21127:
21125:
21123:
21121:
21119:
16126:{\displaystyle \pi _{i}\left(f_{\bullet }\right)\to L_{i}}
12389:{\displaystyle \left(x_{h_{n}}\right)_{n\in \mathbb {N} }}
9664:
may have distinct limit points even in a
Hausdorff space.
8434:{\displaystyle D\triangleq \{J\subset I:|J|<\infty \}.}
21106:
21104:
20402:
because it is not even a sequence since its domain is an
8953:
automatically assumed to be a directed set) and also let
5781:{\displaystyle \left(f\left(x_{a}\right)\right)_{a\in A}}
5007:{\displaystyle \lim {}f\left(x_{\bullet }\right)\to f(x)}
781:{\displaystyle \left\langle x_{a}\right\rangle _{a\in A}}
21326:
18387:
in the usual sense (meaning that for every neighborhood
17048:{\displaystyle m_{\bullet }=\left(m_{i}\right)_{a\in A}}
14163:{\displaystyle f_{\bullet }=\left(f_{a}\right)_{a\in A}}
13794:
has a limit if and only if each projection has a limit.
11876:
This pointwise comparison is a partial order that makes
8906:{\displaystyle x_{\bullet }=\left(x_{a}\right)_{a\in A}}
7924:{\displaystyle x_{\bullet }=\left(x_{a}\right)_{a\in A}}
7553:
is compact. We will need the following observation (see
7438:{\displaystyle x_{\bullet }=\left(x_{a}\right)_{a\in A}}
6528:{\displaystyle x_{\bullet }=\left(x_{a}\right)_{a\in A}}
6058:{\displaystyle x_{\bullet }=\left(x_{a}\right)_{a\in A}}
5360:{\displaystyle x_{\bullet }=\left(x_{a}\right)_{a\in A}}
4889:{\displaystyle x_{\bullet }=\left(x_{a}\right)_{a\in A}}
4254:{\displaystyle s_{\bullet }=\left(s_{a}\right)_{a\in A}}
3839:{\displaystyle x_{\bullet }=\left(x_{a}\right)_{a\in A}}
3531:{\displaystyle x_{\bullet }=\left(x_{a}\right)_{a\in A}}
2681:{\displaystyle s_{\bullet }=\left(s_{i}\right)_{i\in I}}
2620:{\displaystyle x_{\bullet }=\left(x_{a}\right)_{a\in A}}
1980:{\displaystyle x_{\bullet }=\left(x_{a}\right)_{a\in A}}
1896:, but this can lead to ambiguities if the ambient space
880:{\displaystyle x_{\bullet }=\left(x_{a}\right)_{a\in A}}
546:{\displaystyle x_{\bullet }=\left(x_{a}\right)_{a\in A}}
21188:
21186:
21184:
21182:
21180:
21178:
21150:
21148:
21146:
21144:
21116:
19702:
Pages displaying short descriptions of redirect targets
19669:
Characterizations of the category of topological spaces
18855:
Function from a well-ordered set to a topological space
15976:
Tychonoff's theorem and relation to the axiom of choice
12678:{\displaystyle S=\{x\}\cup \left\{x_{a}:a\in A\right\}}
7845:
with no finite subcover contrary to the compactness of
7765:{\displaystyle \bigcap _{i\in I}C_{i}\neq \varnothing }
7687:{\displaystyle \bigcap _{i\in J}C_{i}\neq \varnothing }
458:, since the strict inequalities cannot be satisfied if
21419:(3rd ed.). Berlin: Springer. pp. xxii, 703.
21255:
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if and only if it is eventually in every neighborhood
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20847:
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85:, where they are used to characterize many important
21175:
21141:
19694: – Topological space characterized by sequences
17962:
can be canonically interpreted as a net directed by
13603:
for a topology is also a subbase) and given a point
12835:
In this way, the question of whether or not the net
21417:
Infinite dimensional analysis: A hitchhiker's guide
10725:contains infinitely many elements of the sequence.
8179:and this is precisely the set of cluster points of
168:, typically automatically assumed to be denoted by
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17718:being the origin, for example) and direct the set
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17465:{\displaystyle \left\|m-m_{\bullet }\right\|\to 0}
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15792:{\displaystyle X_{1}\times X_{2}=\mathbb {R} ^{2}}
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12572:{\displaystyle h_{n}:=\inf\{a\in A:a>h_{n-1}\}}
12571:
12498:
12462:
12442:
12415:
12388:
12331:
12276:
12242:
12222:
12200:
12166:
12144:
12124:
12099:{\displaystyle \operatorname {Id} :(E,\geq )\to E}
12098:
12051:
12022:
11996:
11976:
11935:
11900:
11868:
11845:
11801:
11775:
11744:
11719:
11695:
11670:
11650:
11595:
11561:
11542:{\displaystyle \mathbf {0} :\mathbb {R} \to \{0\}}
11541:
11499:
11479:
11432:
11412:
11366:
11342:
11307:
11254:
11234:
11190:
11154:
11107:
11078:
11058:
11035:
11015:
10992:
10947:
10918:
10898:
10875:
10855:
10832:
10805:
10781:
10761:
10741:
10717:
10697:
10677:
10651:
10628:
10592:
10566:
10538:
10518:
10498:
10475:
10452:
10432:
10387:
10364:
10337:
10308:
10280:
10260:
10229:
10198:
10169:
10146:
10100:
10076:
10051:
10031:
10011:
9961:
9939:
9876:generated by this filter base is called the net's
9868:
9848:
9751:
9731:
9644:
9624:
9600:
9580:
9545:
9522:
9473:
9425:
9401:
9381:
9361:
9329:
9297:
9261:
9235:
9215:
9195:
9175:
9143:
9123:
9093:
9071:
9041:
9021:
8994:
8974:
8945:
8925:
8905:
8812:
8769:
8734:
8714:
8687:
8661:
8635:
8584:
8555:
8515:
8482:
8453:
8433:
8370:
8350:
8301:
8281:
8258:
8231:
8201:
8171:
8116:
8057:
7992:
7966:
7943:
7923:
7860:
7837:
7817:
7764:
7715:
7686:
7637:
7617:
7569:
7545:
7525:
7480:
7457:
7437:
7372:
7339:
7316:
7296:
7273:
7253:
7215:
7192:
7166:
7128:
7099:
7073:
6987:
6944:
6908:
6888:
6868:
6832:
6802:
6779:
6757:
6734:
6696:
6676:
6656:
6636:
6616:
6596:
6570:
6547:
6527:
6463:
6435:
6415:
6386:
6348:
6325:
6303:
6280:
6251:
6228:
6205:
6185:
6100:
6057:
5996:
5976:
5953:
5867:
5844:
5815:
5780:
5719:
5690:
5663:
5640:
5611:
5584:
5564:
5544:
5524:
5477:
5454:
5422:
5402:
5359:
5298:
5275:
5255:
5210:
5190:
5168:{\displaystyle f\left(x_{\bullet }\right)\to f(x)}
5167:
5114:
5094:
5061:
5029:
5006:
4948:
4928:
4888:
4827:
4803:
4759:
4736:
4716:
4690:
4667:
4641:
4601:
4581:
4533:
4484:
4458:
4434:
4406:
4377:
4354:
4312:
4286:
4253:
4192:
4172:
4152:
4132:
4112:
4041:is Cauchy if the filter generated by the net is a
4033:
4002:
3979:
3930:
3895:
3869:
3838:
3759:
3736:
3716:
3685:
3662:
3582:
3550:
3530:
3463:
3434:
3407:
3387:
3361:
3316:
3296:
3262:
3232:
3212:
3185:
3157:
3128:
3093:
3064:
3038:
3009:
2962:
2926:
2894:
2827:
2807:
2778:
2746:
2707:
2680:
2619:
2530:
2453:
2427:
2376:
2356:
2329:
2281:
2258:
2232:
2209:
2186:
2154:
2128:
2092:
2066:
2040:
2014:
1979:
1909:
1888:
1852:
1812:
1789:
1664:
1638:
1602:
1563:
1543:
1267:
1244:
1213:
1193:
1166:
1143:
1116:
1096:
1053:
1027:
991:
962:
936:
910:
879:
810:
780:
732:
708:
684:
660:
611:
588:
545:
484:
450:
424:
398:
372:
326:
306:
277:
251:
225:
182:
156:
19624:The first example is a special case of this with
17627:has at least two points, then we can fix a point
17165:{\displaystyle d\left(m,m_{\bullet }\right)\to 0}
16390:{\displaystyle \pi _{i}\left(f_{\bullet }\right)}
15737:{\displaystyle \pi _{2}\left(f_{\bullet }\right)}
15692:{\displaystyle \pi _{1}\left(f_{\bullet }\right)}
15255:{\displaystyle \pi _{i}\left(f_{\bullet }\right)}
14538:{\displaystyle \pi _{i}\left(f_{\bullet }\right)}
9913:
8820:This is a contradiction and completes the proof.
7818:{\displaystyle \left\{C_{i}^{c}\right\}_{i\in I}}
21696:
21411:
20914:in other words, when considered as functions on
19688: – Reflexive and transitive binary relation
19676: – Family of sets representing "large" sets
19204:
18344:
16896:of integration, partially ordered by inclusion.
16850:
16834:
16799:
16752:
16736:
16707:
16691:
16674:
13785:
13549:according to the definition of net convergence.
12780:
12720:
12526:
12490:
11956:
10402:
7003:
6960:
6115:
6073:
5883:
5375:
4963:
4904:
4500:
4327:
1867:
1831:
1756:
1717:
1679:
1617:
1581:
1484:
1423:
1374:
16892:where the net's directed set is the set of all
15514:{\displaystyle (1,1),(0,0),(1,1),(0,0),\ldots }
15210:{\displaystyle {\textstyle \prod }X_{\bullet }}
14926:{\displaystyle {\textstyle \prod }X_{\bullet }}
14197:{\displaystyle {\textstyle \prod }X_{\bullet }}
8461:is a directed set under inclusion and for each
6604:is a point in this neighborhood that is not in
4534:{\displaystyle \lim _{a\in A}s_{\bullet }\to x}
4058:if every Cauchy net converges to some point. A
21365:Megginson, p. 217, p. 221, Exercises 2.53–2.55
20622:{\displaystyle h:\mathbb {N} \to \mathbb {N} }
19932:{\displaystyle I=\{r\in \mathbb {R} :r>0\}}
15860:{\displaystyle \left(L_{1},L_{2}\right)=(0,1)}
11191:{\displaystyle f:\mathbb {R} \to \mathbb {R} }
6876:the member of the directed set whose index is
406:cannot be replaced by the strict inequalities
97:). Nets are in one-to-one correspondence with
16207:{\displaystyle L=\left(L_{i}\right)_{i\in I}}
12152:-valued net. This net converges pointwise to
11783:pointwise in the usual way by declaring that
11658:This will be proven by constructing a net in
9632:is not Hausdorff, then there exists a net on
7618:{\displaystyle \left\{C_{i}\right\}_{i\in I}}
4613:Open sets and characterizations of topologies
109:The concept of a net was first introduced by
20721:{\displaystyle h(\mathbb {N} )=\mathbb {N} }
20085:
20079:
19926:
19900:
18619:
18613:
18581:
18575:
18497:
18491:
18146:
18140:
18108:
18102:
18024:
18018:
17949:
17943:
17743:
17737:
17407:
17401:
17328:
17322:
16660:{\displaystyle \left(x_{a}\right)_{a\in A},}
12637:
12631:
12566:
12529:
12277:{\displaystyle \mathbb {R} ^{\mathbb {R} }.}
12201:{\displaystyle \mathbb {R} ^{\mathbb {R} },}
11971:
11959:
11596:{\displaystyle \mathbb {R} ^{\mathbb {R} };}
11536:
11530:
11474:
11447:
11407:
11395:
10147:{\displaystyle \mathbb {N} =\{1,2,\ldots \}}
10141:
10123:
8761:
8755:
8636:{\displaystyle \left(x_{C}\right)_{C\in D}.}
8425:
8391:
8111:
8072:
7488:This can be seen as a generalization of the
692:is clear from context it is simply called a
668:. Elements of a net's domain are called its
69:. Nets directly generalize the concept of a
21444:Topologies on closed and closed convex sets
20593:, although it is a subnet, because the map
20375:(where this subnet is not a subsequence of
20091:{\displaystyle \varphi (r)=\lceil r\rceil }
13838:{\displaystyle \left(X_{i}\right)_{i\in I}}
13014:{\displaystyle \left(x_{a}\right)_{a\in A}}
12332:{\displaystyle \left(x_{a}\right)_{a\in A}}
11776:{\displaystyle \mathbb {R} ^{\mathbb {R} }}
11343:{\displaystyle \mathbb {R} ^{\mathbb {R} }}
11308:{\displaystyle (f(x))_{x\in \mathbb {R} },}
11155:{\displaystyle \mathbb {R} ^{\mathbb {R} }}
9947:together with the usual integer comparison
9732:{\displaystyle \left(x_{a}\right)_{a\in A}}
9674:Filters in topology § Filters and nets
9474:{\displaystyle \left(y_{b}\right)_{b\in B}}
8828:
8351:{\displaystyle \left\{U_{i}:i\in I\right\}}
4582:{\displaystyle \left(s_{a}\right)_{a\in A}}
4355:{\displaystyle \lim {}_{}s_{\bullet }\to x}
4078:Characterizations of topological properties
1797:using the equal sign in place of the arrow
619:is some directed set, and whose values are
77:. Nets are primarily used in the fields of
21043:
20689:is an order-preserving map whose image is
20152:{\displaystyle \varphi :I\to \mathbb {N} }
20050:{\displaystyle \varphi :I\to \mathbb {N} }
16888:can be interpreted as a limit of a net of
16601:
15017:
14996:
14080:
14076:
14023:
14019:
13995:
12286:More generally, a subnet of a sequence is
7522:
7518:
6988:{\displaystyle \lim _{}x_{\bullet }\to x.}
6101:{\displaystyle \lim _{}x_{\bullet }\to x,}
5403:{\displaystyle \lim _{}x_{\bullet }\to x.}
5252:
5248:
4089:
1754:
1736:
1716:
1698:
1525:
1524:
1514:
1493:
1470:
1469:
1459:
1438:
1409:
1408:
1398:
1377:
1361:
1360:
1350:
1318:
1317:
1307:
21627:
21596:
21570:
21402:
21396:
21319:
21317:
21315:
21203:
21201:
21169:
21086:
20974:
20922:
20897:
20791:
20714:
20703:
20615:
20607:
20547:
20449:
20145:
20043:
19950:
19946:
19910:
19867:
19857:
19802:
19735:
17668:
17480:
17180:
15779:
15413:
15352:
14736:
14732:
13411:
12805:{\displaystyle \lim _{}x_{\bullet }\to x}
12745:{\displaystyle \lim _{}x_{\bullet }\to x}
12380:
12265:
12259:
12189:
12183:
11767:
11761:
11631:
11625:
11584:
11578:
11523:
11413:{\displaystyle f:\mathbb {R} \to \{0,1\}}
11388:
11334:
11328:
11296:
11228:
11219:
11184:
11176:
11146:
11140:
10560:
10302:
10192:
10116:
10067:
9958:
9954:
9933:
8289:) Conversely, suppose that every net in
6535:such that for every open neighborhood of
4929:{\displaystyle \lim _{}x_{\bullet }\to x}
1921:
723:
179:
175:
21633:Handbook of Analysis and Its Foundations
21603:Handbook of Analysis and Its Foundations
15359:{\displaystyle X_{1}=X_{2}=\mathbb {R} }
5816:{\displaystyle f(\operatorname {int} V)}
3980:{\displaystyle \left(x_{a},x_{b}\right)}
2754:if there exists an order-preserving map
27:A generalization of a sequence of points
21657:
21344:
21332:
21207:
21192:
21154:
21135:
19240:{\displaystyle \lim _{x\to t}f(x)\to L}
18509:{\displaystyle f:M\setminus \{c\}\to X}
18380:{\displaystyle \lim _{m\to c}f(m)\to L}
18036:{\displaystyle f:M\setminus \{c\}\to X}
14545:is equal to the composition of the net
12962:
3774:A Cauchy net generalizes the notion of
14:
21697:
21576:An Introduction to Banach Space Theory
21548:
21514:
21312:
21304:: CS1 maint: archived copy as title (
21241:, New Age International, p. 356,
21198:
21110:
21079:
21051:(1922). "A General Theory of Limits".
19556:if and only if for every neighborhood
19247:if and only if for every neighborhood
18786:if and only if for every neighborhood
11315:and conversely) and endow it with the
10440:if and only if for every neighborhood
9667:
5720:{\displaystyle \operatorname {int} V.}
5641:{\displaystyle \operatorname {int} V,}
1889:{\displaystyle \lim x_{\bullet }\to x}
1639:{\displaystyle \lim x_{\bullet }\to y}
1603:{\displaystyle \lim x_{\bullet }\to x}
117:in 1922. The term "net" was coined by
21582:. Vol. 193. New York: Springer.
21468:
21261:
21234:
18625:{\displaystyle m\in M\setminus \{c\}}
18587:{\displaystyle n\in M\setminus \{c\}}
18152:{\displaystyle m\in M\setminus \{c\}}
18114:{\displaystyle n\in M\setminus \{c\}}
17756:reversely according to distance from
16582:The axiom of choice is equivalent to
15946:does not contain even a single point
13048:come and stay as close as we want to
10433:{\displaystyle \lim {}_{n}a_{n}\to L}
10088:because by definition, a sequence in
9588:can have more than one limit, but if
7625:be a collection of closed subsets of
7193:{\displaystyle \operatorname {int} U}
7100:{\displaystyle \operatorname {int} U}
6423:Now the set of open neighborhoods of
3193:is a cluster point of some subnet of
661:{\displaystyle x_{\bullet }(a)=x_{a}}
21441:
19887:is the constant zero sequence. Let
12930:of (that is, the points of) the net
12613:
10956:(continuous in the sequential sense)
10206:), a net is defined on an arbitrary
4140:if and only if every limit point in
17955:{\displaystyle I:=M\setminus \{c\}}
17901:". Given any function with domain
17749:{\displaystyle I:=M\setminus \{c\}}
17681:{\displaystyle M:=\mathbb {R} ^{n}}
16879:
16478:
14812:
13944:denote the canonical projection to
13845:be topological spaces, endow their
13722:{\displaystyle U\in {\mathcal {B}}}
11208:
10815:continuous in the topological sense
10012:{\displaystyle a_{1},a_{2},\ldots }
9891:, while filters are most useful in
9563:
7254:{\displaystyle f\left(x_{a}\right)}
7167:{\displaystyle f\left(x_{a}\right)}
6735:{\displaystyle f\left(x_{a}\right)}
6004:be a point such that for every net
2217:the net is frequently/cofinally in
2174:of a net if for every neighborhood
1853:{\displaystyle \lim x_{\bullet }=x}
1225:expressed equivalently as: the net
589:{\displaystyle x_{\bullet }:A\to X}
24:
20907:{\displaystyle i\in \mathbb {N} ;}
20801:{\displaystyle i\in \mathbb {N} .}
19812:{\displaystyle i\in \mathbb {N} ,}
19748:{\displaystyle X=\mathbb {R} ^{n}}
18211:{\displaystyle d(m,c)\leq d(n,c),}
17854:{\displaystyle d(j,c)\leq d(i,c).}
13714:
13564:
13377:{\displaystyle \left(x_{S}\right)}
12973:Intuitively, convergence of a net
11703:However, there does not exist any
10506:The net is frequently in a subset
9389:is then cofinal. Moreover, giving
8813:{\displaystyle x_{B}\notin U_{c}.}
8422:
8378:with no finite subcover. Consider
2554:Filters in topology § Subnets
25:
21716:
21635:. San Diego, CA: Academic Press.
18681:{\displaystyle d(m,c)\leq d(n,c)}
18610:
18572:
18488:
18137:
18099:
18083:if and only if there exists some
18015:
17940:
17734:
17378:{\displaystyle m_{\bullet }\to m}
17081:{\displaystyle m_{\bullet }\to m}
16884:The definition of the value of a
13752:This characterization extends to
11352:topology of pointwise convergence
10567:{\displaystyle N\in \mathbb {N} }
10309:{\displaystyle N\in \mathbb {N} }
8770:{\displaystyle B\supseteq \{c\},}
8556:{\displaystyle x_{C}\notin U_{a}}
8166:
7759:
7681:
6767:Now, for every open neighborhood
5410:Then for every open neighborhood
5095:{\displaystyle x_{\bullet }\to x}
4659:
4649:is open if and only if no net in
4200:. Explicitly, this means that if
3452:
821:
65:of this function is usually some
21238:Introduction to General Topology
16899:
16590:and so strictly weaker than the
12216:
12160:
11735:
11686:
11611:
11515:
11374:denote the set of all functions
11162:of all functions with prototype
10926:with this sequence converges to
8913:be a net in a topological space
3442:is eventually in the complement
716:is a directed set with preorder
21377:
21368:
21359:
21350:
21267:
21228:
21053:American Journal of Mathematics
21017:{\displaystyle s_{\bullet }=h.}
20409:
19939:be directed by the usual order
19714:
14933:if and only if for every index
14375:denote the result of "plugging
12470: – that is, let
11908:a directed set since given any
9759:induces a filter base of tails
8282:{\displaystyle \Longleftarrow }
5977:{\displaystyle \Longleftarrow }
2874:
1610:and this limit is unique (i.e.
354:. In this case, the conditions
21484:Science & Business Media.
21092:
21037:
20870:
20864:
20770:{\displaystyle s_{i}=x_{h(i)}}
20762:
20756:
20707:
20699:
20645:
20639:
20611:
20580:
20556:
20500:
20458:
20218:
20212:
20198:
20192:
20141:
20073:
20067:
20039:
19871:
19846:
19839:
19536:is a cluster point of the net
19486:
19480:
19457:
19445:
19419:
19407:
19231:
19228:
19222:
19211:
19158:
19152:
19129:
19117:
19091:
19079:
19008:
18996:
18953:
18941:
18881:
18869:
18766:is a cluster point of the net
18704:
18698:
18675:
18663:
18654:
18642:
18500:
18371:
18368:
18362:
18351:
18234:
18228:
18202:
18190:
18181:
18169:
18027:
17981:
17969:
17845:
17833:
17824:
17812:
17614:
17602:
17560:
17539:
17526:
17505:
17456:
17452:
17431:
17416:{\displaystyle (M,\|\cdot \|)}
17410:
17392:
17369:
17337:{\displaystyle (M,\|\cdot \|)}
17331:
17313:
17156:
17107:
17095:
17072:
16923:
16911:
16828:
16802:
16552:
16341:{\displaystyle L_{i}\in X_{i}}
16110:
16068:{\displaystyle L_{i}\in X_{i}}
15933:
15921:
15894:since the open ball of radius
15854:
15842:
15572:
15560:
15540:
15528:
15502:
15490:
15484:
15472:
15466:
15454:
15448:
15436:
14635:
14568:
14470:
14077:
14020:
13572:{\displaystyle {\mathcal {B}}}
12796:
12736:
12116:
12090:
12087:
12075:
11977:{\displaystyle m:=\min\{f,g\}}
11895:
11883:
11840:
11834:
11825:
11819:
11527:
11487:is finite). Then the constant
11465:
11459:
11392:
11285:
11281:
11275:
11269:
11180:
11115:(continuous in the net sense).
11102:
11096:
10942:
10936:
10424:
9914:As generalization of sequences
9523:{\displaystyle y_{b}=x_{h(b)}}
9515:
9509:
9356:
9344:
9321:
9170:
9158:
8415:
8407:
8276:
7519:
7355:
7123:
7117:
7065:
7059:
7053:
6976:
6372:
6366:
6275:
6269:
6177:
6171:
6165:
6089:
5971:
5961:and this direction is proven.
5945:
5939:
5933:
5839:
5833:
5810:
5798:
5516:
5510:
5446:
5440:
5391:
5249:
5162:
5156:
5150:
5086:
5053:
5001:
4995:
4989:
4920:
4811:between topological spaces is
4795:
4525:
4346:
3769:
3574:
3123:
3117:
3033:
3027:
3010:{\displaystyle h(i)\leq h(j).}
3001:
2995:
2986:
2980:
2918:
2869:
2863:
2802:
2796:
2770:
1880:
1804:
1630:
1594:
1511:
1456:
1395:
1347:
1304:
642:
636:
580:
135:
13:
1:
21606:. San Diego: Academic Press.
21580:Graduate Texts in Mathematics
21524:Graduate Texts in Mathematics
21478:Graduate Texts in Mathematics
21390:
21383:Schechter, Sections 7.43–7.47
20932:{\displaystyle \mathbb {N} ,}
19700: – Maximal proper filter
19026:It is eventually in a subset
17490:{\displaystyle \mathbb {R} ,}
17190:{\displaystyle \mathbb {R} ,}
13786:Limits in a Cartesian product
12953:{\displaystyle x_{\bullet }.}
12886:on this topological subspace
12862:converges to the given point
12499:{\displaystyle h_{1}:=\inf A}
12443:{\displaystyle n^{\text{th}}}
12059:This partial order turns the
11846:{\displaystyle f(x)\geq g(x)}
11745:{\displaystyle \mathbf {0} ,}
11696:{\displaystyle \mathbf {0} .}
11319:. This (product) topology on
10077:{\displaystyle \mathbb {N} .}
9568:In general, a net in a space
9124:{\displaystyle x_{\bullet }.}
9072:{\displaystyle x_{\bullet }.}
8232:{\displaystyle x_{\bullet }.}
8202:{\displaystyle x_{\bullet }.}
7716:{\displaystyle J\subseteq I.}
7465:has a subnet with a limit in
5525:{\displaystyle V:=f^{-1}(U),}
5069:is continuous if and only if
4835:if and only if for every net
4774:
4074:) topological vector spaces.
3464:{\displaystyle X\setminus S.}
3362:{\displaystyle S\subseteq X,}
3263:{\displaystyle x_{\bullet }.}
21474:Modern Analysis and Topology
21030:
20981:{\displaystyle \mathbb {N} }
20966:is just the identity map on
20959:{\displaystyle x_{\bullet }}
20395:{\displaystyle x_{\bullet }}
20368:{\displaystyle x_{\bullet }}
20348:is a subnet of the sequence
16234:{\displaystyle f_{\bullet }}
15966:{\displaystyle f_{\bullet }}
15887:{\displaystyle f_{\bullet }}
15156:{\displaystyle f_{\bullet }}
14872:{\displaystyle f_{\bullet }}
14429:", which results in the net
14395:{\displaystyle f_{\bullet }}
13652:{\displaystyle x_{\bullet }}
13111:in a topological space, let
12855:{\displaystyle x_{\bullet }}
12223:{\displaystyle \mathbf {0} }
12167:{\displaystyle \mathbf {0} }
11480:{\displaystyle \{x:f(x)=0\}}
10316:such that for every integer
10199:{\displaystyle \mathbb {N} }
9940:{\displaystyle \mathbb {N} }
9022:{\displaystyle x_{\bullet }}
8259:{\displaystyle x_{\bullet }}
7555:finite intersection property
6869:{\displaystyle b\geq a_{0},}
5823:and thus also eventually in
5691:{\displaystyle x_{\bullet }}
4717:{\displaystyle S\subseteq X}
4668:{\displaystyle X\setminus S}
4642:{\displaystyle S\subseteq X}
4435:{\displaystyle S\subseteq X}
4113:{\displaystyle S\subseteq X}
4034:{\displaystyle x_{\bullet }}
3744:if and only it converges to
3435:{\displaystyle x_{\bullet }}
3388:{\displaystyle x_{\bullet }}
3297:{\displaystyle x_{\bullet }}
3272:
3213:{\displaystyle x_{\bullet }}
2747:{\displaystyle x_{\bullet }}
2708:{\displaystyle s_{\bullet }}
2357:{\displaystyle x_{\bullet }}
1194:{\displaystyle x_{\bullet }}
1097:{\displaystyle x_{\bullet }}
7:
19662:
15654:are cluster points of both
13915:, and that for every index
13292:{\displaystyle S\in N_{x},}
12608:
11242:(by identifying a function
11086:with this net converges to
10749:between topological spaces
10629:{\displaystyle a_{n}\in S,}
10039:can be considered a net in
9298:{\displaystyle x_{a}\in U.}
9203:is an open neighborhood of
7825:would be an open cover for
7490:Bolzano–Weierstrass theorem
7347:This completes the proof.
7304:This is a contradiction so
7107:is an open neighborhood of
6945:{\displaystyle x_{b}\in W.}
5648:is an open neighborhood of
4744:is eventually contained in
3240:is also a cluster point of
2566:, which is as follows: If
2129:{\displaystyle x_{b}\in S.}
1277:; and variously denoted as:
1028:{\displaystyle x_{b}\in S.}
198:, which means that for any
10:
21721:
21658:Willard, Stephen (2004) .
21413:Aliprantis, Charalambos D.
20876:{\displaystyle s_{i}=h(i)}
19646:{\displaystyle c=\omega .}
19501:{\displaystyle f(s)\in V.}
19463:{\displaystyle s\in [r,t)}
19425:{\displaystyle r\in [0,t)}
19354:is frequently in a subset
19135:{\displaystyle s\in [r,t)}
19097:{\displaystyle r\in [0,t)}
18989:This function is a net on
18516:is frequently in a subset
18043:is eventually in a subset
17990:{\displaystyle (I,\leq ).}
16894:partitions of the interval
16160:then the tuple defined by
15867:is not a cluster point of
13404:increases with respect to
12966:
12230:belongs to the closure of
12125:{\displaystyle f\mapsto f}
11549:belongs to the closure of
10574:there exists some integer
9671:
9002:is a limit of a subnet of
8516:{\displaystyle x_{C}\in X}
6387:{\displaystyle f(x)\in U,}
4616:
4287:{\displaystyle s_{a}\in S}
3931:{\displaystyle a,b\geq c,}
2551:
2545:
2541:
346:. A directed set may have
104:
51:Moore–Smith sequence
29:
21415:; Border, Kim C. (2006).
21208:Willard, Stephen (2012),
19954:{\displaystyle \,\leq \,}
19599:the net is frequently in
19394:if and only if for every
18829:the net is frequently in
18556:if and only if for every
11936:{\displaystyle f,g\in E,}
11901:{\displaystyle (E,\geq )}
11198:as the Cartesian product
10546:if and only if for every
10483:the net is eventually in
10237:is taken from sequences.
9962:{\displaystyle \,\leq \,}
8266:has a convergent subnet.
7577:be any non-empty set and
7526:{\displaystyle \implies }
7384:if and only if every net
6684:is not a neighborhood of
6333:is not a neighborhood of
5256:{\displaystyle \implies }
3129:{\displaystyle b\in h(I)}
2337:of all cluster points of
226:{\displaystyle a,b\in A,}
183:{\displaystyle \,\leq \,}
128:was developed in 1937 by
124:The related concept of a
21356:Aliprantis-Border, p. 32
20808:Indeed, this is because
20513:is not a subsequence of
19707:
19698:Ultrafilter (set theory)
19657:ordinal-indexed sequence
16308:is finite or when every
15647:{\displaystyle L_{2}:=1}
15614:{\displaystyle L_{1}:=0}
15521:that alternates between
14422:{\displaystyle \pi _{i}}
13529:does indeed converge to
13421:{\displaystyle \,\geq ,}
12052:{\displaystyle g\geq m.}
11943:their pointwise minimum
10338:{\displaystyle n\geq N,}
10108:is just a function from
9330:{\displaystyle h:B\to A}
9081:Conversely, assume that
8829:Cluster and limit points
6664:(because by assumption,
5062:{\displaystyle f:X\to Y}
4804:{\displaystyle f:X\to Y}
4675:converges to a point of
4050:topological vector space
3724:an ultranet clusters at
3583:{\displaystyle f:X\to Y}
3158:{\displaystyle b\geq a.}
2927:{\displaystyle h:I\to A}
2779:{\displaystyle h:I\to A}
992:{\displaystyle b\geq a,}
733:{\displaystyle \,\leq .}
307:{\displaystyle b\leq c.}
45:and related branches, a
20834:{\displaystyle x_{i}=i}
20274:{\displaystyle r\in R.}
20016:{\displaystyle r\in R.}
19987:{\displaystyle s_{r}=0}
19781:{\displaystyle x_{i}=0}
18966:to a topological space
18063:of a topological space
17795:{\displaystyle i\leq j}
16602:Limit superior/inferior
16028:{\displaystyle i\in I,}
16006:is given but for every
15311:{\displaystyle i\in I.}
14955:{\displaystyle i\in I,}
14246:{\displaystyle i\in I,}
13937:{\displaystyle l\in I,}
13625:{\displaystyle x\in X,}
13599:(where note that every
13211:{\displaystyle S\geq T}
12023:{\displaystyle f\geq m}
11802:{\displaystyle f\geq g}
10979:
10593:{\displaystyle n\geq N}
10019:in a topological space
8975:{\displaystyle y\in X.}
8585:{\displaystyle a\in C.}
8483:{\displaystyle C\in D,}
7533:) First, suppose that
6416:{\displaystyle x\in V.}
5283:be continuous at point
4407:{\displaystyle x\in S.}
4090:Closed sets and closure
4068:sequential completeness
3717:{\displaystyle x\in X,}
3094:{\displaystyle a\in A,}
2963:{\displaystyle i\leq j}
2815:is a cofinal subset of
2093:{\displaystyle b\geq a}
399:{\displaystyle b\leq c}
373:{\displaystyle a\leq c}
278:{\displaystyle a\leq c}
41:, more specifically in
21018:
20982:
20960:
20933:
20908:
20877:
20835:
20802:
20771:
20722:
20683:
20623:
20587:
20507:
20396:
20369:
20342:
20275:
20246:
20153:
20119:
20092:
20051:
20017:
19988:
19955:
19933:
19881:
19813:
19782:
19749:
19647:
19616:
19593:
19570:
19550:
19530:
19529:{\displaystyle y\in X}
19502:
19464:
19426:
19388:
19368:
19348:
19326:
19303:
19284:
19261:
19241:
19188:
19165:
19136:
19098:
19060:
19040:
19018:
19017:{\displaystyle [0,t).}
18983:
18960:
18928:
18908:
18888:
18846:
18823:
18800:
18780:
18760:
18759:{\displaystyle L\in X}
18740:Consequently, a point
18734:
18711:
18682:
18626:
18588:
18550:
18530:
18510:
18463:
18443:
18424:
18401:
18381:
18330:
18329:{\displaystyle L\in X}
18304:
18284:
18264:
18241:
18212:
18153:
18115:
18077:
18057:
18037:
17991:
17956:
17918:
17895:
17875:
17855:
17796:
17770:
17750:
17712:
17682:
17647:
17646:{\displaystyle c\in M}
17621:
17587:
17491:
17466:
17417:
17379:
17338:
17292:
17191:
17166:
17114:
17082:
17049:
16988:
16987:{\displaystyle m\in M}
16958:
16930:
16870:
16784:
16661:
16569:
16509:
16442:
16418:
16391:
16342:
16302:
16282:
16258:
16235:
16208:
16154:
16127:
16069:
16029:
16000:
15999:{\displaystyle L\in X}
15967:
15940:
15908:
15888:
15861:
15793:
15738:
15693:
15648:
15615:
15582:
15581:{\displaystyle (0,0).}
15547:
15515:
15423:
15360:
15312:
15283:
15256:
15211:
15177:
15157:
15130:
15129:{\displaystyle X_{i}.}
15100:
15073:
14956:
14927:
14893:
14873:
14846:
14754:
14655:
14592:
14539:
14490:
14423:
14396:
14369:
14247:
14218:
14198:
14164:
14101:
13965:
13938:
13905:
13839:
13777:
13746:
13723:
13693:
13673:
13653:
13626:
13593:
13573:
13543:
13523:
13496:
13472:
13449:
13422:
13398:
13378:
13343:
13320:
13293:
13257:
13234:
13212:
13186:
13159:
13138:denote the set of all
13132:
13105:
13085:
13062:
13042:
13021:means that the values
13015:
12954:
12920:
12900:
12876:
12856:
12829:
12806:
12766:
12746:
12706:
12679:
12599:
12598:{\displaystyle n>1}
12573:
12500:
12464:
12444:
12417:
12390:
12333:
12278:
12244:
12224:
12202:
12168:
12146:
12126:
12100:
12053:
12024:
11998:
11978:
11937:
11902:
11870:
11847:
11803:
11777:
11746:
11721:
11697:
11672:
11652:
11597:
11563:
11543:
11501:
11481:
11434:
11414:
11368:
11344:
11309:
11256:
11236:
11192:
11156:
11109:
11080:
11060:
11037:
11017:
10994:
10968:first-countable spaces
10949:
10920:
10900:
10877:
10857:
10834:
10807:
10783:
10763:
10743:
10719:
10699:
10679:
10678:{\displaystyle y\in X}
10653:
10630:
10594:
10568:
10540:
10520:
10500:
10477:
10454:
10434:
10389:
10366:
10339:
10310:
10282:
10262:
10231:
10200:
10171:
10148:
10102:
10078:
10053:
10033:
10013:
9963:
9941:
9870:
9850:
9753:
9733:
9646:
9626:
9602:
9582:
9547:
9524:
9475:
9427:
9413:(the neighborhoods of
9403:
9383:
9363:
9331:
9299:
9263:
9262:{\displaystyle a\in A}
9237:
9217:
9197:
9177:
9145:
9125:
9101:is a cluster point of
9095:
9073:
9049:is a cluster point of
9043:
9023:
8996:
8976:
8947:
8927:
8907:
8814:
8771:
8736:
8722:is a neighbourhood of
8716:
8689:
8688:{\displaystyle c\in I}
8663:
8662:{\displaystyle x\in X}
8637:
8586:
8557:
8517:
8484:
8455:
8435:
8372:
8352:
8303:
8283:
8260:
8233:
8203:
8173:
8118:
8059:
7994:
7993:{\displaystyle a\in A}
7968:
7945:
7925:
7862:
7839:
7819:
7766:
7717:
7688:
7639:
7619:
7571:
7547:
7527:
7482:
7459:
7439:
7374:
7341:
7324:must be continuous at
7318:
7298:
7275:
7255:
7217:
7200:and therefore also in
7194:
7168:
7130:
7101:
7075:
6995:and by our assumption
6989:
6946:
6910:
6890:
6870:
6834:
6833:{\displaystyle a_{0}.}
6804:
6781:
6759:
6736:
6698:
6678:
6658:
6638:
6618:
6598:
6572:
6549:
6529:
6465:
6437:
6417:
6388:
6350:
6327:
6305:
6282:
6253:
6230:
6207:
6187:
6102:
6059:
5998:
5978:
5955:
5869:
5846:
5817:
5782:
5721:
5692:
5665:
5642:
5613:
5586:
5566:
5552:(by the continuity of
5546:
5526:
5479:
5456:
5424:
5404:
5361:
5300:
5277:
5257:
5212:
5192:
5169:
5116:
5096:
5063:
5031:
5008:
4950:
4930:
4890:
4829:
4805:
4761:
4738:
4718:
4692:
4669:
4643:
4603:
4583:
4535:
4486:
4485:{\displaystyle x\in X}
4460:
4436:
4408:
4379:
4356:
4314:
4313:{\displaystyle a\in A}
4288:
4255:
4194:
4174:
4154:
4134:
4114:
4035:
4004:
3981:
3932:
3897:
3896:{\displaystyle c\in A}
3871:
3840:
3761:
3738:
3718:
3687:
3664:
3584:
3552:
3532:
3465:
3436:
3409:
3389:
3363:
3318:
3298:
3264:
3234:
3214:
3187:
3186:{\displaystyle x\in X}
3159:
3130:
3095:
3066:
3040:
3011:
2964:
2928:
2896:
2829:
2809:
2780:
2748:
2709:
2682:
2621:
2532:
2455:
2454:{\displaystyle a\in A}
2429:
2378:
2358:
2331:
2283:
2260:
2259:{\displaystyle x\in X}
2234:
2211:
2188:
2156:
2155:{\displaystyle x\in X}
2130:
2094:
2068:
2067:{\displaystyle b\in A}
2042:
2041:{\displaystyle a\in A}
2016:
1981:
1922:Cluster points of nets
1911:
1890:
1854:
1814:
1791:
1666:
1640:
1604:
1565:
1545:
1269:
1246:
1215:
1195:
1168:
1145:
1118:
1098:
1055:
1054:{\displaystyle x\in X}
1029:
993:
964:
963:{\displaystyle b\in A}
938:
937:{\displaystyle a\in A}
912:
881:
812:
811:{\displaystyle a\in A}
782:
734:
710:
686:
662:
613:
590:
547:
486:
452:
451:{\displaystyle b<c}
426:
425:{\displaystyle a<c}
400:
374:
328:
308:
279:
253:
252:{\displaystyle c\in A}
227:
184:
158:
95:Fréchet–Urysohn spaces
87:topological properties
21442:Beer, Gerald (1993).
21235:Joshi, K. D. (1983),
21019:
20983:
20961:
20934:
20909:
20878:
20836:
20803:
20772:
20723:
20684:
20624:
20588:
20508:
20397:
20370:
20343:
20276:
20247:
20154:
20120:
20093:
20052:
20018:
19989:
19956:
19934:
19882:
19814:
19783:
19750:
19648:
19617:
19594:
19571:
19551:
19531:
19503:
19465:
19427:
19389:
19369:
19349:
19327:
19304:
19285:
19262:
19242:
19189:
19166:
19137:
19099:
19061:
19041:
19019:
18984:
18961:
18959:{\displaystyle [0,t)}
18929:
18909:
18889:
18847:
18824:
18801:
18781:
18761:
18735:
18712:
18683:
18627:
18589:
18551:
18531:
18511:
18464:
18444:
18425:
18402:
18382:
18331:
18305:
18285:
18265:
18242:
18213:
18154:
18116:
18078:
18058:
18038:
17992:
17957:
17919:
17896:
17876:
17856:
17797:
17771:
17751:
17713:
17683:
17648:
17622:
17620:{\displaystyle (M,d)}
17588:
17492:
17467:
17418:
17380:
17339:
17293:
17192:
17167:
17115:
17113:{\displaystyle (M,d)}
17083:
17050:
16989:
16959:
16931:
16929:{\displaystyle (M,d)}
16871:
16785:
16662:
16570:
16510:
16443:
16419:
16417:{\displaystyle X_{i}}
16392:
16343:
16303:
16283:
16259:
16236:
16209:
16155:
16153:{\displaystyle X_{i}}
16128:
16070:
16030:
16001:
15968:
15941:
15939:{\displaystyle (0,1)}
15909:
15889:
15862:
15794:
15739:
15694:
15649:
15616:
15583:
15548:
15546:{\displaystyle (1,1)}
15516:
15424:
15361:
15313:
15284:
15282:{\displaystyle L_{i}}
15257:
15212:
15178:
15158:
15136:And whenever the net
15131:
15101:
15099:{\displaystyle L_{i}}
15074:
14957:
14928:
14899:in the product space
14894:
14874:
14847:
14755:
14656:
14593:
14540:
14491:
14424:
14397:
14370:
14248:
14219:
14199:
14165:
14102:
13966:
13964:{\displaystyle X_{l}}
13939:
13906:
13840:
13778:
13760:) of the given point
13754:neighborhood subbases
13747:
13724:
13694:
13674:
13654:
13627:
13594:
13574:
13544:
13524:
13522:{\displaystyle x_{S}}
13497:
13478:. Therefore, in this
13473:
13450:
13448:{\displaystyle x_{S}}
13423:
13399:
13379:
13344:
13321:
13319:{\displaystyle x_{S}}
13294:
13258:
13235:
13213:
13187:
13185:{\displaystyle N_{x}}
13160:
13133:
13131:{\displaystyle N_{x}}
13106:
13086:
13063:
13043:
13041:{\displaystyle x_{a}}
13016:
12955:
12921:
12901:
12877:
12857:
12830:
12807:
12767:
12747:
12707:
12680:
12600:
12574:
12501:
12465:
12445:
12418:
12416:{\displaystyle h_{n}}
12391:
12339:induces the sequence
12334:
12279:
12245:
12225:
12203:
12169:
12147:
12127:
12101:
12054:
12025:
11999:
11979:
11938:
11903:
11871:
11848:
11804:
11778:
11747:
11722:
11698:
11673:
11653:
11598:
11564:
11544:
11502:
11482:
11435:
11415:
11369:
11345:
11310:
11257:
11237:
11193:
11157:
11110:
11081:
11061:
11038:
11018:
10995:
10950:
10921:
10901:
10878:
10858:
10835:
10808:
10784:
10764:
10744:
10720:
10700:
10680:
10654:
10631:
10595:
10569:
10541:
10521:
10501:
10478:
10455:
10435:
10390:
10367:
10365:{\displaystyle a_{n}}
10340:
10311:
10283:
10263:
10232:
10230:{\displaystyle x_{a}}
10201:
10172:
10149:
10103:
10079:
10054:
10034:
10014:
9964:
9942:
9871:
9851:
9754:
9734:
9647:
9627:
9603:
9583:
9548:
9525:
9476:
9428:
9404:
9384:
9364:
9362:{\displaystyle (U,a)}
9332:
9300:
9264:
9238:
9218:
9198:
9178:
9176:{\displaystyle (U,a)}
9146:
9126:
9096:
9074:
9044:
9024:
8997:
8977:
8948:
8928:
8908:
8815:
8772:
8737:
8717:
8715:{\displaystyle U_{c}}
8690:
8664:
8638:
8587:
8558:
8518:
8485:
8456:
8436:
8373:
8353:
8304:
8284:
8261:
8234:
8204:
8174:
8119:
8060:
7995:
7969:
7946:
7926:
7863:
7840:
7820:
7767:
7718:
7689:
7640:
7620:
7572:
7548:
7528:
7483:
7460:
7440:
7375:
7342:
7319:
7299:
7276:
7256:
7218:
7195:
7169:
7131:
7102:
7076:
6990:
6947:
6911:
6891:
6871:
6835:
6805:
6782:
6760:
6737:
6699:
6679:
6659:
6639:
6619:
6599:
6597:{\displaystyle x_{a}}
6573:
6550:
6530:
6466:
6438:
6418:
6389:
6351:
6328:
6306:
6288:whose preimage under
6283:
6254:
6231:
6213:is not continuous at
6208:
6188:
6103:
6060:
5999:
5979:
5956:
5870:
5852:which is a subset of
5847:
5818:
5783:
5722:
5693:
5666:
5643:
5614:
5587:
5567:
5547:
5532:is a neighborhood of
5527:
5480:
5457:
5455:{\displaystyle f(x),}
5425:
5405:
5362:
5301:
5278:
5258:
5220:first-countable space
5213:
5193:
5170:
5117:
5097:
5064:
5032:
5009:
4951:
4931:
4891:
4830:
4806:
4762:
4739:
4719:
4693:
4670:
4644:
4604:
4584:
4536:
4487:
4466:is the set of points
4461:
4437:
4409:
4380:
4357:
4315:
4289:
4256:
4195:
4175:
4155:
4135:
4115:
4036:
4010:More generally, in a
4005:
3982:
3933:
3898:
3872:
3841:
3762:
3739:
3719:
3688:
3665:
3585:
3553:
3533:
3466:
3437:
3410:
3390:
3364:
3319:
3299:
3265:
3235:
3215:
3188:
3160:
3131:
3096:
3072:means that for every
3067:
3041:
3012:
2965:
2929:
2897:
2830:
2810:
2781:
2749:
2710:
2683:
2622:
2533:
2456:
2430:
2379:
2359:
2332:
2284:
2261:
2235:
2212:
2189:
2157:
2131:
2095:
2069:
2043:
2017:
1982:
1912:
1891:
1855:
1815:
1813:{\displaystyle \to .}
1792:
1667:
1641:
1605:
1566:
1546:
1270:
1247:
1216:
1196:
1169:
1146:
1119:
1099:
1056:
1030:
994:
965:
939:
918:if there exists some
913:
882:
813:
783:
735:
711:
687:
663:
614:
591:
548:
487:
453:
427:
401:
375:
329:
309:
280:
254:
228:
185:
159:
21572:Megginson, Robert E.
20992:
20970:
20943:
20918:
20887:
20845:
20812:
20781:
20732:
20693:
20633:
20597:
20517:
20419:
20379:
20352:
20285:
20256:
20163:
20129:
20106:
20061:
20027:
19998:
19965:
19943:
19891:
19823:
19792:
19759:
19724:
19720:For an example, let
19628:
19603:
19580:
19560:
19540:
19514:
19474:
19436:
19398:
19378:
19358:
19338:
19313:
19293:
19271:
19251:
19200:
19175:
19164:{\displaystyle f(s)}
19146:
19108:
19104:such that for every
19070:
19050:
19030:
18993:
18970:
18938:
18918:
18898:
18866:
18833:
18810:
18790:
18770:
18744:
18721:
18710:{\displaystyle f(m)}
18692:
18636:
18598:
18560:
18540:
18520:
18476:
18453:
18433:
18411:
18391:
18340:
18314:
18294:
18274:
18251:
18240:{\displaystyle f(m)}
18222:
18163:
18125:
18121:such that for every
18087:
18067:
18047:
18003:
17966:
17928:
17905:
17885:
17865:
17806:
17780:
17760:
17722:
17711:{\displaystyle c:=0}
17696:
17657:
17631:
17599:
17501:
17476:
17427:
17389:
17356:
17310:
17201:
17176:
17124:
17092:
17059:
16998:
16972:
16964:is endowed with the
16948:
16908:
16796:
16671:
16620:
16519:
16452:
16432:
16401:
16356:
16312:
16292:
16272:
16245:
16218:
16164:
16137:
16079:
16039:
16010:
15984:
15950:
15918:
15898:
15871:
15803:
15748:
15703:
15658:
15625:
15592:
15557:
15525:
15433:
15429:denote the sequence
15370:
15322:
15293:
15266:
15221:
15187:
15167:
15140:
15110:
15083:
14965:
14937:
14903:
14883:
14856:
14766:
14762:For any given point
14665:
14602:
14598:with the projection
14549:
14504:
14498:function composition
14433:
14406:
14379:
14257:
14228:
14224:and for every index
14208:
14174:
14113:
13975:
13948:
13919:
13852:
13801:
13764:
13733:
13703:
13683:
13663:
13636:
13607:
13583:
13579:for the topology on
13559:
13533:
13506:
13486:
13459:
13432:
13408:
13388:
13353:
13330:
13303:
13267:
13244:
13224:
13196:
13169:
13146:
13115:
13095:
13072:
13052:
13025:
12977:
12963:Neighborhood systems
12934:
12910:
12890:
12866:
12839:
12816:
12776:
12756:
12716:
12693:
12685:is endowed with the
12622:
12583:
12510:
12474:
12454:
12427:
12400:
12343:
12295:
12254:
12234:
12212:
12178:
12156:
12136:
12110:
12066:
12034:
12008:
11988:
11947:
11912:
11880:
11857:
11813:
11787:
11756:
11731:
11711:
11682:
11662:
11607:
11573:
11553:
11511:
11491:
11444:
11424:
11378:
11358:
11350:is identical to the
11323:
11266:
11246:
11202:
11166:
11135:
11108:{\displaystyle f(x)}
11090:
11070:
11047:
11027:
11004:
10984:
10948:{\displaystyle f(x)}
10930:
10910:
10887:
10867:
10863:and any sequence in
10844:
10824:
10797:
10773:
10753:
10733:
10709:
10689:
10663:
10640:
10604:
10578:
10550:
10530:
10510:
10487:
10464:
10444:
10399:
10376:
10349:
10320:
10292:
10272:
10252:
10214:
10188:
10158:
10112:
10092:
10063:
10043:
10023:
9977:
9951:
9929:
9860:
9856:where the filter in
9763:
9743:
9695:
9636:
9616:
9592:
9572:
9534:
9485:
9437:
9417:
9393:
9373:
9341:
9309:
9273:
9247:
9227:
9207:
9187:
9155:
9151:be the set of pairs
9135:
9105:
9085:
9053:
9033:
9006:
8986:
8957:
8937:
8917:
8856:
8781:
8746:
8726:
8699:
8673:
8647:
8596:
8567:
8527:
8494:
8465:
8445:
8382:
8362:
8358:be an open cover of
8313:
8293:
8273:
8243:
8213:
8183:
8128:
8069:
8004:
7978:
7955:
7935:
7874:
7849:
7829:
7776:
7772:as well. Otherwise,
7727:
7698:
7649:
7629:
7581:
7561:
7537:
7515:
7469:
7449:
7388:
7364:
7328:
7308:
7285:
7265:
7227:
7223:in contradiction to
7204:
7178:
7140:
7129:{\displaystyle f(x)}
7111:
7085:
6999:
6956:
6920:
6900:
6896:is contained within
6880:
6844:
6814:
6791:
6771:
6746:
6708:
6688:
6668:
6648:
6628:
6608:
6581:
6559:
6539:
6478:
6455:
6427:
6398:
6360:
6337:
6314:
6292:
6281:{\displaystyle f(x)}
6263:
6243:
6217:
6197:
6111:
6069:
6008:
5988:
5968:
5879:
5856:
5845:{\displaystyle f(V)}
5827:
5792:
5731:
5702:
5675:
5652:
5623:
5619:which is denoted by
5600:
5576:
5556:
5536:
5488:
5466:
5434:
5414:
5371:
5310:
5287:
5267:
5245:
5202:
5179:
5126:
5106:
5073:
5041:
5037:Briefly, a function
5018:
4960:
4940:
4900:
4839:
4819:
4783:
4748:
4728:
4702:
4679:
4653:
4627:
4593:
4545:
4496:
4470:
4450:
4420:
4389:
4366:
4324:
4298:
4265:
4204:
4184:
4180:necessarily lies in
4164:
4144:
4124:
4098:
4018:
3991:
3941:
3907:
3881:
3861:
3789:
3748:
3728:
3699:
3674:
3594:
3562:
3542:
3481:
3446:
3419:
3399:
3372:
3344:
3340:if for every subset
3308:
3281:
3244:
3224:
3197:
3171:
3140:
3105:
3076:
3056:
3039:{\displaystyle h(I)}
3021:
2974:
2948:
2906:
2839:
2819:
2808:{\displaystyle h(I)}
2790:
2758:
2731:
2692:
2631:
2570:
2548:Subnet (mathematics)
2465:
2439:
2388:
2368:
2341:
2293:
2270:
2244:
2221:
2198:
2178:
2140:
2104:
2078:
2052:
2026:
2006:
1930:
1901:
1864:
1828:
1801:
1676:
1650:
1614:
1578:
1555:
1281:
1259:
1236:
1205:
1178:
1155:
1135:
1108:
1081:
1039:
1003:
974:
948:
944:such that for every
922:
902:
830:
796:
744:
720:
700:
676:
623:
603:
561:
496:
476:
436:
410:
384:
358:
318:
289:
263:
237:
202:
172:
148:
21172:, pp. 157–168.
19680:Filters in topology
19674:Filter (set theory)
19066:if there exists an
17924:its restriction to
16584:Tychonoff's theorem
16214:will be a limit of
13480:neighborhood system
12969:Neighborhood system
12208:which implies that
11121:neighbourhood basis
11066:the composition of
10906:the composition of
10288:if there exists an
10246:limit of a function
10242:limit of a sequence
9920:totally ordered set
9668:Relation to filters
8742:; however, for all
7798:
7494:Heine–Borel theorem
6704:). It follows that
6474:We construct a net
5462:its preimage under
5367:be a net such that
4442:is any subset, the
4416:More generally, if
4084:limit of a sequence
3778:to nets defined on
3590:is a function then
2877: for all
1665:{\displaystyle x=y}
144:is a non-empty set
21670:Dover Publications
21014:
20978:
20956:
20929:
20904:
20873:
20831:
20798:
20767:
20718:
20679:
20673:
20619:
20583:
20503:
20392:
20365:
20338:
20271:
20242:
20149:
20118:{\displaystyle r.}
20115:
20088:
20047:
20013:
19984:
19951:
19929:
19877:
19809:
19778:
19745:
19643:
19615:{\displaystyle V.}
19612:
19592:{\displaystyle y,}
19589:
19566:
19546:
19526:
19498:
19460:
19432:there exists some
19422:
19384:
19364:
19344:
19325:{\displaystyle V.}
19322:
19299:
19283:{\displaystyle L,}
19280:
19257:
19237:
19218:
19187:{\displaystyle V.}
19184:
19161:
19132:
19094:
19056:
19036:
19014:
18982:{\displaystyle X.}
18979:
18956:
18924:
18904:
18884:
18845:{\displaystyle V.}
18842:
18822:{\displaystyle L,}
18819:
18796:
18776:
18756:
18733:{\displaystyle S.}
18730:
18707:
18678:
18622:
18594:there exists some
18584:
18546:
18526:
18506:
18459:
18439:
18423:{\displaystyle L,}
18420:
18397:
18377:
18358:
18326:
18300:
18280:
18263:{\displaystyle S.}
18260:
18237:
18208:
18149:
18111:
18073:
18053:
18033:
17987:
17952:
17917:{\displaystyle M,}
17914:
17891:
17871:
17851:
17792:
17776:by declaring that
17766:
17746:
17708:
17678:
17643:
17617:
17583:
17487:
17462:
17413:
17375:
17334:
17288:
17187:
17162:
17110:
17078:
17045:
16984:
16954:
16942:pseudometric space
16926:
16866:
16780:
16766:
16750:
16721:
16705:
16657:
16565:
16540:
16505:
16493:
16492:
16460:
16438:
16414:
16387:
16338:
16298:
16278:
16257:{\displaystyle X.}
16254:
16231:
16204:
16150:
16123:
16065:
16035:there exists some
16025:
15996:
15963:
15936:
15904:
15884:
15857:
15789:
15734:
15689:
15644:
15611:
15578:
15543:
15511:
15419:
15356:
15308:
15279:
15252:
15207:
15195:
15173:
15153:
15126:
15096:
15069:
15012:
14952:
14923:
14911:
14889:
14869:
14842:
14827:
14826:
14750:
14714:
14651:
14623:
14588:
14576:
14535:
14486:
14419:
14392:
14365:
14306:
14243:
14214:
14194:
14182:
14160:
14097:
14095:
14004:
13961:
13934:
13901:
13890:
13860:
13835:
13776:{\displaystyle x.}
13773:
13758:neighborhood bases
13745:{\displaystyle x.}
13742:
13719:
13689:
13669:
13649:
13622:
13589:
13569:
13539:
13519:
13492:
13471:{\displaystyle x,}
13468:
13445:
13418:
13394:
13374:
13342:{\displaystyle S.}
13339:
13316:
13289:
13256:{\displaystyle T.}
13253:
13230:
13208:
13182:
13158:{\displaystyle x.}
13155:
13128:
13101:
13084:{\displaystyle a.}
13081:
13058:
13038:
13011:
12950:
12916:
12896:
12872:
12852:
12828:{\displaystyle S.}
12825:
12802:
12785:
12762:
12742:
12725:
12705:{\displaystyle X,}
12702:
12675:
12595:
12579:for every integer
12569:
12496:
12460:
12450:smallest value in
12440:
12423:is defined as the
12413:
12386:
12329:
12274:
12240:
12220:
12198:
12164:
12142:
12122:
12096:
12049:
12020:
11994:
11974:
11933:
11898:
11869:{\displaystyle x.}
11866:
11843:
11799:
11773:
11742:
11727:that converges to
11717:
11693:
11678:that converges to
11668:
11648:
11593:
11559:
11539:
11497:
11477:
11430:
11420:that are equal to
11410:
11364:
11340:
11305:
11252:
11232:
11225:
11224:
11188:
11152:
11105:
11076:
11059:{\displaystyle x,}
11056:
11033:
11016:{\displaystyle X,}
11013:
10990:
10945:
10916:
10899:{\displaystyle x,}
10896:
10873:
10856:{\displaystyle X,}
10853:
10830:
10803:
10779:
10759:
10739:
10715:
10695:
10675:
10652:{\displaystyle S.}
10649:
10626:
10590:
10564:
10536:
10516:
10499:{\displaystyle V.}
10496:
10476:{\displaystyle L,}
10473:
10450:
10430:
10388:{\displaystyle S.}
10385:
10362:
10335:
10306:
10278:
10258:
10227:
10196:
10170:{\displaystyle X.}
10167:
10144:
10098:
10074:
10049:
10029:
10009:
9969:preorder form the
9959:
9937:
9893:algebraic topology
9878:eventuality filter
9866:
9846:
9749:
9729:
9642:
9622:
9598:
9578:
9546:{\displaystyle y.}
9543:
9520:
9471:
9423:
9399:
9379:
9359:
9327:
9295:
9259:
9233:
9213:
9193:
9173:
9141:
9121:
9091:
9069:
9039:
9019:
8992:
8972:
8943:
8923:
8903:
8810:
8767:
8732:
8712:
8685:
8659:
8633:
8582:
8553:
8513:
8480:
8451:
8431:
8368:
8348:
8299:
8279:
8256:
8229:
8199:
8169:
8146:
8114:
8055:
7990:
7967:{\displaystyle A.}
7964:
7941:
7921:
7861:{\displaystyle X.}
7858:
7835:
7815:
7784:
7762:
7745:
7713:
7684:
7667:
7635:
7615:
7567:
7543:
7523:
7481:{\displaystyle X.}
7478:
7455:
7435:
7370:
7340:{\displaystyle x.}
7337:
7314:
7297:{\displaystyle a.}
7294:
7271:
7251:
7216:{\displaystyle U,}
7213:
7190:
7164:
7126:
7097:
7071:
7008:
6985:
6965:
6942:
6906:
6886:
6866:
6830:
6803:{\displaystyle x,}
6800:
6777:
6758:{\displaystyle U.}
6755:
6732:
6694:
6674:
6654:
6634:
6614:
6594:
6571:{\displaystyle a,}
6568:
6545:
6525:
6461:
6433:
6413:
6384:
6349:{\displaystyle x.}
6346:
6326:{\displaystyle V,}
6323:
6304:{\displaystyle f,}
6301:
6278:
6249:
6229:{\displaystyle x.}
6226:
6203:
6183:
6120:
6098:
6078:
6055:
5994:
5974:
5951:
5888:
5868:{\displaystyle U.}
5865:
5842:
5813:
5778:
5717:
5688:
5664:{\displaystyle x,}
5661:
5638:
5612:{\displaystyle V,}
5609:
5582:
5562:
5542:
5522:
5478:{\displaystyle f,}
5475:
5452:
5420:
5400:
5380:
5357:
5299:{\displaystyle x,}
5296:
5273:
5253:
5208:
5191:{\displaystyle Y.}
5188:
5165:
5112:
5092:
5059:
5030:{\displaystyle Y.}
5027:
5004:
4946:
4926:
4909:
4886:
4825:
4801:
4760:{\displaystyle S.}
4757:
4734:
4714:
4691:{\displaystyle S.}
4688:
4665:
4639:
4599:
4579:
4531:
4514:
4482:
4456:
4432:
4404:
4378:{\displaystyle X,}
4375:
4352:
4310:
4284:
4251:
4190:
4170:
4150:
4130:
4110:
4031:
4003:{\displaystyle V.}
4000:
3977:
3928:
3903:such that for all
3893:
3867:
3836:
3760:{\displaystyle x.}
3757:
3734:
3714:
3686:{\displaystyle Y.}
3683:
3670:is an ultranet in
3660:
3580:
3548:
3538:is an ultranet in
3528:
3461:
3432:
3405:
3385:
3359:
3314:
3294:
3260:
3230:
3210:
3183:
3155:
3126:
3101:there exists some
3091:
3062:
3036:
3007:
2960:
2942:order homomorphism
2924:
2892:
2825:
2805:
2776:
2744:
2705:
2678:
2617:
2528:
2451:
2425:
2374:
2354:
2327:
2282:{\displaystyle x.}
2279:
2256:
2233:{\displaystyle U.}
2230:
2210:{\displaystyle x,}
2207:
2184:
2166:accumulation point
2152:
2126:
2090:
2064:
2048:there exists some
2038:
2012:
1977:
1918:is not Hausdorff.
1907:
1886:
1850:
1810:
1787:
1770:
1672:) then one writes:
1662:
1636:
1600:
1561:
1541:
1539:
1492:
1437:
1265:
1242:
1211:
1191:
1167:{\displaystyle x,}
1164:
1141:
1114:
1094:
1051:
1025:
989:
960:
934:
908:
877:
808:
790:algebraic topology
788:. As is common in
778:
730:
706:
696:, and one assumes
682:
658:
609:
586:
543:
482:
448:
422:
396:
370:
324:
304:
275:
249:
233:there exists some
223:
180:
154:
57:whose domain is a
21679:978-0-486-43479-7
21642:978-0-12-622760-4
21533:978-0-387-90125-1
21491:978-0-387-97986-1
21426:978-3-540-32696-0
21335:, pp. 71–72.
21264:, pp. 83–92.
21138:, pp. 73–77.
21113:, pp. 65–72.
21098:Megginson, p. 143
20672:
19569:{\displaystyle V}
19549:{\displaystyle f}
19387:{\displaystyle X}
19367:{\displaystyle V}
19347:{\displaystyle f}
19309:is eventually in
19302:{\displaystyle f}
19260:{\displaystyle V}
19203:
19059:{\displaystyle X}
19039:{\displaystyle V}
18927:{\displaystyle f}
18907:{\displaystyle t}
18894:with limit point
18799:{\displaystyle V}
18779:{\displaystyle f}
18549:{\displaystyle X}
18529:{\displaystyle S}
18462:{\displaystyle V}
18449:is eventually in
18442:{\displaystyle f}
18400:{\displaystyle V}
18343:
18310:to a given point
18303:{\displaystyle X}
18283:{\displaystyle f}
18076:{\displaystyle X}
18056:{\displaystyle S}
17894:{\displaystyle c}
17874:{\displaystyle c}
17769:{\displaystyle c}
16957:{\displaystyle M}
16751:
16735:
16706:
16690:
16588:ultrafilter lemma
16477:
16441:{\displaystyle I}
16352:limit of the net
16301:{\displaystyle I}
16281:{\displaystyle L}
15907:{\displaystyle 1}
15176:{\displaystyle L}
15014:
15010:
14892:{\displaystyle L}
14811:
14721:
14716:
14712:
14698:
14313:
14308:
14304:
14290:
14217:{\displaystyle A}
13875:
13847:Cartesian product
13692:{\displaystyle x}
13672:{\displaystyle X}
13592:{\displaystyle X}
13542:{\displaystyle x}
13495:{\displaystyle x}
13397:{\displaystyle S}
13233:{\displaystyle S}
13104:{\displaystyle x}
13068:for large enough
13061:{\displaystyle x}
12919:{\displaystyle x}
12899:{\displaystyle S}
12875:{\displaystyle x}
12779:
12765:{\displaystyle X}
12719:
12689:induced on it by
12687:subspace topology
12614:Subspace topology
12463:{\displaystyle A}
12437:
12243:{\displaystyle E}
12145:{\displaystyle E}
11997:{\displaystyle E}
11720:{\displaystyle E}
11671:{\displaystyle E}
11562:{\displaystyle E}
11500:{\displaystyle 0}
11433:{\displaystyle 1}
11367:{\displaystyle E}
11255:{\displaystyle f}
11207:
11079:{\displaystyle f}
11036:{\displaystyle X}
10993:{\displaystyle x}
10964:sequential spaces
10919:{\displaystyle f}
10876:{\displaystyle X}
10833:{\displaystyle x}
10806:{\displaystyle f}
10782:{\displaystyle Y}
10762:{\displaystyle X}
10742:{\displaystyle f}
10718:{\displaystyle y}
10698:{\displaystyle V}
10539:{\displaystyle X}
10519:{\displaystyle S}
10453:{\displaystyle V}
10281:{\displaystyle X}
10261:{\displaystyle S}
10240:Similarly, every
10101:{\displaystyle X}
10052:{\displaystyle X}
10032:{\displaystyle X}
9869:{\displaystyle X}
9752:{\displaystyle X}
9645:{\displaystyle X}
9625:{\displaystyle X}
9601:{\displaystyle X}
9581:{\displaystyle X}
9557:
9556:
9426:{\displaystyle y}
9402:{\displaystyle B}
9382:{\displaystyle a}
9236:{\displaystyle X}
9216:{\displaystyle y}
9196:{\displaystyle U}
9144:{\displaystyle B}
9094:{\displaystyle y}
9042:{\displaystyle y}
8995:{\displaystyle y}
8946:{\displaystyle A}
8926:{\displaystyle X}
8825:
8824:
8735:{\displaystyle x}
8592:Consider the net
8454:{\displaystyle D}
8371:{\displaystyle X}
8302:{\displaystyle X}
8131:
7944:{\displaystyle X}
7838:{\displaystyle X}
7730:
7652:
7638:{\displaystyle X}
7570:{\displaystyle I}
7546:{\displaystyle X}
7458:{\displaystyle X}
7373:{\displaystyle X}
7352:
7351:
7317:{\displaystyle f}
7274:{\displaystyle U}
7174:is eventually in
7002:
6959:
6909:{\displaystyle W}
6889:{\displaystyle b}
6780:{\displaystyle W}
6697:{\displaystyle x}
6677:{\displaystyle V}
6657:{\displaystyle V}
6637:{\displaystyle x}
6617:{\displaystyle V}
6548:{\displaystyle x}
6464:{\displaystyle x}
6436:{\displaystyle x}
6252:{\displaystyle U}
6206:{\displaystyle f}
6193:Now suppose that
6114:
6072:
5997:{\displaystyle x}
5882:
5788:is eventually in
5698:is eventually in
5671:and consequently
5585:{\displaystyle x}
5565:{\displaystyle f}
5545:{\displaystyle x}
5423:{\displaystyle U}
5374:
5276:{\displaystyle f}
5211:{\displaystyle X}
5115:{\displaystyle X}
4949:{\displaystyle X}
4903:
4828:{\displaystyle x}
4737:{\displaystyle S}
4602:{\displaystyle S}
4499:
4459:{\displaystyle S}
4193:{\displaystyle S}
4173:{\displaystyle S}
4153:{\displaystyle X}
4133:{\displaystyle X}
3870:{\displaystyle V}
3737:{\displaystyle x}
3551:{\displaystyle X}
3408:{\displaystyle S}
3395:is eventually in
3317:{\displaystyle X}
3233:{\displaystyle x}
3065:{\displaystyle A}
2878:
2828:{\displaystyle A}
2377:{\displaystyle X}
2187:{\displaystyle U}
2162:is said to be an
2015:{\displaystyle S}
1910:{\displaystyle X}
1755:
1753:
1750:
1746:
1742:
1739:
1715:
1712:
1708:
1704:
1701:
1564:{\displaystyle X}
1529:
1483:
1474:
1422:
1413:
1365:
1322:
1268:{\displaystyle x}
1245:{\displaystyle x}
1214:{\displaystyle U}
1201:is eventually in
1144:{\displaystyle U}
1117:{\displaystyle X}
911:{\displaystyle S}
709:{\displaystyle A}
685:{\displaystyle X}
612:{\displaystyle A}
485:{\displaystyle X}
348:greatest elements
327:{\displaystyle A}
157:{\displaystyle A}
91:sequential spaces
67:topological space
16:(Redirected from
21712:
21705:General topology
21691:
21661:General Topology
21654:
21624:
21622:
21620:
21593:
21567:
21554:General Topology
21545:
21520:General Topology
21511:
21472:(23 June 1995).
21470:Howes, Norman R.
21465:
21438:
21408:
21406:
21384:
21381:
21375:
21372:
21366:
21363:
21357:
21354:
21348:
21342:
21336:
21330:
21324:
21321:
21310:
21309:
21303:
21295:
21293:
21292:
21286:
21280:. Archived from
21279:
21271:
21265:
21259:
21253:
21251:
21232:
21226:
21224:
21211:General Topology
21205:
21196:
21190:
21173:
21167:
21158:
21152:
21139:
21133:
21114:
21108:
21099:
21096:
21090:
21083:
21077:
21076:
21041:
21024:
21023:
21021:
21020:
21015:
21004:
21003:
20987:
20985:
20984:
20979:
20977:
20965:
20963:
20962:
20957:
20955:
20954:
20938:
20936:
20935:
20930:
20925:
20913:
20911:
20910:
20905:
20900:
20882:
20880:
20879:
20874:
20857:
20856:
20840:
20838:
20837:
20832:
20824:
20823:
20807:
20805:
20804:
20799:
20794:
20776:
20774:
20773:
20768:
20766:
20765:
20744:
20743:
20727:
20725:
20724:
20719:
20717:
20706:
20688:
20686:
20685:
20680:
20678:
20674:
20668:
20657:
20628:
20626:
20625:
20620:
20618:
20610:
20592:
20590:
20589:
20584:
20552:
20551:
20550:
20538:
20534:
20533:
20512:
20510:
20509:
20504:
20454:
20453:
20452:
20440:
20436:
20435:
20413:
20407:
20401:
20399:
20398:
20393:
20391:
20390:
20374:
20372:
20371:
20366:
20364:
20363:
20347:
20345:
20344:
20339:
20331:
20330:
20318:
20317:
20306:
20302:
20301:
20281:This shows that
20280:
20278:
20277:
20272:
20252:holds for every
20251:
20249:
20248:
20243:
20241:
20240:
20222:
20221:
20191:
20187:
20180:
20179:
20158:
20156:
20155:
20150:
20148:
20124:
20122:
20121:
20116:
20097:
20095:
20094:
20089:
20056:
20054:
20053:
20048:
20046:
20022:
20020:
20019:
20014:
19993:
19991:
19990:
19985:
19977:
19976:
19960:
19958:
19957:
19952:
19938:
19936:
19935:
19930:
19913:
19886:
19884:
19883:
19878:
19870:
19862:
19861:
19860:
19835:
19834:
19818:
19816:
19815:
19810:
19805:
19787:
19785:
19784:
19779:
19771:
19770:
19754:
19752:
19751:
19746:
19744:
19743:
19738:
19718:
19703:
19692:Sequential space
19652:
19650:
19649:
19644:
19621:
19619:
19618:
19613:
19598:
19596:
19595:
19590:
19575:
19573:
19572:
19567:
19555:
19553:
19552:
19547:
19535:
19533:
19532:
19527:
19507:
19505:
19504:
19499:
19469:
19467:
19466:
19461:
19431:
19429:
19428:
19423:
19393:
19391:
19390:
19385:
19373:
19371:
19370:
19365:
19353:
19351:
19350:
19345:
19331:
19329:
19328:
19323:
19308:
19306:
19305:
19300:
19289:
19287:
19286:
19281:
19266:
19264:
19263:
19258:
19246:
19244:
19243:
19238:
19217:
19193:
19191:
19190:
19185:
19170:
19168:
19167:
19162:
19141:
19139:
19138:
19133:
19103:
19101:
19100:
19095:
19065:
19063:
19062:
19057:
19045:
19043:
19042:
19037:
19023:
19021:
19020:
19015:
18988:
18986:
18985:
18980:
18965:
18963:
18962:
18957:
18933:
18931:
18930:
18925:
18913:
18911:
18910:
18905:
18893:
18891:
18890:
18887:{\displaystyle }
18885:
18861:well-ordered set
18851:
18849:
18848:
18843:
18828:
18826:
18825:
18820:
18805:
18803:
18802:
18797:
18785:
18783:
18782:
18777:
18765:
18763:
18762:
18757:
18739:
18737:
18736:
18731:
18716:
18714:
18713:
18708:
18687:
18685:
18684:
18679:
18631:
18629:
18628:
18623:
18593:
18591:
18590:
18585:
18555:
18553:
18552:
18547:
18535:
18533:
18532:
18527:
18515:
18513:
18512:
18507:
18468:
18466:
18465:
18460:
18448:
18446:
18445:
18440:
18429:
18427:
18426:
18421:
18406:
18404:
18403:
18398:
18386:
18384:
18383:
18378:
18357:
18335:
18333:
18332:
18327:
18309:
18307:
18306:
18301:
18289:
18287:
18286:
18281:
18269:
18267:
18266:
18261:
18246:
18244:
18243:
18238:
18217:
18215:
18214:
18209:
18158:
18156:
18155:
18150:
18120:
18118:
18117:
18112:
18082:
18080:
18079:
18074:
18062:
18060:
18059:
18054:
18042:
18040:
18039:
18034:
17996:
17994:
17993:
17988:
17961:
17959:
17958:
17953:
17923:
17921:
17920:
17915:
17900:
17898:
17897:
17892:
17880:
17878:
17877:
17872:
17860:
17858:
17857:
17852:
17801:
17799:
17798:
17793:
17775:
17773:
17772:
17767:
17755:
17753:
17752:
17747:
17717:
17715:
17714:
17709:
17690:Euclidean metric
17687:
17685:
17684:
17679:
17677:
17676:
17671:
17652:
17650:
17649:
17644:
17626:
17624:
17623:
17618:
17592:
17590:
17589:
17584:
17579:
17578:
17567:
17563:
17559:
17558:
17557:
17529:
17525:
17524:
17523:
17496:
17494:
17493:
17488:
17483:
17471:
17469:
17468:
17463:
17455:
17451:
17450:
17449:
17422:
17420:
17419:
17414:
17384:
17382:
17381:
17376:
17368:
17367:
17350:seminormed space
17343:
17341:
17340:
17335:
17297:
17295:
17294:
17289:
17287:
17286:
17275:
17271:
17270:
17266:
17265:
17264:
17232:
17228:
17227:
17226:
17196:
17194:
17193:
17188:
17183:
17171:
17169:
17168:
17163:
17155:
17151:
17150:
17149:
17119:
17117:
17116:
17111:
17087:
17085:
17084:
17079:
17071:
17070:
17054:
17052:
17051:
17046:
17044:
17043:
17032:
17028:
17027:
17010:
17009:
16993:
16991:
16990:
16985:
16963:
16961:
16960:
16955:
16935:
16933:
16932:
16927:
16886:Riemann integral
16880:Riemann integral
16875:
16873:
16872:
16867:
16862:
16861:
16846:
16845:
16827:
16826:
16814:
16813:
16789:
16787:
16786:
16781:
16776:
16775:
16765:
16749:
16731:
16730:
16720:
16704:
16686:
16685:
16666:
16664:
16663:
16658:
16653:
16652:
16641:
16637:
16636:
16574:
16572:
16571:
16566:
16564:
16563:
16551:
16550:
16541:
16531:
16530:
16514:
16512:
16511:
16506:
16504:
16503:
16494:
16491:
16471:
16470:
16461:
16448:is infinite and
16447:
16445:
16444:
16439:
16423:
16421:
16420:
16415:
16413:
16412:
16396:
16394:
16393:
16388:
16386:
16382:
16381:
16368:
16367:
16347:
16345:
16344:
16339:
16337:
16336:
16324:
16323:
16307:
16305:
16304:
16299:
16287:
16285:
16284:
16279:
16263:
16261:
16260:
16255:
16240:
16238:
16237:
16232:
16230:
16229:
16213:
16211:
16210:
16205:
16203:
16202:
16191:
16187:
16186:
16159:
16157:
16156:
16151:
16149:
16148:
16132:
16130:
16129:
16124:
16122:
16121:
16109:
16105:
16104:
16091:
16090:
16074:
16072:
16071:
16066:
16064:
16063:
16051:
16050:
16034:
16032:
16031:
16026:
16005:
16003:
16002:
15997:
15972:
15970:
15969:
15964:
15962:
15961:
15945:
15943:
15942:
15937:
15913:
15911:
15910:
15905:
15893:
15891:
15890:
15885:
15883:
15882:
15866:
15864:
15863:
15858:
15838:
15834:
15833:
15832:
15820:
15819:
15798:
15796:
15795:
15790:
15788:
15787:
15782:
15773:
15772:
15760:
15759:
15743:
15741:
15740:
15735:
15733:
15729:
15728:
15715:
15714:
15698:
15696:
15695:
15690:
15688:
15684:
15683:
15670:
15669:
15653:
15651:
15650:
15645:
15637:
15636:
15620:
15618:
15617:
15612:
15604:
15603:
15587:
15585:
15584:
15579:
15552:
15550:
15549:
15544:
15520:
15518:
15517:
15512:
15428:
15426:
15425:
15420:
15418:
15417:
15416:
15404:
15400:
15399:
15382:
15381:
15365:
15363:
15362:
15357:
15355:
15347:
15346:
15334:
15333:
15317:
15315:
15314:
15309:
15289:for every index
15288:
15286:
15285:
15280:
15278:
15277:
15261:
15259:
15258:
15253:
15251:
15247:
15246:
15233:
15232:
15216:
15214:
15213:
15208:
15206:
15205:
15196:
15182:
15180:
15179:
15174:
15162:
15160:
15159:
15154:
15152:
15151:
15135:
15133:
15132:
15127:
15122:
15121:
15105:
15103:
15102:
15097:
15095:
15094:
15078:
15076:
15075:
15070:
15068:
15067:
15056:
15052:
15051:
15047:
15046:
15033:
15032:
15016:
15015:
15013:
15011:
15008:
15004:
14999:
14995:
14991:
14990:
14977:
14976:
14961:
14959:
14958:
14953:
14932:
14930:
14929:
14924:
14922:
14921:
14912:
14898:
14896:
14895:
14890:
14878:
14876:
14875:
14870:
14868:
14867:
14851:
14849:
14848:
14843:
14838:
14837:
14828:
14825:
14805:
14804:
14793:
14789:
14788:
14759:
14757:
14756:
14751:
14746:
14745:
14731:
14730:
14719:
14718:
14717:
14715:
14713:
14710:
14706:
14701:
14696:
14695:
14691:
14690:
14677:
14676:
14660:
14658:
14657:
14652:
14647:
14646:
14634:
14633:
14624:
14614:
14613:
14597:
14595:
14594:
14589:
14587:
14586:
14577:
14561:
14560:
14544:
14542:
14541:
14536:
14534:
14530:
14529:
14516:
14515:
14495:
14493:
14492:
14487:
14482:
14481:
14463:
14459:
14458:
14445:
14444:
14428:
14426:
14425:
14420:
14418:
14417:
14401:
14399:
14398:
14393:
14391:
14390:
14374:
14372:
14371:
14366:
14364:
14363:
14352:
14348:
14347:
14343:
14342:
14329:
14328:
14311:
14310:
14309:
14307:
14305:
14302:
14298:
14293:
14288:
14287:
14283:
14282:
14269:
14268:
14252:
14250:
14249:
14244:
14223:
14221:
14220:
14215:
14203:
14201:
14200:
14195:
14193:
14192:
14183:
14169:
14167:
14166:
14161:
14159:
14158:
14147:
14143:
14142:
14125:
14124:
14106:
14104:
14103:
14098:
14096:
14092:
14091:
14074:
14072:
14071:
14060:
14056:
14055:
14040:
14039:
14035:
14034:
14017:
14015:
14014:
14005:
13997:
13991:
13990:
13970:
13968:
13967:
13962:
13960:
13959:
13943:
13941:
13940:
13935:
13913:product topology
13910:
13908:
13907:
13902:
13900:
13899:
13889:
13871:
13870:
13861:
13844:
13842:
13841:
13836:
13834:
13833:
13822:
13818:
13817:
13797:Explicitly, let
13782:
13780:
13779:
13774:
13751:
13749:
13748:
13743:
13728:
13726:
13725:
13720:
13718:
13717:
13698:
13696:
13695:
13690:
13678:
13676:
13675:
13670:
13658:
13656:
13655:
13650:
13648:
13647:
13631:
13629:
13628:
13623:
13598:
13596:
13595:
13590:
13578:
13576:
13575:
13570:
13568:
13567:
13548:
13546:
13545:
13540:
13528:
13526:
13525:
13520:
13518:
13517:
13501:
13499:
13498:
13493:
13477:
13475:
13474:
13469:
13454:
13452:
13451:
13446:
13444:
13443:
13427:
13425:
13424:
13419:
13403:
13401:
13400:
13395:
13383:
13381:
13380:
13375:
13373:
13369:
13368:
13348:
13346:
13345:
13340:
13325:
13323:
13322:
13317:
13315:
13314:
13298:
13296:
13295:
13290:
13285:
13284:
13262:
13260:
13259:
13254:
13240:is contained in
13239:
13237:
13236:
13231:
13217:
13215:
13214:
13209:
13191:
13189:
13188:
13183:
13181:
13180:
13164:
13162:
13161:
13156:
13137:
13135:
13134:
13129:
13127:
13126:
13110:
13108:
13107:
13102:
13090:
13088:
13087:
13082:
13067:
13065:
13064:
13059:
13047:
13045:
13044:
13039:
13037:
13036:
13020:
13018:
13017:
13012:
13010:
13009:
12998:
12994:
12993:
12959:
12957:
12956:
12951:
12946:
12945:
12925:
12923:
12922:
12917:
12905:
12903:
12902:
12897:
12881:
12879:
12878:
12873:
12861:
12859:
12858:
12853:
12851:
12850:
12834:
12832:
12831:
12826:
12811:
12809:
12808:
12803:
12795:
12794:
12784:
12771:
12769:
12768:
12763:
12751:
12749:
12748:
12743:
12735:
12734:
12724:
12711:
12709:
12708:
12703:
12684:
12682:
12681:
12676:
12674:
12670:
12657:
12656:
12604:
12602:
12601:
12596:
12578:
12576:
12575:
12570:
12565:
12564:
12522:
12521:
12505:
12503:
12502:
12497:
12486:
12485:
12469:
12467:
12466:
12461:
12449:
12447:
12446:
12441:
12439:
12438:
12435:
12422:
12420:
12419:
12414:
12412:
12411:
12395:
12393:
12392:
12387:
12385:
12384:
12383:
12371:
12367:
12366:
12365:
12364:
12338:
12336:
12335:
12330:
12328:
12327:
12316:
12312:
12311:
12283:
12281:
12280:
12275:
12270:
12269:
12268:
12262:
12249:
12247:
12246:
12241:
12229:
12227:
12226:
12221:
12219:
12207:
12205:
12204:
12199:
12194:
12193:
12192:
12186:
12173:
12171:
12170:
12165:
12163:
12151:
12149:
12148:
12143:
12131:
12129:
12128:
12123:
12105:
12103:
12102:
12097:
12058:
12056:
12055:
12050:
12029:
12027:
12026:
12021:
12003:
12001:
12000:
11995:
11983:
11981:
11980:
11975:
11942:
11940:
11939:
11934:
11907:
11905:
11904:
11899:
11875:
11873:
11872:
11867:
11852:
11850:
11849:
11844:
11808:
11806:
11805:
11800:
11782:
11780:
11779:
11774:
11772:
11771:
11770:
11764:
11751:
11749:
11748:
11743:
11738:
11726:
11724:
11723:
11718:
11702:
11700:
11699:
11694:
11689:
11677:
11675:
11674:
11669:
11657:
11655:
11654:
11649:
11638:
11637:
11636:
11635:
11634:
11628:
11614:
11602:
11600:
11599:
11594:
11589:
11588:
11587:
11581:
11568:
11566:
11565:
11560:
11548:
11546:
11545:
11540:
11526:
11518:
11506:
11504:
11503:
11498:
11486:
11484:
11483:
11478:
11439:
11437:
11436:
11431:
11419:
11417:
11416:
11411:
11391:
11373:
11371:
11370:
11365:
11349:
11347:
11346:
11341:
11339:
11338:
11337:
11331:
11317:product topology
11314:
11312:
11311:
11306:
11301:
11300:
11299:
11261:
11259:
11258:
11253:
11241:
11239:
11238:
11233:
11231:
11226:
11223:
11222:
11197:
11195:
11194:
11189:
11187:
11179:
11161:
11159:
11158:
11153:
11151:
11150:
11149:
11143:
11114:
11112:
11111:
11106:
11085:
11083:
11082:
11077:
11065:
11063:
11062:
11057:
11042:
11040:
11039:
11034:
11022:
11020:
11019:
11014:
10999:
10997:
10996:
10991:
10980:Given any point
10954:
10952:
10951:
10946:
10925:
10923:
10922:
10917:
10905:
10903:
10902:
10897:
10882:
10880:
10879:
10874:
10862:
10860:
10859:
10854:
10839:
10837:
10836:
10831:
10820:Given any point
10812:
10810:
10809:
10804:
10788:
10786:
10785:
10780:
10768:
10766:
10765:
10760:
10748:
10746:
10745:
10740:
10724:
10722:
10721:
10716:
10704:
10702:
10701:
10696:
10684:
10682:
10681:
10676:
10658:
10656:
10655:
10650:
10635:
10633:
10632:
10627:
10616:
10615:
10599:
10597:
10596:
10591:
10573:
10571:
10570:
10565:
10563:
10545:
10543:
10542:
10537:
10525:
10523:
10522:
10517:
10505:
10503:
10502:
10497:
10482:
10480:
10479:
10474:
10459:
10457:
10456:
10451:
10439:
10437:
10436:
10431:
10423:
10422:
10413:
10412:
10407:
10394:
10392:
10391:
10386:
10371:
10369:
10368:
10363:
10361:
10360:
10344:
10342:
10341:
10336:
10315:
10313:
10312:
10307:
10305:
10287:
10285:
10284:
10279:
10267:
10265:
10264:
10259:
10236:
10234:
10233:
10228:
10226:
10225:
10205:
10203:
10202:
10197:
10195:
10182:linearly ordered
10176:
10174:
10173:
10168:
10153:
10151:
10150:
10145:
10119:
10107:
10105:
10104:
10099:
10083:
10081:
10080:
10075:
10070:
10058:
10056:
10055:
10050:
10038:
10036:
10035:
10030:
10018:
10016:
10015:
10010:
10002:
10001:
9989:
9988:
9968:
9966:
9965:
9960:
9946:
9944:
9943:
9938:
9936:
9918:Every non-empty
9897:general topology
9884:Robert G. Bartle
9875:
9873:
9872:
9867:
9855:
9853:
9852:
9847:
9845:
9841:
9834:
9833:
9821:
9817:
9810:
9809:
9785:
9784:
9758:
9756:
9755:
9750:
9738:
9736:
9735:
9730:
9728:
9727:
9716:
9712:
9711:
9651:
9649:
9648:
9643:
9631:
9629:
9628:
9623:
9607:
9605:
9604:
9599:
9587:
9585:
9584:
9579:
9564:Other properties
9552:
9550:
9549:
9544:
9529:
9527:
9526:
9521:
9519:
9518:
9497:
9496:
9480:
9478:
9477:
9472:
9470:
9469:
9458:
9454:
9453:
9432:
9430:
9429:
9424:
9408:
9406:
9405:
9400:
9388:
9386:
9385:
9380:
9368:
9366:
9365:
9360:
9336:
9334:
9333:
9328:
9304:
9302:
9301:
9296:
9285:
9284:
9268:
9266:
9265:
9260:
9242:
9240:
9239:
9234:
9222:
9220:
9219:
9214:
9202:
9200:
9199:
9194:
9182:
9180:
9179:
9174:
9150:
9148:
9147:
9142:
9130:
9128:
9127:
9122:
9117:
9116:
9100:
9098:
9097:
9092:
9078:
9076:
9075:
9070:
9065:
9064:
9048:
9046:
9045:
9040:
9028:
9026:
9025:
9020:
9018:
9017:
9001:
8999:
8998:
8993:
8981:
8979:
8978:
8973:
8952:
8950:
8949:
8944:
8933:(where as usual
8932:
8930:
8929:
8924:
8912:
8910:
8909:
8904:
8902:
8901:
8890:
8886:
8885:
8868:
8867:
8841:
8840:
8819:
8817:
8816:
8811:
8806:
8805:
8793:
8792:
8776:
8774:
8773:
8768:
8741:
8739:
8738:
8733:
8721:
8719:
8718:
8713:
8711:
8710:
8694:
8692:
8691:
8686:
8668:
8666:
8665:
8660:
8642:
8640:
8639:
8634:
8629:
8628:
8617:
8613:
8612:
8591:
8589:
8588:
8583:
8562:
8560:
8559:
8554:
8552:
8551:
8539:
8538:
8522:
8520:
8519:
8514:
8506:
8505:
8490:there exists an
8489:
8487:
8486:
8481:
8460:
8458:
8457:
8452:
8440:
8438:
8437:
8432:
8418:
8410:
8377:
8375:
8374:
8369:
8357:
8355:
8354:
8349:
8347:
8343:
8330:
8329:
8308:
8306:
8305:
8300:
8288:
8286:
8285:
8280:
8265:
8263:
8262:
8257:
8255:
8254:
8238:
8236:
8235:
8230:
8225:
8224:
8208:
8206:
8205:
8200:
8195:
8194:
8178:
8176:
8175:
8170:
8162:
8161:
8145:
8123:
8121:
8120:
8115:
8098:
8094:
8093:
8064:
8062:
8061:
8056:
8051:
8047:
8034:
8033:
8016:
8015:
7999:
7997:
7996:
7991:
7973:
7971:
7970:
7965:
7950:
7948:
7947:
7942:
7930:
7928:
7927:
7922:
7920:
7919:
7908:
7904:
7903:
7886:
7885:
7867:
7865:
7864:
7859:
7844:
7842:
7841:
7836:
7824:
7822:
7821:
7816:
7814:
7813:
7802:
7797:
7792:
7771:
7769:
7768:
7763:
7755:
7754:
7744:
7722:
7720:
7719:
7714:
7694:for each finite
7693:
7691:
7690:
7685:
7677:
7676:
7666:
7644:
7642:
7641:
7636:
7624:
7622:
7621:
7616:
7614:
7613:
7602:
7598:
7597:
7576:
7574:
7573:
7568:
7552:
7550:
7549:
7544:
7532:
7530:
7529:
7524:
7500:
7499:
7487:
7485:
7484:
7479:
7464:
7462:
7461:
7456:
7444:
7442:
7441:
7436:
7434:
7433:
7422:
7418:
7417:
7400:
7399:
7379:
7377:
7376:
7371:
7346:
7344:
7343:
7338:
7323:
7321:
7320:
7315:
7303:
7301:
7300:
7295:
7280:
7278:
7277:
7272:
7260:
7258:
7257:
7252:
7250:
7246:
7245:
7222:
7220:
7219:
7214:
7199:
7197:
7196:
7191:
7173:
7171:
7170:
7165:
7163:
7159:
7158:
7135:
7133:
7132:
7127:
7106:
7104:
7103:
7098:
7080:
7078:
7077:
7072:
7052:
7051:
7040:
7036:
7035:
7031:
7030:
7007:
6994:
6992:
6991:
6986:
6975:
6974:
6964:
6951:
6949:
6948:
6943:
6932:
6931:
6915:
6913:
6912:
6907:
6895:
6893:
6892:
6887:
6875:
6873:
6872:
6867:
6862:
6861:
6839:
6837:
6836:
6831:
6826:
6825:
6809:
6807:
6806:
6801:
6786:
6784:
6783:
6778:
6764:
6762:
6761:
6756:
6741:
6739:
6738:
6733:
6731:
6727:
6726:
6703:
6701:
6700:
6695:
6683:
6681:
6680:
6675:
6663:
6661:
6660:
6655:
6643:
6641:
6640:
6635:
6623:
6621:
6620:
6615:
6603:
6601:
6600:
6595:
6593:
6592:
6577:
6575:
6574:
6569:
6554:
6552:
6551:
6546:
6534:
6532:
6531:
6526:
6524:
6523:
6512:
6508:
6507:
6490:
6489:
6470:
6468:
6467:
6462:
6442:
6440:
6439:
6434:
6422:
6420:
6419:
6414:
6393:
6391:
6390:
6385:
6355:
6353:
6352:
6347:
6332:
6330:
6329:
6324:
6310:
6308:
6307:
6302:
6287:
6285:
6284:
6279:
6258:
6256:
6255:
6250:
6236:Then there is a
6235:
6233:
6232:
6227:
6212:
6210:
6209:
6204:
6192:
6190:
6189:
6184:
6164:
6163:
6152:
6148:
6147:
6143:
6142:
6119:
6107:
6105:
6104:
6099:
6088:
6087:
6077:
6064:
6062:
6061:
6056:
6054:
6053:
6042:
6038:
6037:
6020:
6019:
6003:
6001:
6000:
5995:
5983:
5981:
5980:
5975:
5960:
5958:
5957:
5952:
5932:
5931:
5920:
5916:
5915:
5911:
5910:
5887:
5874:
5872:
5871:
5866:
5851:
5849:
5848:
5843:
5822:
5820:
5819:
5814:
5787:
5785:
5784:
5779:
5777:
5776:
5765:
5761:
5760:
5756:
5755:
5726:
5724:
5723:
5718:
5697:
5695:
5694:
5689:
5687:
5686:
5670:
5668:
5667:
5662:
5647:
5645:
5644:
5639:
5618:
5616:
5615:
5610:
5591:
5589:
5588:
5583:
5571:
5569:
5568:
5563:
5551:
5549:
5548:
5543:
5531:
5529:
5528:
5523:
5509:
5508:
5484:
5482:
5481:
5476:
5461:
5459:
5458:
5453:
5429:
5427:
5426:
5421:
5409:
5407:
5406:
5401:
5390:
5389:
5379:
5366:
5364:
5363:
5358:
5356:
5355:
5344:
5340:
5339:
5322:
5321:
5305:
5303:
5302:
5297:
5282:
5280:
5279:
5274:
5262:
5260:
5259:
5254:
5230:
5229:
5224:sequential space
5217:
5215:
5214:
5209:
5197:
5195:
5194:
5189:
5174:
5172:
5171:
5166:
5149:
5145:
5144:
5121:
5119:
5118:
5113:
5101:
5099:
5098:
5093:
5085:
5084:
5068:
5066:
5065:
5060:
5036:
5034:
5033:
5028:
5013:
5011:
5010:
5005:
4988:
4984:
4983:
4967:
4955:
4953:
4952:
4947:
4935:
4933:
4932:
4927:
4919:
4918:
4908:
4895:
4893:
4892:
4887:
4885:
4884:
4873:
4869:
4868:
4851:
4850:
4834:
4832:
4831:
4826:
4810:
4808:
4807:
4802:
4766:
4764:
4763:
4758:
4743:
4741:
4740:
4735:
4723:
4721:
4720:
4715:
4697:
4695:
4694:
4689:
4674:
4672:
4671:
4666:
4648:
4646:
4645:
4640:
4608:
4606:
4605:
4600:
4588:
4586:
4585:
4580:
4578:
4577:
4566:
4562:
4561:
4540:
4538:
4537:
4532:
4524:
4523:
4513:
4491:
4489:
4488:
4483:
4465:
4463:
4462:
4457:
4441:
4439:
4438:
4433:
4413:
4411:
4410:
4405:
4384:
4382:
4381:
4376:
4361:
4359:
4358:
4353:
4345:
4344:
4335:
4334:
4332:
4319:
4317:
4316:
4311:
4293:
4291:
4290:
4285:
4277:
4276:
4260:
4258:
4257:
4252:
4250:
4249:
4238:
4234:
4233:
4216:
4215:
4199:
4197:
4196:
4191:
4179:
4177:
4176:
4171:
4159:
4157:
4156:
4151:
4139:
4137:
4136:
4131:
4119:
4117:
4116:
4111:
4052:(TVS) is called
4040:
4038:
4037:
4032:
4030:
4029:
4009:
4007:
4006:
4001:
3986:
3984:
3983:
3978:
3976:
3972:
3971:
3970:
3958:
3957:
3937:
3935:
3934:
3929:
3902:
3900:
3899:
3894:
3876:
3874:
3873:
3868:
3852:
3851:
3845:
3843:
3842:
3837:
3835:
3834:
3823:
3819:
3818:
3801:
3800:
3766:
3764:
3763:
3758:
3743:
3741:
3740:
3735:
3723:
3721:
3720:
3715:
3692:
3690:
3689:
3684:
3669:
3667:
3666:
3661:
3659:
3658:
3647:
3643:
3642:
3638:
3637:
3612:
3611:
3589:
3587:
3586:
3581:
3557:
3555:
3554:
3549:
3537:
3535:
3534:
3529:
3527:
3526:
3515:
3511:
3510:
3493:
3492:
3470:
3468:
3467:
3462:
3441:
3439:
3438:
3433:
3431:
3430:
3414:
3412:
3411:
3406:
3394:
3392:
3391:
3386:
3384:
3383:
3368:
3366:
3365:
3360:
3338:
3337:
3330:
3329:
3323:
3321:
3320:
3315:
3303:
3301:
3300:
3295:
3293:
3292:
3269:
3267:
3266:
3261:
3256:
3255:
3239:
3237:
3236:
3231:
3219:
3217:
3216:
3211:
3209:
3208:
3192:
3190:
3189:
3184:
3164:
3162:
3161:
3156:
3135:
3133:
3132:
3127:
3100:
3098:
3097:
3092:
3071:
3069:
3068:
3063:
3045:
3043:
3042:
3037:
3016:
3014:
3013:
3008:
2969:
2967:
2966:
2961:
2937:order-preserving
2933:
2931:
2930:
2925:
2901:
2899:
2898:
2893:
2879:
2876:
2873:
2872:
2851:
2850:
2834:
2832:
2831:
2826:
2814:
2812:
2811:
2806:
2785:
2783:
2782:
2777:
2753:
2751:
2750:
2745:
2743:
2742:
2725:
2724:
2714:
2712:
2711:
2706:
2704:
2703:
2687:
2685:
2684:
2679:
2677:
2676:
2665:
2661:
2660:
2643:
2642:
2626:
2624:
2623:
2618:
2616:
2615:
2604:
2600:
2599:
2582:
2581:
2537:
2535:
2534:
2529:
2527:
2523:
2498:
2497:
2480:
2479:
2460:
2458:
2457:
2452:
2434:
2432:
2431:
2426:
2424:
2420:
2419:
2400:
2399:
2383:
2381:
2380:
2375:
2363:
2361:
2360:
2355:
2353:
2352:
2336:
2334:
2333:
2328:
2326:
2322:
2321:
2305:
2304:
2288:
2286:
2285:
2280:
2265:
2263:
2262:
2257:
2239:
2237:
2236:
2231:
2216:
2214:
2213:
2208:
2193:
2191:
2190:
2185:
2168:
2167:
2161:
2159:
2158:
2153:
2135:
2133:
2132:
2127:
2116:
2115:
2099:
2097:
2096:
2091:
2073:
2071:
2070:
2065:
2047:
2045:
2044:
2039:
2021:
2019:
2018:
2013:
2001:
2000:
1993:
1992:
1986:
1984:
1983:
1978:
1976:
1975:
1964:
1960:
1959:
1942:
1941:
1916:
1914:
1913:
1908:
1895:
1893:
1892:
1887:
1879:
1878:
1859:
1857:
1856:
1851:
1843:
1842:
1819:
1817:
1816:
1811:
1796:
1794:
1793:
1788:
1780:
1779:
1769:
1751:
1748:
1747:
1744:
1740:
1737:
1729:
1728:
1713:
1710:
1709:
1706:
1702:
1699:
1691:
1690:
1671:
1669:
1668:
1663:
1645:
1643:
1642:
1637:
1629:
1628:
1609:
1607:
1606:
1601:
1593:
1592:
1570:
1568:
1567:
1562:
1550:
1548:
1547:
1542:
1540:
1530:
1527:
1522:
1516:
1507:
1505:
1504:
1491:
1475:
1472:
1467:
1461:
1452:
1450:
1449:
1436:
1414:
1411:
1406:
1400:
1391:
1389:
1388:
1366:
1363:
1358:
1352:
1343:
1341:
1340:
1330:
1323:
1320:
1315:
1309:
1300:
1298:
1297:
1287:
1274:
1272:
1271:
1266:
1251:
1249:
1248:
1243:
1231:
1230:
1220:
1218:
1217:
1212:
1200:
1198:
1197:
1192:
1190:
1189:
1173:
1171:
1170:
1165:
1150:
1148:
1147:
1142:
1123:
1121:
1120:
1115:
1103:
1101:
1100:
1095:
1093:
1092:
1075:
1074:
1067:
1066:
1060:
1058:
1057:
1052:
1034:
1032:
1031:
1026:
1015:
1014:
998:
996:
995:
990:
969:
967:
966:
961:
943:
941:
940:
935:
917:
915:
914:
909:
886:
884:
883:
878:
876:
875:
864:
860:
859:
842:
841:
817:
815:
814:
809:
787:
785:
784:
779:
777:
776:
765:
761:
760:
739:
737:
736:
731:
715:
713:
712:
707:
691:
689:
688:
683:
667:
665:
664:
659:
657:
656:
635:
634:
618:
616:
615:
610:
595:
593:
592:
587:
573:
572:
552:
550:
549:
544:
542:
541:
530:
526:
525:
508:
507:
491:
489:
488:
483:
457:
455:
454:
449:
431:
429:
428:
423:
405:
403:
402:
397:
379:
377:
376:
371:
352:maximal elements
333:
331:
330:
325:
313:
311:
310:
305:
284:
282:
281:
276:
258:
256:
255:
250:
232:
230:
229:
224:
189:
187:
186:
181:
164:together with a
163:
161:
160:
155:
43:general topology
32:Net (polyhedron)
21:
21720:
21719:
21715:
21714:
21713:
21711:
21710:
21709:
21695:
21694:
21680:
21643:
21629:Schechter, Eric
21618:
21616:
21614:
21598:Schechter, Eric
21590:
21564:
21550:Kelley, John L.
21534:
21516:Kelley, John L.
21492:
21482:Springer-Verlag
21454:
21427:
21393:
21388:
21387:
21382:
21378:
21373:
21369:
21364:
21360:
21355:
21351:
21343:
21339:
21331:
21327:
21322:
21313:
21297:
21296:
21290:
21288:
21284:
21277:
21275:"Archived copy"
21273:
21272:
21268:
21260:
21256:
21249:
21233:
21229:
21222:
21206:
21199:
21191:
21176:
21168:
21161:
21153:
21142:
21134:
21117:
21109:
21102:
21097:
21093:
21084:
21080:
21065:10.2307/2370388
21042:
21038:
21033:
21028:
21027:
20999:
20995:
20993:
20990:
20989:
20973:
20971:
20968:
20967:
20950:
20946:
20944:
20941:
20940:
20921:
20919:
20916:
20915:
20896:
20888:
20885:
20884:
20852:
20848:
20846:
20843:
20842:
20819:
20815:
20813:
20810:
20809:
20790:
20782:
20779:
20778:
20752:
20748:
20739:
20735:
20733:
20730:
20729:
20713:
20702:
20694:
20691:
20690:
20658:
20655:
20651:
20634:
20631:
20630:
20614:
20606:
20598:
20595:
20594:
20546:
20539:
20529:
20525:
20521:
20520:
20518:
20515:
20514:
20448:
20441:
20431:
20427:
20423:
20422:
20420:
20417:
20416:
20414:
20410:
20404:uncountable set
20386:
20382:
20380:
20377:
20376:
20359:
20355:
20353:
20350:
20349:
20326:
20322:
20307:
20297:
20293:
20289:
20288:
20286:
20283:
20282:
20257:
20254:
20253:
20236:
20232:
20208:
20204:
20175:
20171:
20170:
20166:
20164:
20161:
20160:
20144:
20130:
20127:
20126:
20107:
20104:
20103:
20062:
20059:
20058:
20042:
20028:
20025:
20024:
19999:
19996:
19995:
19972:
19968:
19966:
19963:
19962:
19944:
19941:
19940:
19909:
19892:
19889:
19888:
19866:
19856:
19849:
19845:
19830:
19826:
19824:
19821:
19820:
19801:
19793:
19790:
19789:
19766:
19762:
19760:
19757:
19756:
19739:
19734:
19733:
19725:
19722:
19721:
19719:
19715:
19710:
19701:
19665:
19629:
19626:
19625:
19604:
19601:
19600:
19581:
19578:
19577:
19561:
19558:
19557:
19541:
19538:
19537:
19515:
19512:
19511:
19475:
19472:
19471:
19437:
19434:
19433:
19399:
19396:
19395:
19379:
19376:
19375:
19359:
19356:
19355:
19339:
19336:
19335:
19314:
19311:
19310:
19294:
19291:
19290:
19272:
19269:
19268:
19252:
19249:
19248:
19207:
19201:
19198:
19197:
19176:
19173:
19172:
19147:
19144:
19143:
19109:
19106:
19105:
19071:
19068:
19067:
19051:
19048:
19047:
19031:
19028:
19027:
18994:
18991:
18990:
18971:
18968:
18967:
18939:
18936:
18935:
18919:
18916:
18915:
18914:and a function
18899:
18896:
18895:
18867:
18864:
18863:
18857:
18834:
18831:
18830:
18811:
18808:
18807:
18791:
18788:
18787:
18771:
18768:
18767:
18745:
18742:
18741:
18722:
18719:
18718:
18693:
18690:
18689:
18637:
18634:
18633:
18599:
18596:
18595:
18561:
18558:
18557:
18541:
18538:
18537:
18521:
18518:
18517:
18477:
18474:
18473:
18454:
18451:
18450:
18434:
18431:
18430:
18412:
18409:
18408:
18392:
18389:
18388:
18347:
18341:
18338:
18337:
18336:if and only if
18315:
18312:
18311:
18295:
18292:
18291:
18275:
18272:
18271:
18252:
18249:
18248:
18223:
18220:
18219:
18164:
18161:
18160:
18126:
18123:
18122:
18088:
18085:
18084:
18068:
18065:
18064:
18048:
18045:
18044:
18004:
18001:
18000:
17967:
17964:
17963:
17929:
17926:
17925:
17906:
17903:
17902:
17886:
17883:
17882:
17866:
17863:
17862:
17807:
17804:
17803:
17802:if and only if
17781:
17778:
17777:
17761:
17758:
17757:
17723:
17720:
17719:
17697:
17694:
17693:
17672:
17667:
17666:
17658:
17655:
17654:
17632:
17629:
17628:
17600:
17597:
17596:
17568:
17553:
17549:
17542:
17538:
17534:
17533:
17519:
17515:
17508:
17504:
17502:
17499:
17498:
17479:
17477:
17474:
17473:
17445:
17441:
17434:
17430:
17428:
17425:
17424:
17423:if and only if
17390:
17387:
17386:
17363:
17359:
17357:
17354:
17353:
17311:
17308:
17307:
17276:
17260:
17256:
17249:
17245:
17241:
17237:
17236:
17222:
17218:
17211:
17207:
17202:
17199:
17198:
17179:
17177:
17174:
17173:
17145:
17141:
17134:
17130:
17125:
17122:
17121:
17120:if and only if
17093:
17090:
17089:
17066:
17062:
17060:
17057:
17056:
17055:is a net, then
17033:
17023:
17019:
17015:
17014:
17005:
17001:
16999:
16996:
16995:
16994:is a point and
16973:
16970:
16969:
16966:metric topology
16949:
16946:
16945:
16909:
16906:
16905:
16902:
16882:
16857:
16853:
16841:
16837:
16822:
16818:
16809:
16805:
16797:
16794:
16793:
16771:
16767:
16755:
16739:
16726:
16722:
16710:
16694:
16681:
16677:
16672:
16669:
16668:
16642:
16632:
16628:
16624:
16623:
16621:
16618:
16617:
16604:
16592:axiom of choice
16577:surjective maps
16559:
16555:
16546:
16542:
16535:
16526:
16522:
16520:
16517:
16516:
16499:
16495:
16481:
16475:
16466:
16462:
16455:
16453:
16450:
16449:
16433:
16430:
16429:
16426:Hausdorff space
16408:
16404:
16402:
16399:
16398:
16377:
16373:
16369:
16363:
16359:
16357:
16354:
16353:
16332:
16328:
16319:
16315:
16313:
16310:
16309:
16293:
16290:
16289:
16273:
16270:
16269:
16266:axiom of choice
16246:
16243:
16242:
16225:
16221:
16219:
16216:
16215:
16192:
16182:
16178:
16174:
16173:
16165:
16162:
16161:
16144:
16140:
16138:
16135:
16134:
16117:
16113:
16100:
16096:
16092:
16086:
16082:
16080:
16077:
16076:
16059:
16055:
16046:
16042:
16040:
16037:
16036:
16011:
16008:
16007:
15985:
15982:
15981:
15978:
15957:
15953:
15951:
15948:
15947:
15919:
15916:
15915:
15899:
15896:
15895:
15878:
15874:
15872:
15869:
15868:
15828:
15824:
15815:
15811:
15810:
15806:
15804:
15801:
15800:
15783:
15778:
15777:
15768:
15764:
15755:
15751:
15749:
15746:
15745:
15724:
15720:
15716:
15710:
15706:
15704:
15701:
15700:
15679:
15675:
15671:
15665:
15661:
15659:
15656:
15655:
15632:
15628:
15626:
15623:
15622:
15599:
15595:
15593:
15590:
15589:
15558:
15555:
15554:
15526:
15523:
15522:
15434:
15431:
15430:
15412:
15405:
15395:
15391:
15387:
15386:
15377:
15373:
15371:
15368:
15367:
15351:
15342:
15338:
15329:
15325:
15323:
15320:
15319:
15294:
15291:
15290:
15273:
15269:
15267:
15264:
15263:
15242:
15238:
15234:
15228:
15224:
15222:
15219:
15218:
15201:
15197:
15190:
15188:
15185:
15184:
15168:
15165:
15164:
15147:
15143:
15141:
15138:
15137:
15117:
15113:
15111:
15108:
15107:
15090:
15086:
15084:
15081:
15080:
15057:
15042:
15038:
15034:
15028:
15024:
15023:
15019:
15018:
15007:
15005:
15000:
14998:
14997:
14986:
14982:
14978:
14972:
14968:
14966:
14963:
14962:
14938:
14935:
14934:
14917:
14913:
14906:
14904:
14901:
14900:
14884:
14881:
14880:
14863:
14859:
14857:
14854:
14853:
14833:
14829:
14815:
14809:
14794:
14784:
14780:
14776:
14775:
14767:
14764:
14763:
14741:
14737:
14726:
14722:
14709:
14707:
14702:
14700:
14699:
14686:
14682:
14678:
14672:
14668:
14666:
14663:
14662:
14642:
14638:
14629:
14625:
14618:
14609:
14605:
14603:
14600:
14599:
14582:
14578:
14571:
14556:
14552:
14550:
14547:
14546:
14525:
14521:
14517:
14511:
14507:
14505:
14502:
14501:
14477:
14473:
14454:
14450:
14446:
14440:
14436:
14434:
14431:
14430:
14413:
14409:
14407:
14404:
14403:
14386:
14382:
14380:
14377:
14376:
14353:
14338:
14334:
14330:
14324:
14320:
14319:
14315:
14314:
14301:
14299:
14294:
14292:
14291:
14278:
14274:
14270:
14264:
14260:
14258:
14255:
14254:
14229:
14226:
14225:
14209:
14206:
14205:
14188:
14184:
14177:
14175:
14172:
14171:
14148:
14138:
14134:
14130:
14129:
14120:
14116:
14114:
14111:
14110:
14094:
14093:
14087:
14083:
14081:
14073:
14061:
14051:
14047:
14043:
14042:
14037:
14036:
14030:
14026:
14024:
14016:
14010:
14006:
13999:
13996:
13986:
13982:
13978:
13976:
13973:
13972:
13955:
13951:
13949:
13946:
13945:
13920:
13917:
13916:
13895:
13891:
13879:
13866:
13862:
13855:
13853:
13850:
13849:
13823:
13813:
13809:
13805:
13804:
13802:
13799:
13798:
13788:
13765:
13762:
13761:
13734:
13731:
13730:
13713:
13712:
13704:
13701:
13700:
13684:
13681:
13680:
13664:
13661:
13660:
13643:
13639:
13637:
13634:
13633:
13608:
13605:
13604:
13584:
13581:
13580:
13563:
13562:
13560:
13557:
13556:
13534:
13531:
13530:
13513:
13509:
13507:
13504:
13503:
13487:
13484:
13483:
13460:
13457:
13456:
13439:
13435:
13433:
13430:
13429:
13409:
13406:
13405:
13389:
13386:
13385:
13364:
13360:
13356:
13354:
13351:
13350:
13331:
13328:
13327:
13310:
13306:
13304:
13301:
13300:
13280:
13276:
13268:
13265:
13264:
13245:
13242:
13241:
13225:
13222:
13221:
13197:
13194:
13193:
13176:
13172:
13170:
13167:
13166:
13147:
13144:
13143:
13122:
13118:
13116:
13113:
13112:
13096:
13093:
13092:
13073:
13070:
13069:
13053:
13050:
13049:
13032:
13028:
13026:
13023:
13022:
12999:
12989:
12985:
12981:
12980:
12978:
12975:
12974:
12971:
12965:
12941:
12937:
12935:
12932:
12931:
12911:
12908:
12907:
12891:
12888:
12887:
12867:
12864:
12863:
12846:
12842:
12840:
12837:
12836:
12817:
12814:
12813:
12790:
12786:
12783:
12777:
12774:
12773:
12772:if and only if
12757:
12754:
12753:
12730:
12726:
12723:
12717:
12714:
12713:
12694:
12691:
12690:
12652:
12648:
12647:
12643:
12623:
12620:
12619:
12616:
12611:
12584:
12581:
12580:
12554:
12550:
12517:
12513:
12511:
12508:
12507:
12481:
12477:
12475:
12472:
12471:
12455:
12452:
12451:
12434:
12430:
12428:
12425:
12424:
12407:
12403:
12401:
12398:
12397:
12379:
12372:
12360:
12356:
12355:
12351:
12347:
12346:
12344:
12341:
12340:
12317:
12307:
12303:
12299:
12298:
12296:
12293:
12292:
12264:
12263:
12258:
12257:
12255:
12252:
12251:
12235:
12232:
12231:
12215:
12213:
12210:
12209:
12188:
12187:
12182:
12181:
12179:
12176:
12175:
12159:
12157:
12154:
12153:
12137:
12134:
12133:
12111:
12108:
12107:
12067:
12064:
12063:
12035:
12032:
12031:
12009:
12006:
12005:
11989:
11986:
11985:
11948:
11945:
11944:
11913:
11910:
11909:
11881:
11878:
11877:
11858:
11855:
11854:
11814:
11811:
11810:
11809:if and only if
11788:
11785:
11784:
11766:
11765:
11760:
11759:
11757:
11754:
11753:
11734:
11732:
11729:
11728:
11712:
11709:
11708:
11685:
11683:
11680:
11679:
11663:
11660:
11659:
11630:
11629:
11624:
11623:
11622:
11618:
11610:
11608:
11605:
11604:
11583:
11582:
11577:
11576:
11574:
11571:
11570:
11554:
11551:
11550:
11522:
11514:
11512:
11509:
11508:
11492:
11489:
11488:
11445:
11442:
11441:
11425:
11422:
11421:
11387:
11379:
11376:
11375:
11359:
11356:
11355:
11333:
11332:
11327:
11326:
11324:
11321:
11320:
11295:
11288:
11284:
11267:
11264:
11263:
11262:with the tuple
11247:
11244:
11243:
11227:
11218:
11211:
11205:
11203:
11200:
11199:
11183:
11175:
11167:
11164:
11163:
11145:
11144:
11139:
11138:
11136:
11133:
11132:
11091:
11088:
11087:
11071:
11068:
11067:
11048:
11045:
11044:
11028:
11025:
11024:
11023:and any net in
11005:
11002:
11001:
10985:
10982:
10981:
10931:
10928:
10927:
10911:
10908:
10907:
10888:
10885:
10884:
10868:
10865:
10864:
10845:
10842:
10841:
10825:
10822:
10821:
10798:
10795:
10794:
10774:
10771:
10770:
10754:
10751:
10750:
10734:
10731:
10730:
10710:
10707:
10706:
10690:
10687:
10686:
10664:
10661:
10660:
10641:
10638:
10637:
10611:
10607:
10605:
10602:
10601:
10579:
10576:
10575:
10559:
10551:
10548:
10547:
10531:
10528:
10527:
10511:
10508:
10507:
10488:
10485:
10484:
10465:
10462:
10461:
10445:
10442:
10441:
10418:
10414:
10408:
10406:
10405:
10400:
10397:
10396:
10377:
10374:
10373:
10356:
10352:
10350:
10347:
10346:
10321:
10318:
10317:
10301:
10293:
10290:
10289:
10273:
10270:
10269:
10253:
10250:
10249:
10221:
10217:
10215:
10212:
10211:
10191:
10189:
10186:
10185:
10159:
10156:
10155:
10115:
10113:
10110:
10109:
10093:
10090:
10089:
10066:
10064:
10061:
10060:
10044:
10041:
10040:
10024:
10021:
10020:
9997:
9993:
9984:
9980:
9978:
9975:
9974:
9952:
9949:
9948:
9932:
9930:
9927:
9926:
9924:natural numbers
9916:
9861:
9858:
9857:
9829:
9825:
9805:
9801:
9780:
9776:
9775:
9771:
9770:
9766:
9764:
9761:
9760:
9744:
9741:
9740:
9717:
9707:
9703:
9699:
9698:
9696:
9693:
9692:
9676:
9670:
9637:
9634:
9633:
9617:
9614:
9613:
9610:Hausdorff space
9593:
9590:
9589:
9573:
9570:
9569:
9566:
9558:
9535:
9532:
9531:
9505:
9501:
9492:
9488:
9486:
9483:
9482:
9459:
9449:
9445:
9441:
9440:
9438:
9435:
9434:
9418:
9415:
9414:
9394:
9391:
9390:
9374:
9371:
9370:
9342:
9339:
9338:
9310:
9307:
9306:
9280:
9276:
9274:
9271:
9270:
9248:
9245:
9244:
9228:
9225:
9224:
9208:
9205:
9204:
9188:
9185:
9184:
9156:
9153:
9152:
9136:
9133:
9132:
9112:
9108:
9106:
9103:
9102:
9086:
9083:
9082:
9060:
9056:
9054:
9051:
9050:
9034:
9031:
9030:
9013:
9009:
9007:
9004:
9003:
8987:
8984:
8983:
8958:
8955:
8954:
8938:
8935:
8934:
8918:
8915:
8914:
8891:
8881:
8877:
8873:
8872:
8863:
8859:
8857:
8854:
8853:
8846:
8831:
8826:
8801:
8797:
8788:
8784:
8782:
8779:
8778:
8747:
8744:
8743:
8727:
8724:
8723:
8706:
8702:
8700:
8697:
8696:
8674:
8671:
8670:
8648:
8645:
8644:
8618:
8608:
8604:
8600:
8599:
8597:
8594:
8593:
8568:
8565:
8564:
8547:
8543:
8534:
8530:
8528:
8525:
8524:
8501:
8497:
8495:
8492:
8491:
8466:
8463:
8462:
8446:
8443:
8442:
8414:
8406:
8383:
8380:
8379:
8363:
8360:
8359:
8325:
8321:
8320:
8316:
8314:
8311:
8310:
8294:
8291:
8290:
8274:
8271:
8270:
8250:
8246:
8244:
8241:
8240:
8220:
8216:
8214:
8211:
8210:
8190:
8186:
8184:
8181:
8180:
8157:
8153:
8135:
8129:
8126:
8125:
8089:
8085:
8081:
8070:
8067:
8066:
8065:The collection
8029:
8025:
8024:
8020:
8011:
8007:
8005:
8002:
8001:
7979:
7976:
7975:
7956:
7953:
7952:
7936:
7933:
7932:
7909:
7899:
7895:
7891:
7890:
7881:
7877:
7875:
7872:
7871:
7850:
7847:
7846:
7830:
7827:
7826:
7803:
7793:
7788:
7780:
7779:
7777:
7774:
7773:
7750:
7746:
7734:
7728:
7725:
7724:
7699:
7696:
7695:
7672:
7668:
7656:
7650:
7647:
7646:
7630:
7627:
7626:
7603:
7593:
7589:
7585:
7584:
7582:
7579:
7578:
7562:
7559:
7558:
7538:
7535:
7534:
7516:
7513:
7512:
7505:
7470:
7467:
7466:
7450:
7447:
7446:
7423:
7413:
7409:
7405:
7404:
7395:
7391:
7389:
7386:
7385:
7365:
7362:
7361:
7358:
7353:
7329:
7326:
7325:
7309:
7306:
7305:
7286:
7283:
7282:
7266:
7263:
7262:
7241:
7237:
7233:
7228:
7225:
7224:
7205:
7202:
7201:
7179:
7176:
7175:
7154:
7150:
7146:
7141:
7138:
7137:
7112:
7109:
7108:
7086:
7083:
7082:
7041:
7026:
7022:
7018:
7014:
7010:
7009:
7006:
7000:
6997:
6996:
6970:
6966:
6963:
6957:
6954:
6953:
6927:
6923:
6921:
6918:
6917:
6901:
6898:
6897:
6881:
6878:
6877:
6857:
6853:
6845:
6842:
6841:
6821:
6817:
6815:
6812:
6811:
6792:
6789:
6788:
6772:
6769:
6768:
6747:
6744:
6743:
6722:
6718:
6714:
6709:
6706:
6705:
6689:
6686:
6685:
6669:
6666:
6665:
6649:
6646:
6645:
6644:is included in
6629:
6626:
6625:
6609:
6606:
6605:
6588:
6584:
6582:
6579:
6578:
6560:
6557:
6556:
6555:whose index is
6540:
6537:
6536:
6513:
6503:
6499:
6495:
6494:
6485:
6481:
6479:
6476:
6475:
6456:
6453:
6452:
6428:
6425:
6424:
6399:
6396:
6395:
6361:
6358:
6357:
6338:
6335:
6334:
6315:
6312:
6311:
6293:
6290:
6289:
6264:
6261:
6260:
6244:
6241:
6240:
6218:
6215:
6214:
6198:
6195:
6194:
6153:
6138:
6134:
6130:
6126:
6122:
6121:
6118:
6112:
6109:
6108:
6083:
6079:
6076:
6070:
6067:
6066:
6043:
6033:
6029:
6025:
6024:
6015:
6011:
6009:
6006:
6005:
5989:
5986:
5985:
5969:
5966:
5965:
5921:
5906:
5902:
5898:
5894:
5890:
5889:
5886:
5880:
5877:
5876:
5857:
5854:
5853:
5828:
5825:
5824:
5793:
5790:
5789:
5766:
5751:
5747:
5743:
5739:
5735:
5734:
5732:
5729:
5728:
5703:
5700:
5699:
5682:
5678:
5676:
5673:
5672:
5653:
5650:
5649:
5624:
5621:
5620:
5601:
5598:
5597:
5577:
5574:
5573:
5557:
5554:
5553:
5537:
5534:
5533:
5501:
5497:
5489:
5486:
5485:
5467:
5464:
5463:
5435:
5432:
5431:
5415:
5412:
5411:
5385:
5381:
5378:
5372:
5369:
5368:
5345:
5335:
5331:
5327:
5326:
5317:
5313:
5311:
5308:
5307:
5288:
5285:
5284:
5268:
5265:
5264:
5246:
5243:
5242:
5235:
5203:
5200:
5199:
5180:
5177:
5176:
5140:
5136:
5132:
5127:
5124:
5123:
5107:
5104:
5103:
5080:
5076:
5074:
5071:
5070:
5042:
5039:
5038:
5019:
5016:
5015:
4979:
4975:
4971:
4966:
4961:
4958:
4957:
4941:
4938:
4937:
4914:
4910:
4907:
4901:
4898:
4897:
4896:in the domain,
4874:
4864:
4860:
4856:
4855:
4846:
4842:
4840:
4837:
4836:
4820:
4817:
4816:
4784:
4781:
4780:
4777:
4749:
4746:
4745:
4729:
4726:
4725:
4703:
4700:
4699:
4680:
4677:
4676:
4654:
4651:
4650:
4628:
4625:
4624:
4621:
4615:
4594:
4591:
4590:
4567:
4557:
4553:
4549:
4548:
4546:
4543:
4542:
4519:
4515:
4503:
4497:
4494:
4493:
4471:
4468:
4467:
4451:
4448:
4447:
4421:
4418:
4417:
4390:
4387:
4386:
4367:
4364:
4363:
4340:
4336:
4333:
4331:
4330:
4325:
4322:
4321:
4299:
4296:
4295:
4272:
4268:
4266:
4263:
4262:
4239:
4229:
4225:
4221:
4220:
4211:
4207:
4205:
4202:
4201:
4185:
4182:
4181:
4165:
4162:
4161:
4145:
4142:
4141:
4125:
4122:
4121:
4099:
4096:
4095:
4092:
4080:
4025:
4021:
4019:
4016:
4015:
3992:
3989:
3988:
3987:is a member of
3966:
3962:
3953:
3949:
3948:
3944:
3942:
3939:
3938:
3908:
3905:
3904:
3882:
3879:
3878:
3862:
3859:
3858:
3849:
3848:
3824:
3814:
3810:
3806:
3805:
3796:
3792:
3790:
3787:
3786:
3776:Cauchy sequence
3772:
3749:
3746:
3745:
3729:
3726:
3725:
3700:
3697:
3696:
3675:
3672:
3671:
3648:
3633:
3629:
3625:
3621:
3617:
3616:
3607:
3603:
3595:
3592:
3591:
3563:
3560:
3559:
3543:
3540:
3539:
3516:
3506:
3502:
3498:
3497:
3488:
3484:
3482:
3479:
3478:
3475:axiom of choice
3447:
3444:
3443:
3426:
3422:
3420:
3417:
3416:
3400:
3397:
3396:
3379:
3375:
3373:
3370:
3369:
3345:
3342:
3341:
3335:
3334:
3327:
3326:
3309:
3306:
3305:
3288:
3284:
3282:
3279:
3278:
3275:
3251:
3247:
3245:
3242:
3241:
3225:
3222:
3221:
3204:
3200:
3198:
3195:
3194:
3172:
3169:
3168:
3141:
3138:
3137:
3106:
3103:
3102:
3077:
3074:
3073:
3057:
3054:
3053:
3022:
3019:
3018:
2975:
2972:
2971:
2949:
2946:
2945:
2907:
2904:
2903:
2875:
2859:
2855:
2846:
2842:
2840:
2837:
2836:
2820:
2817:
2816:
2791:
2788:
2787:
2759:
2756:
2755:
2738:
2734:
2732:
2729:
2728:
2722:
2721:
2699:
2695:
2693:
2690:
2689:
2666:
2656:
2652:
2648:
2647:
2638:
2634:
2632:
2629:
2628:
2605:
2595:
2591:
2587:
2586:
2577:
2573:
2571:
2568:
2567:
2564:Stephen Willard
2556:
2550:
2544:
2493:
2489:
2488:
2484:
2472:
2468:
2466:
2463:
2462:
2440:
2437:
2436:
2412:
2408:
2404:
2395:
2391:
2389:
2386:
2385:
2369:
2366:
2365:
2348:
2344:
2342:
2339:
2338:
2317:
2313:
2309:
2300:
2296:
2294:
2291:
2290:
2271:
2268:
2267:
2245:
2242:
2241:
2222:
2219:
2218:
2199:
2196:
2195:
2179:
2176:
2175:
2165:
2164:
2141:
2138:
2137:
2111:
2107:
2105:
2102:
2101:
2079:
2076:
2075:
2053:
2050:
2049:
2027:
2024:
2023:
2007:
2004:
2003:
1998:
1997:
1990:
1989:
1965:
1955:
1951:
1947:
1946:
1937:
1933:
1931:
1928:
1927:
1924:
1902:
1899:
1898:
1874:
1870:
1865:
1862:
1861:
1838:
1834:
1829:
1826:
1825:
1822:Hausdorff space
1802:
1799:
1798:
1775:
1771:
1759:
1743:
1724:
1720:
1705:
1686:
1682:
1677:
1674:
1673:
1651:
1648:
1647:
1624:
1620:
1615:
1612:
1611:
1588:
1584:
1579:
1576:
1575:
1556:
1553:
1552:
1538:
1537:
1526:
1521:
1515:
1506:
1500:
1496:
1494:
1487:
1480:
1479:
1471:
1466:
1460:
1451:
1445:
1441:
1439:
1426:
1419:
1418:
1410:
1405:
1399:
1390:
1384:
1380:
1378:
1371:
1370:
1362:
1357:
1351:
1342:
1336:
1332:
1328:
1327:
1319:
1314:
1308:
1299:
1293:
1289:
1284:
1282:
1279:
1278:
1260:
1257:
1256:
1237:
1234:
1233:
1228:
1227:
1206:
1203:
1202:
1185:
1181:
1179:
1176:
1175:
1156:
1153:
1152:
1136:
1133:
1132:
1128:for every open
1109:
1106:
1105:
1088:
1084:
1082:
1079:
1078:
1072:
1071:
1064:
1063:
1040:
1037:
1036:
1010:
1006:
1004:
1001:
1000:
975:
972:
971:
949:
946:
945:
923:
920:
919:
903:
900:
899:
865:
855:
851:
847:
846:
837:
833:
831:
828:
827:
824:
797:
794:
793:
766:
756:
752:
748:
747:
745:
742:
741:
721:
718:
717:
701:
698:
697:
677:
674:
673:
672:. When the set
652:
648:
630:
626:
624:
621:
620:
604:
601:
600:
568:
564:
562:
559:
558:
531:
521:
517:
513:
512:
503:
499:
497:
494:
493:
477:
474:
473:
437:
434:
433:
411:
408:
407:
385:
382:
381:
359:
356:
355:
338:required to be
319:
316:
315:
290:
287:
286:
264:
261:
260:
238:
235:
234:
203:
200:
199:
173:
170:
169:
149:
146:
145:
138:
115:Herman L. Smith
107:
35:
28:
23:
22:
15:
12:
11:
5:
21718:
21708:
21707:
21693:
21692:
21678:
21655:
21641:
21625:
21612:
21594:
21588:
21568:
21562:
21546:
21532:
21512:
21490:
21466:
21452:
21439:
21425:
21409:
21392:
21389:
21386:
21385:
21376:
21367:
21358:
21349:
21337:
21325:
21311:
21266:
21254:
21247:
21227:
21220:
21197:
21174:
21170:Schechter 1996
21159:
21140:
21115:
21100:
21091:
21089:, p. 16n)
21087:Sundström 2010
21078:
21059:(2): 102–121.
21035:
21034:
21032:
21029:
21026:
21025:
21013:
21010:
21007:
21002:
20998:
20976:
20953:
20949:
20928:
20924:
20903:
20899:
20895:
20892:
20872:
20869:
20866:
20863:
20860:
20855:
20851:
20830:
20827:
20822:
20818:
20797:
20793:
20789:
20786:
20764:
20761:
20758:
20755:
20751:
20747:
20742:
20738:
20728:and satisfies
20716:
20712:
20709:
20705:
20701:
20698:
20677:
20671:
20667:
20664:
20661:
20654:
20650:
20647:
20644:
20641:
20638:
20617:
20613:
20609:
20605:
20602:
20582:
20579:
20576:
20573:
20570:
20567:
20564:
20561:
20558:
20555:
20549:
20545:
20542:
20537:
20532:
20528:
20524:
20502:
20499:
20496:
20493:
20490:
20487:
20484:
20481:
20478:
20475:
20472:
20469:
20466:
20463:
20460:
20457:
20451:
20447:
20444:
20439:
20434:
20430:
20426:
20408:
20389:
20385:
20362:
20358:
20337:
20334:
20329:
20325:
20321:
20316:
20313:
20310:
20305:
20300:
20296:
20292:
20270:
20267:
20264:
20261:
20239:
20235:
20231:
20228:
20225:
20220:
20217:
20214:
20211:
20207:
20203:
20200:
20197:
20194:
20190:
20186:
20183:
20178:
20174:
20169:
20147:
20143:
20140:
20137:
20134:
20114:
20111:
20087:
20084:
20081:
20078:
20075:
20072:
20069:
20066:
20045:
20041:
20038:
20035:
20032:
20012:
20009:
20006:
20003:
19983:
19980:
19975:
19971:
19949:
19928:
19925:
19922:
19919:
19916:
19912:
19908:
19905:
19902:
19899:
19896:
19876:
19873:
19869:
19865:
19859:
19855:
19852:
19848:
19844:
19841:
19838:
19833:
19829:
19808:
19804:
19800:
19797:
19777:
19774:
19769:
19765:
19742:
19737:
19732:
19729:
19712:
19711:
19709:
19706:
19705:
19704:
19695:
19689:
19683:
19677:
19671:
19664:
19661:
19642:
19639:
19636:
19633:
19611:
19608:
19588:
19585:
19565:
19545:
19525:
19522:
19519:
19497:
19494:
19491:
19488:
19485:
19482:
19479:
19459:
19456:
19453:
19450:
19447:
19444:
19441:
19421:
19418:
19415:
19412:
19409:
19406:
19403:
19383:
19363:
19343:
19321:
19318:
19298:
19279:
19276:
19256:
19236:
19233:
19230:
19227:
19224:
19221:
19216:
19213:
19210:
19206:
19183:
19180:
19160:
19157:
19154:
19151:
19131:
19128:
19125:
19122:
19119:
19116:
19113:
19093:
19090:
19087:
19084:
19081:
19078:
19075:
19055:
19035:
19013:
19010:
19007:
19004:
19001:
18998:
18978:
18975:
18955:
18952:
18949:
18946:
18943:
18923:
18903:
18883:
18880:
18877:
18874:
18871:
18856:
18853:
18841:
18838:
18818:
18815:
18795:
18775:
18755:
18752:
18749:
18729:
18726:
18706:
18703:
18700:
18697:
18677:
18674:
18671:
18668:
18665:
18662:
18659:
18656:
18653:
18650:
18647:
18644:
18641:
18621:
18618:
18615:
18612:
18609:
18606:
18603:
18583:
18580:
18577:
18574:
18571:
18568:
18565:
18545:
18525:
18505:
18502:
18499:
18496:
18493:
18490:
18487:
18484:
18481:
18458:
18438:
18419:
18416:
18396:
18376:
18373:
18370:
18367:
18364:
18361:
18356:
18353:
18350:
18346:
18325:
18322:
18319:
18299:
18279:
18259:
18256:
18236:
18233:
18230:
18227:
18207:
18204:
18201:
18198:
18195:
18192:
18189:
18186:
18183:
18180:
18177:
18174:
18171:
18168:
18148:
18145:
18142:
18139:
18136:
18133:
18130:
18110:
18107:
18104:
18101:
18098:
18095:
18092:
18072:
18052:
18032:
18029:
18026:
18023:
18020:
18017:
18014:
18011:
18008:
17986:
17983:
17980:
17977:
17974:
17971:
17951:
17948:
17945:
17942:
17939:
17936:
17933:
17913:
17910:
17890:
17870:
17850:
17847:
17844:
17841:
17838:
17835:
17832:
17829:
17826:
17823:
17820:
17817:
17814:
17811:
17791:
17788:
17785:
17765:
17745:
17742:
17739:
17736:
17733:
17730:
17727:
17707:
17704:
17701:
17675:
17670:
17665:
17662:
17642:
17639:
17636:
17616:
17613:
17610:
17607:
17604:
17582:
17577:
17574:
17571:
17566:
17562:
17556:
17552:
17548:
17545:
17541:
17537:
17532:
17528:
17522:
17518:
17514:
17511:
17507:
17486:
17482:
17461:
17458:
17454:
17448:
17444:
17440:
17437:
17433:
17412:
17409:
17406:
17403:
17400:
17397:
17394:
17374:
17371:
17366:
17362:
17333:
17330:
17327:
17324:
17321:
17318:
17315:
17285:
17282:
17279:
17274:
17269:
17263:
17259:
17255:
17252:
17248:
17244:
17240:
17235:
17231:
17225:
17221:
17217:
17214:
17210:
17206:
17186:
17182:
17161:
17158:
17154:
17148:
17144:
17140:
17137:
17133:
17129:
17109:
17106:
17103:
17100:
17097:
17077:
17074:
17069:
17065:
17042:
17039:
17036:
17031:
17026:
17022:
17018:
17013:
17008:
17004:
16983:
16980:
16977:
16953:
16925:
16922:
16919:
16916:
16913:
16901:
16898:
16881:
16878:
16865:
16860:
16856:
16852:
16851:lim sup
16849:
16844:
16840:
16836:
16835:lim sup
16833:
16830:
16825:
16821:
16817:
16812:
16808:
16804:
16801:
16800:lim sup
16779:
16774:
16770:
16764:
16761:
16758:
16754:
16748:
16745:
16742:
16738:
16734:
16729:
16725:
16719:
16716:
16713:
16709:
16703:
16700:
16697:
16693:
16689:
16684:
16680:
16676:
16675:lim sup
16656:
16651:
16648:
16645:
16640:
16635:
16631:
16627:
16611:limit inferior
16607:Limit superior
16603:
16600:
16562:
16558:
16554:
16549:
16545:
16539:
16534:
16529:
16525:
16502:
16498:
16490:
16487:
16484:
16480:
16474:
16469:
16465:
16459:
16437:
16411:
16407:
16385:
16380:
16376:
16372:
16366:
16362:
16351:
16335:
16331:
16327:
16322:
16318:
16297:
16277:
16253:
16250:
16228:
16224:
16201:
16198:
16195:
16190:
16185:
16181:
16177:
16172:
16169:
16147:
16143:
16120:
16116:
16112:
16108:
16103:
16099:
16095:
16089:
16085:
16062:
16058:
16054:
16049:
16045:
16024:
16021:
16018:
16015:
15995:
15992:
15989:
15977:
15974:
15960:
15956:
15935:
15932:
15929:
15926:
15923:
15903:
15881:
15877:
15856:
15853:
15850:
15847:
15844:
15841:
15837:
15831:
15827:
15823:
15818:
15814:
15809:
15786:
15781:
15776:
15771:
15767:
15763:
15758:
15754:
15732:
15727:
15723:
15719:
15713:
15709:
15687:
15682:
15678:
15674:
15668:
15664:
15643:
15640:
15635:
15631:
15610:
15607:
15602:
15598:
15577:
15574:
15571:
15568:
15565:
15562:
15542:
15539:
15536:
15533:
15530:
15510:
15507:
15504:
15501:
15498:
15495:
15492:
15489:
15486:
15483:
15480:
15477:
15474:
15471:
15468:
15465:
15462:
15459:
15456:
15453:
15450:
15447:
15444:
15441:
15438:
15415:
15411:
15408:
15403:
15398:
15394:
15390:
15385:
15380:
15376:
15354:
15350:
15345:
15341:
15337:
15332:
15328:
15307:
15304:
15301:
15298:
15276:
15272:
15250:
15245:
15241:
15237:
15231:
15227:
15204:
15200:
15194:
15172:
15150:
15146:
15125:
15120:
15116:
15093:
15089:
15066:
15063:
15060:
15055:
15050:
15045:
15041:
15037:
15031:
15027:
15022:
15003:
14994:
14989:
14985:
14981:
14975:
14971:
14951:
14948:
14945:
14942:
14920:
14916:
14910:
14888:
14866:
14862:
14841:
14836:
14832:
14824:
14821:
14818:
14814:
14808:
14803:
14800:
14797:
14792:
14787:
14783:
14779:
14774:
14771:
14749:
14744:
14740:
14735:
14729:
14725:
14705:
14694:
14689:
14685:
14681:
14675:
14671:
14650:
14645:
14641:
14637:
14632:
14628:
14622:
14617:
14612:
14608:
14585:
14581:
14575:
14570:
14567:
14564:
14559:
14555:
14533:
14528:
14524:
14520:
14514:
14510:
14485:
14480:
14476:
14472:
14469:
14466:
14462:
14457:
14453:
14449:
14443:
14439:
14416:
14412:
14389:
14385:
14362:
14359:
14356:
14351:
14346:
14341:
14337:
14333:
14327:
14323:
14318:
14297:
14286:
14281:
14277:
14273:
14267:
14263:
14242:
14239:
14236:
14233:
14213:
14191:
14187:
14181:
14157:
14154:
14151:
14146:
14141:
14137:
14133:
14128:
14123:
14119:
14090:
14086:
14082:
14079:
14075:
14070:
14067:
14064:
14059:
14054:
14050:
14046:
14041:
14038:
14033:
14029:
14025:
14022:
14018:
14013:
14009:
14003:
13998:
13994:
13989:
13985:
13981:
13980:
13958:
13954:
13933:
13930:
13927:
13924:
13898:
13894:
13888:
13885:
13882:
13878:
13874:
13869:
13865:
13859:
13832:
13829:
13826:
13821:
13816:
13812:
13808:
13787:
13784:
13772:
13769:
13741:
13738:
13716:
13711:
13708:
13688:
13668:
13646:
13642:
13621:
13618:
13615:
13612:
13588:
13566:
13538:
13516:
13512:
13491:
13467:
13464:
13442:
13438:
13417:
13414:
13393:
13372:
13367:
13363:
13359:
13338:
13335:
13326:be a point in
13313:
13309:
13288:
13283:
13279:
13275:
13272:
13252:
13249:
13229:
13219:if and only if
13207:
13204:
13201:
13179:
13175:
13154:
13151:
13140:neighbourhoods
13125:
13121:
13100:
13091:Given a point
13080:
13077:
13057:
13035:
13031:
13008:
13005:
13002:
12997:
12992:
12988:
12984:
12967:Main article:
12964:
12961:
12949:
12944:
12940:
12915:
12906:consisting of
12895:
12885:
12871:
12849:
12845:
12824:
12821:
12801:
12798:
12793:
12789:
12782:
12761:
12741:
12738:
12733:
12729:
12722:
12701:
12698:
12673:
12669:
12666:
12663:
12660:
12655:
12651:
12646:
12642:
12639:
12636:
12633:
12630:
12627:
12615:
12612:
12610:
12607:
12594:
12591:
12588:
12568:
12563:
12560:
12557:
12553:
12549:
12546:
12543:
12540:
12537:
12534:
12531:
12528:
12525:
12520:
12516:
12495:
12492:
12489:
12484:
12480:
12459:
12433:
12410:
12406:
12382:
12378:
12375:
12370:
12363:
12359:
12354:
12350:
12326:
12323:
12320:
12315:
12310:
12306:
12302:
12289:
12273:
12267:
12261:
12239:
12218:
12197:
12191:
12185:
12162:
12141:
12121:
12118:
12115:
12095:
12092:
12089:
12086:
12083:
12080:
12077:
12074:
12071:
12048:
12045:
12042:
12039:
12019:
12016:
12013:
12004:and satisfies
11993:
11973:
11970:
11967:
11964:
11961:
11958:
11955:
11952:
11932:
11929:
11926:
11923:
11920:
11917:
11897:
11894:
11891:
11888:
11885:
11865:
11862:
11842:
11839:
11836:
11833:
11830:
11827:
11824:
11821:
11818:
11798:
11795:
11792:
11769:
11763:
11741:
11737:
11716:
11706:
11692:
11688:
11667:
11647:
11644:
11641:
11633:
11627:
11621:
11617:
11613:
11592:
11586:
11580:
11558:
11538:
11535:
11532:
11529:
11525:
11521:
11517:
11496:
11476:
11473:
11470:
11467:
11464:
11461:
11458:
11455:
11452:
11449:
11429:
11409:
11406:
11403:
11400:
11397:
11394:
11390:
11386:
11383:
11363:
11336:
11330:
11304:
11298:
11294:
11291:
11287:
11283:
11280:
11277:
11274:
11271:
11251:
11230:
11221:
11217:
11214:
11210:
11186:
11182:
11178:
11174:
11171:
11148:
11142:
11117:
11116:
11104:
11101:
11098:
11095:
11075:
11055:
11052:
11043:converging to
11032:
11012:
11009:
10989:
10978:
10960:
10959:
10944:
10941:
10938:
10935:
10915:
10895:
10892:
10883:converging to
10872:
10852:
10849:
10829:
10818:
10802:
10778:
10758:
10738:
10714:
10694:
10674:
10671:
10668:
10648:
10645:
10625:
10622:
10619:
10614:
10610:
10589:
10586:
10583:
10562:
10558:
10555:
10535:
10515:
10495:
10492:
10472:
10469:
10449:
10429:
10426:
10421:
10417:
10411:
10404:
10384:
10381:
10359:
10355:
10334:
10331:
10328:
10325:
10304:
10300:
10297:
10277:
10257:
10224:
10220:
10194:
10166:
10163:
10143:
10140:
10137:
10134:
10131:
10128:
10125:
10122:
10118:
10097:
10073:
10069:
10048:
10028:
10008:
10005:
10000:
9996:
9992:
9987:
9983:
9957:
9935:
9915:
9912:
9879:
9865:
9844:
9840:
9837:
9832:
9828:
9824:
9820:
9816:
9813:
9808:
9804:
9800:
9797:
9794:
9791:
9788:
9783:
9779:
9774:
9769:
9748:
9726:
9723:
9720:
9715:
9710:
9706:
9702:
9690:
9689:associated net
9669:
9666:
9655:
9641:
9621:
9597:
9577:
9565:
9562:
9555:
9554:
9542:
9539:
9517:
9514:
9511:
9508:
9504:
9500:
9495:
9491:
9468:
9465:
9462:
9457:
9452:
9448:
9444:
9422:
9398:
9378:
9358:
9355:
9352:
9349:
9346:
9326:
9323:
9320:
9317:
9314:
9294:
9291:
9288:
9283:
9279:
9258:
9255:
9252:
9232:
9212:
9192:
9172:
9169:
9166:
9163:
9160:
9140:
9120:
9115:
9111:
9090:
9068:
9063:
9059:
9038:
9016:
9012:
8991:
8971:
8968:
8965:
8962:
8942:
8922:
8900:
8897:
8894:
8889:
8884:
8880:
8876:
8871:
8866:
8862:
8848:
8847:
8844:
8839:
8830:
8827:
8823:
8822:
8809:
8804:
8800:
8796:
8791:
8787:
8766:
8763:
8760:
8757:
8754:
8751:
8731:
8709:
8705:
8684:
8681:
8678:
8658:
8655:
8652:
8632:
8627:
8624:
8621:
8616:
8611:
8607:
8603:
8581:
8578:
8575:
8572:
8550:
8546:
8542:
8537:
8533:
8512:
8509:
8504:
8500:
8479:
8476:
8473:
8470:
8450:
8430:
8427:
8424:
8421:
8417:
8413:
8409:
8405:
8402:
8399:
8396:
8393:
8390:
8387:
8367:
8346:
8342:
8339:
8336:
8333:
8328:
8324:
8319:
8298:
8278:
8253:
8249:
8228:
8223:
8219:
8198:
8193:
8189:
8168:
8165:
8160:
8156:
8152:
8149:
8144:
8141:
8138:
8134:
8113:
8110:
8107:
8104:
8101:
8097:
8092:
8088:
8084:
8080:
8077:
8074:
8054:
8050:
8046:
8043:
8040:
8037:
8032:
8028:
8023:
8019:
8014:
8010:
7989:
7986:
7983:
7963:
7960:
7940:
7918:
7915:
7912:
7907:
7902:
7898:
7894:
7889:
7884:
7880:
7857:
7854:
7834:
7812:
7809:
7806:
7801:
7796:
7791:
7787:
7783:
7761:
7758:
7753:
7749:
7743:
7740:
7737:
7733:
7712:
7709:
7706:
7703:
7683:
7680:
7675:
7671:
7665:
7662:
7659:
7655:
7634:
7612:
7609:
7606:
7601:
7596:
7592:
7588:
7566:
7542:
7521:
7507:
7506:
7503:
7498:
7477:
7474:
7454:
7432:
7429:
7426:
7421:
7416:
7412:
7408:
7403:
7398:
7394:
7369:
7357:
7354:
7350:
7349:
7336:
7333:
7313:
7293:
7290:
7270:
7249:
7244:
7240:
7236:
7232:
7212:
7209:
7189:
7186:
7183:
7162:
7157:
7153:
7149:
7145:
7125:
7122:
7119:
7116:
7096:
7093:
7090:
7070:
7067:
7064:
7061:
7058:
7055:
7050:
7047:
7044:
7039:
7034:
7029:
7025:
7021:
7017:
7013:
7005:
6984:
6981:
6978:
6973:
6969:
6962:
6941:
6938:
6935:
6930:
6926:
6905:
6885:
6865:
6860:
6856:
6852:
6849:
6829:
6824:
6820:
6799:
6796:
6776:
6754:
6751:
6730:
6725:
6721:
6717:
6713:
6693:
6673:
6653:
6633:
6613:
6591:
6587:
6567:
6564:
6544:
6522:
6519:
6516:
6511:
6506:
6502:
6498:
6493:
6488:
6484:
6460:
6447:preorder is a
6432:
6412:
6409:
6406:
6403:
6383:
6380:
6377:
6374:
6371:
6368:
6365:
6345:
6342:
6322:
6319:
6300:
6297:
6277:
6274:
6271:
6268:
6248:
6225:
6222:
6202:
6182:
6179:
6176:
6173:
6170:
6167:
6162:
6159:
6156:
6151:
6146:
6141:
6137:
6133:
6129:
6125:
6117:
6097:
6094:
6091:
6086:
6082:
6075:
6052:
6049:
6046:
6041:
6036:
6032:
6028:
6023:
6018:
6014:
5993:
5973:
5950:
5947:
5944:
5941:
5938:
5935:
5930:
5927:
5924:
5919:
5914:
5909:
5905:
5901:
5897:
5893:
5885:
5864:
5861:
5841:
5838:
5835:
5832:
5812:
5809:
5806:
5803:
5800:
5797:
5775:
5772:
5769:
5764:
5759:
5754:
5750:
5746:
5742:
5738:
5716:
5713:
5710:
5707:
5685:
5681:
5660:
5657:
5637:
5634:
5631:
5628:
5608:
5605:
5581:
5561:
5541:
5521:
5518:
5515:
5512:
5507:
5504:
5500:
5496:
5493:
5474:
5471:
5451:
5448:
5445:
5442:
5439:
5419:
5399:
5396:
5393:
5388:
5384:
5377:
5354:
5351:
5348:
5343:
5338:
5334:
5330:
5325:
5320:
5316:
5295:
5292:
5272:
5251:
5237:
5236:
5233:
5228:
5207:
5187:
5184:
5164:
5161:
5158:
5155:
5152:
5148:
5143:
5139:
5135:
5131:
5111:
5091:
5088:
5083:
5079:
5058:
5055:
5052:
5049:
5046:
5026:
5023:
5003:
5000:
4997:
4994:
4991:
4987:
4982:
4978:
4974:
4970:
4965:
4945:
4925:
4922:
4917:
4913:
4906:
4883:
4880:
4877:
4872:
4867:
4863:
4859:
4854:
4849:
4845:
4824:
4800:
4797:
4794:
4791:
4788:
4776:
4773:
4756:
4753:
4733:
4713:
4710:
4707:
4687:
4684:
4664:
4661:
4658:
4638:
4635:
4632:
4614:
4611:
4598:
4576:
4573:
4570:
4565:
4560:
4556:
4552:
4530:
4527:
4522:
4518:
4512:
4509:
4506:
4502:
4481:
4478:
4475:
4455:
4431:
4428:
4425:
4403:
4400:
4397:
4394:
4374:
4371:
4351:
4348:
4343:
4339:
4329:
4309:
4306:
4303:
4283:
4280:
4275:
4271:
4261:is a net with
4248:
4245:
4242:
4237:
4232:
4228:
4224:
4219:
4214:
4210:
4189:
4169:
4149:
4129:
4109:
4106:
4103:
4091:
4088:
4079:
4076:
4069:
4057:
4028:
4024:
3999:
3996:
3975:
3969:
3965:
3961:
3956:
3952:
3947:
3927:
3924:
3921:
3918:
3915:
3912:
3892:
3889:
3886:
3866:
3853:
3833:
3830:
3827:
3822:
3817:
3813:
3809:
3804:
3799:
3795:
3780:uniform spaces
3771:
3768:
3756:
3753:
3733:
3713:
3710:
3707:
3704:
3682:
3679:
3657:
3654:
3651:
3646:
3641:
3636:
3632:
3628:
3624:
3620:
3615:
3610:
3606:
3602:
3599:
3579:
3576:
3573:
3570:
3567:
3547:
3525:
3522:
3519:
3514:
3509:
3505:
3501:
3496:
3491:
3487:
3460:
3457:
3454:
3451:
3429:
3425:
3404:
3382:
3378:
3358:
3355:
3352:
3349:
3339:
3331:
3313:
3291:
3287:
3274:
3271:
3259:
3254:
3250:
3229:
3207:
3203:
3182:
3179:
3176:
3154:
3151:
3148:
3145:
3125:
3122:
3119:
3116:
3113:
3110:
3090:
3087:
3084:
3081:
3061:
3051:
3035:
3032:
3029:
3026:
3006:
3003:
3000:
2997:
2994:
2991:
2988:
2985:
2982:
2979:
2959:
2956:
2953:
2943:
2939:
2923:
2920:
2917:
2914:
2911:
2891:
2888:
2885:
2882:
2871:
2868:
2865:
2862:
2858:
2854:
2849:
2845:
2824:
2804:
2801:
2798:
2795:
2775:
2772:
2769:
2766:
2763:
2741:
2737:
2726:
2723:Willard-subnet
2718:
2702:
2698:
2688:are nets then
2675:
2672:
2669:
2664:
2659:
2655:
2651:
2646:
2641:
2637:
2614:
2611:
2608:
2603:
2598:
2594:
2590:
2585:
2580:
2576:
2546:Main article:
2543:
2540:
2526:
2522:
2519:
2516:
2513:
2510:
2507:
2504:
2501:
2496:
2492:
2487:
2483:
2478:
2475:
2471:
2450:
2447:
2444:
2423:
2418:
2415:
2411:
2407:
2403:
2398:
2394:
2373:
2351:
2347:
2325:
2320:
2316:
2312:
2308:
2303:
2299:
2278:
2275:
2255:
2252:
2249:
2229:
2226:
2206:
2203:
2183:
2169:
2151:
2148:
2145:
2125:
2122:
2119:
2114:
2110:
2089:
2086:
2083:
2063:
2060:
2057:
2037:
2034:
2031:
2011:
2002:
1994:
1987:is said to be
1974:
1971:
1968:
1963:
1958:
1954:
1950:
1945:
1940:
1936:
1923:
1920:
1906:
1885:
1882:
1877:
1873:
1869:
1849:
1846:
1841:
1837:
1833:
1809:
1806:
1786:
1783:
1778:
1774:
1768:
1765:
1762:
1758:
1745: or
1735:
1732:
1727:
1723:
1719:
1707: or
1697:
1694:
1689:
1685:
1681:
1661:
1658:
1655:
1635:
1632:
1627:
1623:
1619:
1599:
1596:
1591:
1587:
1583:
1560:
1536:
1533:
1528: in
1523:
1520:
1517:
1513:
1510:
1508:
1503:
1499:
1495:
1490:
1486:
1482:
1481:
1478:
1473: in
1468:
1465:
1462:
1458:
1455:
1453:
1448:
1444:
1440:
1435:
1432:
1429:
1425:
1421:
1420:
1417:
1412: in
1407:
1404:
1401:
1397:
1394:
1392:
1387:
1383:
1379:
1376:
1373:
1372:
1369:
1364: in
1359:
1356:
1353:
1349:
1346:
1344:
1339:
1335:
1331:
1329:
1326:
1321: in
1316:
1313:
1310:
1306:
1303:
1301:
1296:
1292:
1288:
1286:
1276:
1264:
1252:
1241:
1223:
1222:
1210:
1188:
1184:
1163:
1160:
1140:
1113:
1091:
1087:
1076:
1068:
1050:
1047:
1044:
1024:
1021:
1018:
1013:
1009:
988:
985:
982:
979:
959:
956:
953:
933:
930:
927:
907:
897:
894:
890:
887:is said to be
874:
871:
868:
863:
858:
854:
850:
845:
840:
836:
823:
822:Limits of nets
820:
807:
804:
801:
775:
772:
769:
764:
759:
755:
751:
729:
726:
705:
681:
671:
655:
651:
647:
644:
641:
638:
633:
629:
608:
585:
582:
579:
576:
571:
567:
540:
537:
534:
529:
524:
520:
516:
511:
506:
502:
481:
447:
444:
441:
421:
418:
415:
395:
392:
389:
369:
366:
363:
344:partial orders
337:
323:
303:
300:
297:
294:
274:
271:
268:
248:
245:
242:
222:
219:
216:
213:
210:
207:
197:
193:
178:
153:
137:
134:
119:John L. Kelley
106:
103:
26:
18:Convergent net
9:
6:
4:
3:
2:
21717:
21706:
21703:
21702:
21700:
21689:
21685:
21681:
21675:
21671:
21667:
21666:Mineola, N.Y.
21663:
21662:
21656:
21652:
21648:
21644:
21638:
21634:
21630:
21626:
21615:
21613:9780080532998
21609:
21605:
21604:
21599:
21595:
21591:
21589:0-387-98431-3
21585:
21581:
21577:
21573:
21569:
21565:
21563:3-540-90125-6
21559:
21555:
21551:
21547:
21543:
21539:
21535:
21529:
21525:
21521:
21517:
21513:
21509:
21505:
21501:
21497:
21493:
21487:
21483:
21479:
21475:
21471:
21467:
21463:
21459:
21455:
21453:0-7923-2531-1
21449:
21445:
21440:
21436:
21432:
21428:
21422:
21418:
21414:
21410:
21405:
21400:
21395:
21394:
21380:
21371:
21362:
21353:
21347:, p. 76.
21346:
21341:
21334:
21329:
21320:
21318:
21316:
21307:
21301:
21287:on 2015-04-24
21283:
21276:
21270:
21263:
21258:
21250:
21248:9780852264447
21244:
21240:
21239:
21231:
21223:
21221:9780486131788
21217:
21213:
21212:
21204:
21202:
21195:, p. 77.
21194:
21189:
21187:
21185:
21183:
21181:
21179:
21171:
21166:
21164:
21157:, p. 75.
21156:
21151:
21149:
21147:
21145:
21137:
21132:
21130:
21128:
21126:
21124:
21122:
21120:
21112:
21107:
21105:
21095:
21088:
21082:
21074:
21070:
21066:
21062:
21058:
21054:
21050:
21046:
21040:
21036:
21011:
21008:
21005:
21000:
20996:
20951:
20947:
20939:the sequence
20926:
20901:
20893:
20890:
20867:
20861:
20858:
20853:
20849:
20828:
20825:
20820:
20816:
20795:
20787:
20784:
20759:
20753:
20749:
20745:
20740:
20736:
20710:
20696:
20675:
20669:
20665:
20662:
20659:
20652:
20648:
20642:
20636:
20603:
20600:
20577:
20574:
20571:
20568:
20565:
20562:
20559:
20553:
20543:
20540:
20535:
20530:
20526:
20522:
20497:
20494:
20491:
20488:
20485:
20482:
20479:
20476:
20473:
20470:
20467:
20464:
20461:
20455:
20445:
20442:
20437:
20432:
20428:
20424:
20415:The sequence
20412:
20405:
20387:
20383:
20360:
20356:
20335:
20332:
20327:
20323:
20319:
20314:
20311:
20308:
20303:
20298:
20294:
20290:
20268:
20265:
20262:
20259:
20237:
20233:
20229:
20226:
20223:
20215:
20209:
20205:
20201:
20195:
20188:
20184:
20181:
20176:
20172:
20167:
20138:
20135:
20132:
20112:
20109:
20101:
20082:
20076:
20070:
20064:
20036:
20033:
20030:
20010:
20007:
20004:
20001:
19981:
19978:
19973:
19969:
19947:
19923:
19920:
19917:
19914:
19906:
19903:
19897:
19894:
19874:
19863:
19853:
19850:
19842:
19836:
19831:
19827:
19806:
19798:
19795:
19775:
19772:
19767:
19763:
19740:
19730:
19727:
19717:
19713:
19699:
19696:
19693:
19690:
19687:
19684:
19681:
19678:
19675:
19672:
19670:
19667:
19666:
19660:
19658:
19653:
19640:
19637:
19634:
19631:
19622:
19609:
19606:
19586:
19583:
19563:
19543:
19523:
19520:
19517:
19508:
19495:
19492:
19489:
19483:
19477:
19454:
19451:
19448:
19442:
19439:
19416:
19413:
19410:
19404:
19401:
19381:
19361:
19341:
19332:
19319:
19316:
19296:
19277:
19274:
19254:
19234:
19225:
19219:
19214:
19208:
19194:
19181:
19178:
19155:
19149:
19126:
19123:
19120:
19114:
19111:
19088:
19085:
19082:
19076:
19073:
19053:
19033:
19024:
19011:
19005:
19002:
18999:
18976:
18973:
18950:
18947:
18944:
18921:
18901:
18878:
18875:
18872:
18862:
18852:
18839:
18836:
18816:
18813:
18793:
18773:
18753:
18750:
18747:
18727:
18724:
18701:
18695:
18672:
18669:
18666:
18660:
18657:
18651:
18648:
18645:
18639:
18616:
18607:
18604:
18601:
18578:
18569:
18566:
18563:
18543:
18523:
18503:
18494:
18485:
18482:
18479:
18470:
18456:
18436:
18417:
18414:
18394:
18374:
18365:
18359:
18354:
18348:
18323:
18320:
18317:
18297:
18290:converges in
18277:
18257:
18254:
18231:
18225:
18205:
18199:
18196:
18193:
18187:
18184:
18178:
18175:
18172:
18166:
18143:
18134:
18131:
18128:
18105:
18096:
18093:
18090:
18070:
18050:
18030:
18021:
18012:
18009:
18006:
17997:
17984:
17978:
17975:
17972:
17946:
17937:
17934:
17931:
17911:
17908:
17888:
17868:
17848:
17842:
17839:
17836:
17830:
17827:
17821:
17818:
17815:
17809:
17789:
17786:
17783:
17763:
17740:
17731:
17728:
17725:
17705:
17702:
17699:
17691:
17673:
17663:
17660:
17640:
17637:
17634:
17611:
17608:
17605:
17593:
17580:
17575:
17572:
17569:
17564:
17554:
17550:
17546:
17543:
17535:
17530:
17520:
17516:
17512:
17509:
17484:
17459:
17446:
17442:
17438:
17435:
17404:
17398:
17395:
17372:
17364:
17360:
17351:
17347:
17325:
17319:
17316:
17305:
17304:plain English
17301:
17283:
17280:
17277:
17272:
17267:
17261:
17257:
17253:
17250:
17246:
17242:
17238:
17233:
17229:
17223:
17219:
17215:
17212:
17208:
17204:
17184:
17159:
17152:
17146:
17142:
17138:
17135:
17131:
17127:
17104:
17101:
17098:
17075:
17067:
17063:
17040:
17037:
17034:
17029:
17024:
17020:
17016:
17011:
17006:
17002:
16981:
16978:
16975:
16967:
16951:
16943:
16939:
16920:
16917:
16914:
16900:Metric spaces
16897:
16895:
16891:
16887:
16877:
16863:
16858:
16854:
16847:
16842:
16838:
16831:
16823:
16819:
16815:
16810:
16806:
16790:
16777:
16772:
16768:
16762:
16759:
16756:
16746:
16743:
16740:
16732:
16727:
16723:
16717:
16714:
16711:
16701:
16698:
16695:
16687:
16682:
16678:
16654:
16649:
16646:
16643:
16638:
16633:
16629:
16625:
16614:
16612:
16608:
16599:
16597:
16593:
16589:
16585:
16580:
16578:
16560:
16556:
16547:
16543:
16537:
16532:
16527:
16523:
16500:
16496:
16488:
16485:
16482:
16472:
16467:
16463:
16457:
16435:
16427:
16409:
16405:
16383:
16378:
16374:
16370:
16364:
16360:
16349:
16333:
16329:
16325:
16320:
16316:
16295:
16275:
16267:
16264:However, the
16251:
16248:
16226:
16222:
16199:
16196:
16193:
16188:
16183:
16179:
16175:
16170:
16167:
16145:
16141:
16118:
16114:
16106:
16101:
16097:
16093:
16087:
16083:
16060:
16056:
16052:
16047:
16043:
16022:
16019:
16016:
16013:
15993:
15990:
15987:
15973:
15958:
15954:
15930:
15927:
15924:
15901:
15879:
15875:
15851:
15848:
15845:
15839:
15835:
15829:
15825:
15821:
15816:
15812:
15807:
15784:
15774:
15769:
15765:
15761:
15756:
15752:
15730:
15725:
15721:
15717:
15711:
15707:
15685:
15680:
15676:
15672:
15666:
15662:
15641:
15638:
15633:
15629:
15608:
15605:
15600:
15596:
15575:
15569:
15566:
15563:
15537:
15534:
15531:
15508:
15505:
15499:
15496:
15493:
15487:
15481:
15478:
15475:
15469:
15463:
15460:
15457:
15451:
15445:
15442:
15439:
15409:
15406:
15401:
15396:
15392:
15388:
15383:
15378:
15374:
15348:
15343:
15339:
15335:
15330:
15326:
15305:
15302:
15299:
15296:
15274:
15270:
15248:
15243:
15239:
15235:
15229:
15225:
15202:
15198:
15192:
15170:
15148:
15144:
15123:
15118:
15114:
15091:
15087:
15079:converges to
15064:
15061:
15058:
15053:
15048:
15043:
15039:
15035:
15029:
15025:
15020:
15001:
14992:
14987:
14983:
14979:
14973:
14969:
14949:
14946:
14943:
14940:
14918:
14914:
14908:
14886:
14879:converges to
14864:
14860:
14839:
14834:
14830:
14822:
14819:
14816:
14806:
14801:
14798:
14795:
14790:
14785:
14781:
14777:
14772:
14769:
14760:
14747:
14742:
14738:
14733:
14727:
14723:
14703:
14692:
14687:
14683:
14679:
14673:
14669:
14648:
14643:
14639:
14630:
14626:
14620:
14615:
14610:
14606:
14583:
14579:
14573:
14565:
14562:
14557:
14553:
14531:
14526:
14522:
14518:
14512:
14508:
14499:
14483:
14478:
14474:
14467:
14464:
14460:
14455:
14451:
14447:
14441:
14437:
14414:
14410:
14387:
14383:
14360:
14357:
14354:
14349:
14344:
14339:
14335:
14331:
14325:
14321:
14316:
14295:
14284:
14279:
14275:
14271:
14265:
14261:
14240:
14237:
14234:
14231:
14211:
14189:
14185:
14179:
14155:
14152:
14149:
14144:
14139:
14135:
14131:
14126:
14121:
14117:
14107:
14088:
14084:
14068:
14065:
14062:
14057:
14052:
14048:
14044:
14031:
14027:
14011:
14007:
14001:
13992:
13987:
13983:
13956:
13952:
13931:
13928:
13925:
13922:
13914:
13896:
13892:
13886:
13883:
13880:
13876:
13872:
13867:
13863:
13857:
13848:
13830:
13827:
13824:
13819:
13814:
13810:
13806:
13795:
13793:
13792:product space
13790:A net in the
13783:
13770:
13767:
13759:
13756:(and so also
13755:
13739:
13736:
13709:
13706:
13686:
13679:converges to
13666:
13644:
13640:
13619:
13616:
13613:
13610:
13602:
13586:
13555:
13550:
13536:
13514:
13510:
13489:
13481:
13465:
13462:
13440:
13436:
13415:
13412:
13391:
13384:is a net. As
13370:
13365:
13361:
13357:
13336:
13333:
13311:
13307:
13286:
13281:
13277:
13273:
13270:
13250:
13247:
13227:
13220:
13205:
13202:
13199:
13177:
13173:
13152:
13149:
13141:
13123:
13119:
13098:
13078:
13075:
13055:
13033:
13029:
13006:
13003:
13000:
12995:
12990:
12986:
12982:
12970:
12960:
12947:
12942:
12938:
12929:
12913:
12893:
12883:
12869:
12847:
12843:
12822:
12819:
12799:
12791:
12787:
12759:
12739:
12731:
12727:
12699:
12696:
12688:
12671:
12667:
12664:
12661:
12658:
12653:
12649:
12644:
12640:
12634:
12628:
12625:
12606:
12592:
12589:
12586:
12561:
12558:
12555:
12551:
12547:
12544:
12541:
12538:
12535:
12532:
12523:
12518:
12514:
12493:
12487:
12482:
12478:
12457:
12431:
12408:
12404:
12376:
12373:
12368:
12361:
12357:
12352:
12348:
12324:
12321:
12318:
12313:
12308:
12304:
12300:
12287:
12284:
12271:
12237:
12195:
12139:
12119:
12113:
12093:
12084:
12081:
12078:
12072:
12069:
12062:
12046:
12043:
12040:
12037:
12017:
12014:
12011:
11991:
11968:
11965:
11962:
11953:
11950:
11930:
11927:
11924:
11921:
11918:
11915:
11892:
11889:
11886:
11863:
11860:
11837:
11831:
11828:
11822:
11816:
11796:
11793:
11790:
11739:
11714:
11704:
11690:
11665:
11645:
11642:
11639:
11619:
11615:
11590:
11556:
11533:
11519:
11494:
11471:
11468:
11462:
11456:
11453:
11450:
11427:
11404:
11401:
11398:
11384:
11381:
11361:
11353:
11318:
11302:
11292:
11289:
11278:
11272:
11249:
11215:
11212:
11172:
11169:
11129:
11128:in behavior.
11127:
11126:directed sets
11122:
11099:
11093:
11073:
11053:
11050:
11030:
11010:
11007:
10987:
10977:
10976:
10975:
10973:
10972:metric spaces
10969:
10965:
10957:
10939:
10933:
10913:
10893:
10890:
10870:
10850:
10847:
10827:
10819:
10816:
10800:
10792:
10791:
10790:
10776:
10756:
10736:
10726:
10712:
10692:
10672:
10669:
10666:
10659:Thus a point
10646:
10643:
10623:
10620:
10617:
10612:
10608:
10587:
10584:
10581:
10556:
10553:
10533:
10513:
10493:
10490:
10470:
10467:
10447:
10427:
10419:
10415:
10409:
10382:
10379:
10357:
10353:
10332:
10329:
10326:
10323:
10298:
10295:
10275:
10255:
10247:
10243:
10238:
10222:
10218:
10209:
10183:
10180:
10164:
10161:
10138:
10135:
10132:
10129:
10126:
10120:
10095:
10087:
10071:
10046:
10026:
10006:
10003:
9998:
9994:
9990:
9985:
9981:
9972:
9955:
9925:
9921:
9911:
9909:
9905:
9900:
9898:
9894:
9890:
9885:
9881:
9877:
9863:
9842:
9838:
9835:
9830:
9826:
9822:
9818:
9814:
9811:
9806:
9802:
9798:
9795:
9792:
9789:
9786:
9781:
9777:
9772:
9767:
9746:
9724:
9721:
9718:
9713:
9708:
9704:
9700:
9688:
9686:
9681:
9675:
9665:
9663:
9662:partial order
9659:
9653:
9639:
9619:
9611:
9595:
9575:
9561:
9553:
9540:
9537:
9530:converges to
9512:
9506:
9502:
9498:
9493:
9489:
9466:
9463:
9460:
9455:
9450:
9446:
9442:
9420:
9412:
9411:product order
9396:
9376:
9353:
9350:
9347:
9324:
9318:
9315:
9312:
9292:
9289:
9286:
9281:
9277:
9269:is such that
9256:
9253:
9250:
9230:
9210:
9190:
9167:
9164:
9161:
9138:
9118:
9113:
9109:
9088:
9079:
9066:
9061:
9057:
9036:
9014:
9010:
8989:
8969:
8966:
8963:
8960:
8940:
8920:
8898:
8895:
8892:
8887:
8882:
8878:
8874:
8869:
8864:
8860:
8850:
8849:
8843:
8842:
8838:
8836:
8821:
8807:
8802:
8798:
8794:
8789:
8785:
8777:we have that
8764:
8758:
8752:
8749:
8729:
8707:
8703:
8682:
8679:
8676:
8669:there exists
8656:
8653:
8650:
8630:
8625:
8622:
8619:
8614:
8609:
8605:
8601:
8579:
8576:
8573:
8570:
8548:
8544:
8540:
8535:
8531:
8510:
8507:
8502:
8498:
8477:
8474:
8471:
8468:
8448:
8441:Observe that
8428:
8419:
8411:
8403:
8400:
8397:
8394:
8388:
8385:
8365:
8344:
8340:
8337:
8334:
8331:
8326:
8322:
8317:
8296:
8267:
8251:
8247:
8226:
8221:
8217:
8196:
8191:
8187:
8163:
8158:
8154:
8150:
8147:
8142:
8139:
8136:
8132:
8108:
8105:
8102:
8099:
8095:
8090:
8086:
8082:
8078:
8075:
8052:
8048:
8044:
8041:
8038:
8035:
8030:
8026:
8021:
8017:
8012:
8008:
7987:
7984:
7981:
7961:
7958:
7938:
7916:
7913:
7910:
7905:
7900:
7896:
7892:
7887:
7882:
7878:
7868:
7855:
7852:
7832:
7810:
7807:
7804:
7799:
7794:
7789:
7785:
7781:
7756:
7751:
7747:
7741:
7738:
7735:
7731:
7710:
7707:
7704:
7701:
7678:
7673:
7669:
7663:
7660:
7657:
7653:
7632:
7610:
7607:
7604:
7599:
7594:
7590:
7586:
7564:
7556:
7540:
7509:
7508:
7502:
7501:
7497:
7495:
7491:
7475:
7472:
7452:
7430:
7427:
7424:
7419:
7414:
7410:
7406:
7401:
7396:
7392:
7383:
7367:
7348:
7334:
7331:
7311:
7291:
7288:
7268:
7261:not being in
7247:
7242:
7238:
7234:
7230:
7210:
7207:
7187:
7184:
7181:
7160:
7155:
7151:
7147:
7143:
7120:
7114:
7094:
7091:
7088:
7068:
7062:
7056:
7048:
7045:
7042:
7037:
7032:
7027:
7023:
7019:
7015:
7011:
6982:
6979:
6971:
6967:
6939:
6936:
6933:
6928:
6924:
6903:
6883:
6863:
6858:
6854:
6850:
6847:
6827:
6822:
6818:
6797:
6794:
6774:
6765:
6752:
6749:
6728:
6723:
6719:
6715:
6711:
6691:
6671:
6651:
6631:
6611:
6589:
6585:
6565:
6562:
6542:
6520:
6517:
6514:
6509:
6504:
6500:
6496:
6491:
6486:
6482:
6472:
6458:
6450:
6446:
6430:
6410:
6407:
6404:
6401:
6381:
6378:
6375:
6369:
6363:
6343:
6340:
6320:
6317:
6298:
6295:
6272:
6266:
6246:
6239:
6223:
6220:
6200:
6180:
6174:
6168:
6160:
6157:
6154:
6149:
6144:
6139:
6135:
6131:
6127:
6123:
6095:
6092:
6084:
6080:
6050:
6047:
6044:
6039:
6034:
6030:
6026:
6021:
6016:
6012:
5991:
5962:
5948:
5942:
5936:
5928:
5925:
5922:
5917:
5912:
5907:
5903:
5899:
5895:
5891:
5862:
5859:
5836:
5830:
5807:
5804:
5801:
5795:
5773:
5770:
5767:
5762:
5757:
5752:
5748:
5744:
5740:
5736:
5714:
5711:
5708:
5705:
5683:
5679:
5658:
5655:
5635:
5632:
5629:
5626:
5606:
5603:
5595:
5579:
5559:
5539:
5519:
5513:
5505:
5502:
5498:
5494:
5491:
5472:
5469:
5449:
5443:
5437:
5417:
5397:
5394:
5386:
5382:
5352:
5349:
5346:
5341:
5336:
5332:
5328:
5323:
5318:
5314:
5293:
5290:
5270:
5239:
5238:
5232:
5231:
5227:
5225:
5221:
5205:
5185:
5182:
5159:
5153:
5146:
5141:
5137:
5133:
5129:
5109:
5089:
5081:
5077:
5056:
5050:
5047:
5044:
5024:
5021:
4998:
4992:
4985:
4980:
4976:
4972:
4968:
4943:
4923:
4915:
4911:
4881:
4878:
4875:
4870:
4865:
4861:
4857:
4852:
4847:
4843:
4822:
4814:
4798:
4792:
4789:
4786:
4772:
4770:
4754:
4751:
4731:
4711:
4708:
4705:
4698:Also, subset
4685:
4682:
4662:
4656:
4636:
4633:
4630:
4620:
4610:
4596:
4574:
4571:
4568:
4563:
4558:
4554:
4550:
4541:for some net
4528:
4520:
4516:
4510:
4507:
4504:
4479:
4476:
4473:
4453:
4445:
4429:
4426:
4423:
4414:
4401:
4398:
4395:
4392:
4372:
4369:
4349:
4341:
4337:
4307:
4304:
4301:
4281:
4278:
4273:
4269:
4246:
4243:
4240:
4235:
4230:
4226:
4222:
4217:
4212:
4208:
4187:
4167:
4147:
4127:
4120:is closed in
4107:
4104:
4101:
4087:
4085:
4075:
4073:
4067:
4065:
4061:
4056:
4053:
4051:
4046:
4044:
4043:Cauchy filter
4026:
4022:
4013:
3997:
3994:
3973:
3967:
3963:
3959:
3954:
3950:
3945:
3925:
3922:
3919:
3916:
3913:
3910:
3890:
3887:
3884:
3877:there exists
3864:
3857:
3854:if for every
3847:
3831:
3828:
3825:
3820:
3815:
3811:
3807:
3802:
3797:
3793:
3783:
3781:
3777:
3767:
3754:
3751:
3731:
3711:
3708:
3705:
3702:
3693:
3680:
3677:
3655:
3652:
3649:
3644:
3639:
3634:
3630:
3626:
3622:
3618:
3613:
3608:
3604:
3600:
3597:
3577:
3571:
3568:
3565:
3545:
3523:
3520:
3517:
3512:
3507:
3503:
3499:
3494:
3489:
3485:
3476:
3471:
3458:
3455:
3449:
3427:
3423:
3402:
3380:
3376:
3356:
3353:
3350:
3347:
3333:
3328:universal net
3325:
3311:
3289:
3285:
3270:
3257:
3252:
3248:
3227:
3205:
3201:
3180:
3177:
3174:
3165:
3152:
3149:
3146:
3143:
3120:
3114:
3111:
3108:
3088:
3085:
3082:
3079:
3059:
3050:
3047:
3030:
3024:
3004:
2998:
2992:
2989:
2983:
2977:
2957:
2954:
2951:
2941:
2938:
2935:
2921:
2915:
2912:
2909:
2889:
2886:
2883:
2880:
2866:
2860:
2856:
2852:
2847:
2843:
2822:
2799:
2793:
2773:
2767:
2764:
2761:
2739:
2735:
2720:
2716:
2700:
2696:
2673:
2670:
2667:
2662:
2657:
2653:
2649:
2644:
2639:
2635:
2612:
2609:
2606:
2601:
2596:
2592:
2588:
2583:
2578:
2574:
2565:
2561:
2555:
2549:
2539:
2524:
2520:
2517:
2514:
2511:
2508:
2505:
2502:
2499:
2494:
2490:
2485:
2481:
2476:
2473:
2469:
2448:
2445:
2442:
2421:
2416:
2413:
2409:
2405:
2401:
2396:
2392:
2371:
2349:
2345:
2323:
2318:
2314:
2310:
2306:
2301:
2297:
2276:
2273:
2253:
2250:
2247:
2227:
2224:
2204:
2201:
2181:
2173:
2172:cluster point
2163:
2149:
2146:
2143:
2123:
2120:
2117:
2112:
2108:
2087:
2084:
2081:
2061:
2058:
2055:
2035:
2032:
2029:
2022:if for every
2009:
1996:
1988:
1972:
1969:
1966:
1961:
1956:
1952:
1948:
1943:
1938:
1934:
1919:
1917:
1904:
1883:
1875:
1871:
1847:
1844:
1839:
1835:
1823:
1807:
1784:
1781:
1776:
1772:
1766:
1763:
1760:
1733:
1730:
1725:
1721:
1695:
1692:
1687:
1683:
1659:
1656:
1653:
1633:
1625:
1621:
1597:
1589:
1585:
1572:
1558:
1534:
1531:
1518:
1509:
1501:
1497:
1488:
1476:
1463:
1454:
1446:
1442:
1433:
1430:
1427:
1415:
1402:
1393:
1385:
1381:
1367:
1354:
1345:
1337:
1333:
1324:
1311:
1302:
1294:
1290:
1262:
1254:
1239:
1226:
1208:
1186:
1182:
1161:
1158:
1138:
1131:
1127:
1126:
1125:
1111:
1089:
1085:
1070:
1062:
1048:
1045:
1042:
1022:
1019:
1016:
1011:
1007:
986:
983:
980:
977:
957:
954:
951:
931:
928:
925:
905:
895:
892:
888:
872:
869:
866:
861:
856:
852:
848:
843:
838:
834:
819:
805:
802:
799:
791:
773:
770:
767:
762:
757:
753:
749:
727:
724:
703:
695:
679:
669:
653:
649:
645:
639:
631:
627:
606:
599:
583:
577:
574:
569:
565:
556:
538:
535:
532:
527:
522:
518:
514:
509:
504:
500:
479:
472:
467:
465:
461:
445:
442:
439:
419:
416:
413:
393:
390:
387:
367:
364:
361:
353:
349:
345:
341:
335:
321:
301:
298:
295:
292:
272:
269:
266:
246:
243:
240:
220:
217:
214:
211:
208:
205:
195:
191:
176:
167:
151:
143:
133:
131:
127:
122:
120:
116:
112:
102:
100:
96:
92:
88:
84:
80:
76:
72:
68:
64:
60:
56:
52:
48:
44:
40:
33:
19:
21660:
21632:
21617:. Retrieved
21602:
21575:
21556:. Springer.
21553:
21519:
21480:. New York:
21473:
21443:
21416:
21379:
21370:
21361:
21352:
21345:Willard 2004
21340:
21333:Willard 2004
21328:
21289:. Retrieved
21282:the original
21269:
21257:
21237:
21230:
21210:
21193:Willard 2004
21155:Willard 2004
21136:Willard 2004
21094:
21081:
21056:
21052:
21049:Smith, H. L.
21045:Moore, E. H.
21039:
20411:
19716:
19654:
19623:
19509:
19333:
19195:
19025:
18858:
18471:
17998:
17594:
17346:normed space
17300:real numbers
17298:is a net of
16938:metric space
16903:
16890:Riemann sums
16883:
16791:
16615:
16605:
16581:
15979:
15914:centered at
15262:clusters at
15163:clusters at
14761:
14204:directed by
14170:be a net in
14108:
13796:
13789:
13551:
12972:
12617:
12285:
12106:(defined by
12061:identity map
11130:
11118:
10970:, including
10961:
10727:
10239:
10208:directed set
9971:archetypical
9917:
9908:ultrafilters
9901:
9882:
9677:
9567:
9559:
9080:
8851:
8832:
8268:
7951:directed by
7931:be a net in
7869:
7510:
7359:
6916:; therefore
6766:
6473:
6449:directed set
6394:necessarily
6238:neighborhood
5963:
5592:). Thus the
5240:
4778:
4622:
4415:
4160:of a net in
4093:
4081:
4064:Banach space
4060:normed space
4047:
4012:Cauchy space
3784:
3773:
3694:
3472:
3324:is called a
3276:
3166:
2944:if whenever
2715:is called a
2557:
2384:is equal to
2171:
1999:cofinally in
1925:
1897:
1573:
1224:
1130:neighborhood
1061:is called a
825:
693:
557:of the form
470:
468:
466:is maximal.
463:
459:
340:total orders
142:directed set
139:
130:Henri Cartan
123:
108:
75:metric space
59:directed set
50:
46:
36:
21404:1006.4131v1
21111:Kelley 1975
20629:defined by
20057:by letting
18859:Consider a
18270:Such a net
18159:satisfying
13482:of a point
13428:the points
13142:containing
12618:If the set
11984:belongs to
10059:defined on
9687:induces an
9685:filter base
9481:defined by
7356:Compactness
6445:containment
4815:at a point
4779:A function
3770:Cauchy nets
1232:to/towards
1077:of the net
1065:limit point
136:Definitions
111:E. H. Moore
39:mathematics
21391:References
21374:Beer, p. 2
21291:2013-01-15
21262:Howes 1995
20883:for every
19788:for every
19470:such that
19142:the point
18688:such that
18218:the point
16616:For a net
16075:such that
14500:: the net
12132:) into an
10600:such that
10345:the point
9672:See also:
9654:equivalent
8695:such that
8523:such that
7974:For every
7645:such that
7281:for every
6840:For every
6742:is not in
6471:as well).
6065:such that
5727:Therefore
5222:(or not a
4813:continuous
4775:Continuity
4769:topologies
4617:See also:
3850:Cauchy net
3136:such that
2934:is called
2786:such that
2552:See also:
2074:such that
1991:frequently
1275:as a limit
1124:whenever:
999:the point
893:residually
889:eventually
492:, denoted
259:such that
21651:175294365
21031:Citations
21001:∙
20952:∙
20894:∈
20788:∈
20612:→
20578:…
20544:∈
20498:…
20446:∈
20388:∙
20361:∙
20336:φ
20333:∘
20328:∙
20312:∈
20263:∈
20210:φ
20185:φ
20182:∘
20177:∙
20142:→
20133:φ
20086:⌉
20080:⌈
20065:φ
20040:→
20031:φ
20005:∈
19994:for each
19948:≤
19907:∈
19872:→
19854:∈
19832:∙
19799:∈
19655:See also
19638:ω
19521:∈
19490:∈
19443:∈
19405:∈
19232:→
19212:→
19115:∈
19077:∈
18751:∈
18658:≤
18611:∖
18605:∈
18573:∖
18567:∈
18501:→
18489:∖
18372:→
18352:→
18321:∈
18185:≤
18138:∖
18132:∈
18100:∖
18094:∈
18028:→
18016:∖
17979:≤
17941:∖
17828:≤
17787:≤
17735:∖
17688:with the
17653:(such as
17638:∈
17573:∈
17547:−
17521:∙
17513:−
17457:→
17447:∙
17439:−
17408:‖
17405:⋅
17402:‖
17370:→
17365:∙
17329:‖
17326:⋅
17323:‖
17281:∈
17224:∙
17157:→
17147:∙
17073:→
17068:∙
17038:∈
17007:∙
16979:∈
16832:≤
16760:⪰
16744:∈
16715:⪰
16699:∈
16647:∈
16553:→
16548:∙
16538:∏
16524:π
16486:∈
16479:∏
16468:∙
16458:∏
16379:∙
16361:π
16326:∈
16227:∙
16197:∈
16111:→
16102:∙
16084:π
16053:∈
16017:∈
15991:∈
15959:∙
15880:∙
15762:×
15726:∙
15708:π
15681:∙
15663:π
15509:…
15410:∈
15379:∙
15300:∈
15244:∙
15226:π
15203:∙
15193:∏
15149:∙
15062:∈
15026:π
14988:∙
14970:π
14944:∈
14919:∙
14909:∏
14865:∙
14820:∈
14813:∏
14807:∈
14799:∈
14743:∙
14734:∘
14724:π
14688:∙
14670:π
14661:that is,
14636:→
14631:∙
14621:∏
14607:π
14584:∙
14574:∏
14569:→
14558:∙
14527:∙
14509:π
14471:→
14456:∙
14438:π
14411:π
14388:∙
14358:∈
14322:π
14280:∙
14262:π
14235:∈
14190:∙
14180:∏
14153:∈
14122:∙
14078:↦
14066:∈
14021:→
14012:∙
14002:∏
13984:π
13926:∈
13911:with the
13884:∈
13877:∏
13868:∙
13858:∏
13828:∈
13710:∈
13645:∙
13614:∈
13413:≥
13274:∈
13203:≥
13004:∈
12943:∙
12848:∙
12797:→
12792:∙
12737:→
12732:∙
12665:∈
12641:∪
12559:−
12536:∈
12377:∈
12322:∈
12117:↦
12091:→
12085:≥
12041:≥
12015:≥
11925:∈
11893:≥
11829:≥
11794:≥
11640:
11616:∈
11603:that is,
11528:→
11507:function
11393:→
11293:∈
11216:∈
11209:∏
11181:→
10670:∈
10618:∈
10585:≥
10557:∈
10425:→
10327:≥
10299:∈
10179:countable
10139:…
10007:…
9956:≤
9836:∈
9812:≤
9793:∈
9722:∈
9464:∈
9322:→
9287:∈
9254:∈
9114:∙
9062:∙
9015:∙
8964:∈
8896:∈
8865:∙
8795:∉
8753:⊇
8680:∈
8654:∈
8623:∈
8574:∈
8541:∉
8508:∈
8472:∈
8423:∞
8398:⊂
8389:≜
8338:∈
8277:⟸
8252:∙
8222:∙
8192:∙
8167:∅
8164:≠
8151:
8140:∈
8133:⋂
8106:∈
8079:
8042:≥
8018:≜
7985:∈
7914:∈
7883:∙
7808:∈
7760:∅
7757:≠
7739:∈
7732:⋂
7705:⊆
7682:∅
7679:≠
7661:∈
7654:⋂
7608:∈
7520:⟹
7428:∈
7397:∙
7185:
7136:and thus
7092:
7054:→
7046:∈
6977:→
6972:∙
6934:∈
6851:≥
6518:∈
6487:∙
6443:with the
6405:∈
6376:∈
6166:→
6158:∈
6090:→
6085:∙
6048:∈
6017:∙
5972:⟸
5934:→
5926:∈
5805:
5771:∈
5709:
5684:∙
5630:
5503:−
5392:→
5387:∙
5350:∈
5319:∙
5250:⟹
5218:is not a
5151:→
5142:∙
5087:→
5082:∙
5054:→
4990:→
4981:∙
4921:→
4916:∙
4879:∈
4848:∙
4796:→
4709:⊆
4660:∖
4634:⊆
4623:A subset
4572:∈
4526:→
4521:∙
4508:∈
4477:∈
4427:⊆
4396:∈
4347:→
4342:∙
4305:∈
4279:∈
4244:∈
4213:∙
4105:⊆
4094:A subset
4027:∙
3920:≥
3888:∈
3856:entourage
3829:∈
3798:∙
3706:∈
3653:∈
3609:∙
3601:∘
3575:→
3521:∈
3490:∙
3453:∖
3428:∙
3381:∙
3351:⊆
3290:∙
3273:Ultranets
3253:∙
3206:∙
3178:∈
3147:≥
3112:∈
3083:∈
2990:≤
2955:≤
2919:→
2884:∈
2771:→
2740:∙
2701:∙
2671:∈
2640:∙
2610:∈
2579:∙
2518:∈
2506:≥
2474:≥
2446:∈
2435:for each
2414:≥
2402:
2350:∙
2319:∙
2307:
2251:∈
2240:In fact,
2147:∈
2118:∈
2085:≥
2059:∈
2033:∈
1970:∈
1939:∙
1881:→
1876:∙
1840:∙
1805:→
1764:∈
1688:∙
1646:only for
1631:→
1626:∙
1595:→
1590:∙
1512:→
1457:→
1431:∈
1396:→
1386:∙
1348:→
1305:→
1295:∙
1229:converges
1187:∙
1090:∙
1046:∈
1017:∈
981:≥
955:∈
929:∈
870:∈
839:∙
803:∈
771:∈
725:≤
632:∙
581:→
570:∙
536:∈
505:∙
391:≤
365:≤
296:≤
270:≤
244:∈
215:∈
177:≤
21699:Category
21631:(1996).
21600:(1997).
21574:(1998).
21552:(1991).
21518:(1975).
21508:1272666M
21500:31969970
21300:cite web
20777:for all
20676:⌋
20653:⌊
20125:The map
19961:and let
19819:so that
19755:and let
19686:Preorder
19663:See also
19510:A point
19334:The net
18472:The net
17561:‖
17540:‖
17527:‖
17506:‖
17453:‖
17432:‖
16904:Suppose
15366:and let
14852:the net
13552:Given a
12926:and the
12882:depends
12609:Examples
12506:and let
11853:for all
11705:sequence
10793:The map
10086:sequence
9904:analysts
9889:analysis
9658:preorder
9337:mapping
9305:The map
8563:for all
7360:A space
6356:Because
5594:interior
5306:and let
5122:implies
4956:implies
4294:for all
4072:normable
4055:complete
4014:, a net
3336:ultranet
3017:The set
2902:The map
2461:, where
2289:The set
2136:A point
1174:the net
1035:A point
763:⟩
750:⟨
555:function
342:or even
196:directed
166:preorder
83:topology
79:analysis
71:sequence
63:codomain
55:function
21619:22 June
21462:1269778
21435:2378491
21073:2370388
20100:ceiling
20098:be the
20023:Define
17352:) then
16348:is the
13554:subbase
8835:subnets
8000:define
7557:). Let
7382:compact
5984:) Let
5263:) Let
4444:closure
3304:in set
3049:cofinal
2940:and an
2542:Subnets
670:indices
553:, is a
350:and/or
105:History
99:filters
21688:115240
21686:
21676:
21649:
21639:
21610:
21586:
21560:
21542:338047
21540:
21530:
21506:
21498:
21488:
21460:
21450:
21433:
21423:
21245:
21218:
21071:
20988:while
19171:is in
18717:is in
18247:is in
17999:A net
17497:where
17348:(or a
17302:. In
17197:where
16944:) and
16940:(or a
16596:subnet
16350:unique
15980:If no
14720:
14697:
14312:
14289:
13632:a net
12884:solely
12396:where
11354:. Let
10966:. All
10372:is in
9680:filter
9183:where
4320:, and
3785:A net
3695:Given
3332:or an
3277:A net
3046:being
2717:subnet
1926:A net
1752:
1749:
1741:
1738:
1714:
1711:
1703:
1700:
898:a set
826:A net
598:domain
596:whose
471:net in
192:upward
126:filter
61:. The
21399:arXiv
21285:(PDF)
21278:(PDF)
21069:JSTOR
19708:Notes
18934:from
18632:with
17692:with
17344:is a
16968:. If
16936:is a
16428:. If
16424:is a
15588:Then
15217:then
14402:into
14253:let
13349:Then
13165:Then
12928:image
12712:then
10184:set (
10154:into
9608:is a
9029:then
8845:Proof
8239:Thus
7723:Then
7504:Proof
6952:Thus
5875:Thus
5234:Proof
4492:with
4385:then
3846:is a
3220:then
2970:then
2835:and
1820:In a
1073:limit
970:with
73:in a
53:is a
21684:OCLC
21674:ISBN
21647:OCLC
21637:ISBN
21621:2013
21608:ISBN
21584:ISBN
21558:ISBN
21538:OCLC
21528:ISBN
21496:OCLC
21486:ISBN
21448:ISBN
21421:ISBN
21306:link
21243:ISBN
21216:ISBN
20841:and
19921:>
18469:).
16667:put
16609:and
16575:are
15799:but
15699:and
15621:and
15553:and
14109:Let
13601:base
13299:let
13263:For
12590:>
12548:>
12030:and
10769:and
10244:and
9409:the
9243:and
9131:Let
8852:Let
8420:<
7870:Let
7492:and
7081:But
3558:and
2627:and
2560:1970
2100:and
1860:and
1255:has
443:<
432:and
417:<
380:and
285:and
113:and
93:and
81:and
21061:doi
20102:of
19576:of
19374:of
19267:of
19205:lim
19196:So
19046:of
18806:of
18536:of
18407:of
18345:lim
17595:If
17472:in
17385:in
17172:in
17088:in
16753:sup
16737:inf
16708:sup
16692:lim
16241:in
16133:in
15744:in
15183:in
15106:in
15009:def
14711:def
14303:def
13971:by
13729:of
13659:in
12812:in
12781:lim
12752:in
12721:lim
12527:inf
12491:inf
12288:not
12250:in
12174:in
11957:min
11707:in
11569:in
11000:in
10840:in
10813:is
10705:of
10526:of
10460:of
10403:lim
10395:So
10268:of
9739:in
9660:or
9369:to
9223:in
8982:If
7445:in
7380:is
7182:int
7089:int
7004:lim
6961:lim
6787:of
6259:of
6116:lim
6074:lim
5884:lim
5802:int
5706:int
5627:int
5596:of
5572:at
5430:of
5376:lim
5226:).
5175:in
5102:in
5014:in
4964:lim
4936:in
4905:lim
4609:.
4589:in
4501:lim
4446:of
4362:in
4328:lim
3415:or
3167:If
3052:in
2727:of
2719:or
2562:by
2364:in
2194:of
2170:or
1995:or
1868:lim
1832:lim
1757:lim
1718:lim
1680:lim
1618:lim
1582:lim
1574:If
1551:If
1485:lim
1424:lim
1375:lim
1253:or
1151:of
1104:in
1069:or
891:or
694:net
462:or
336:not
49:or
47:net
37:In
21701::
21682:.
21672:.
21668::
21664:.
21645:.
21578:.
21536:.
21522:.
21504:OL
21502:.
21494:.
21476:.
21458:MR
21456:.
21431:MR
21429:.
21314:^
21302:}}
21298:{{
21200:^
21177:^
21162:^
21143:^
21118:^
21103:^
21067:.
21057:44
21055:.
21047:;
20649::=
20554::=
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17935::=
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16598:.
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12524::=
12488::=
12436:th
12070:Id
11954::=
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9899:.
9678:A
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8076:cl
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18640:d
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17581:.
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