7731:
5298:
7514:
7752:
7720:
5105:
31:
7789:
7762:
7742:
5293:{\displaystyle {\begin{alignedat}{4}\operatorname {SeqInt} _{X}A:&=\{a\in A~:~{\text{ whenever a sequence in }}X{\text{ converges to }}a{\text{ in }}(X,\tau ),{\text{ then that sequence is eventually in }}A\}\\&=\{a\in A~:~{\text{ there does NOT exist a sequence in }}X\setminus A{\text{ that converges in }}(X,\tau ){\text{ to a point in }}A\}\\\end{alignedat}}}
6693:
Moreover, a subset is a regular open set if and only if it is preopen and semi-closed. The intersection of an α-open set and a semi-preopen (resp. semi-open, preopen, b-open) set is a semi-preopen (resp. semi-open, preopen, b-open) set. Preopen sets need not be semi-open and semi-open sets need not
4389:
3904:
4502:
5869:
6697:
Arbitrary unions of preopen (resp. α-open, b-open, semi-preopen) sets are once again preopen (resp. α-open, b-open, semi-preopen). However, finite intersections of preopen sets need not be preopen. The set of all α-open subsets of a space
6610:
4006:
3064:
5794:
6534:
199:
In practice, however, open sets are usually chosen to provide a notion of nearness that is similar to that of metric spaces, without having a notion of distance defined. In particular, a topology allows defining properties such as
4278:
2735:
1408:
A set might be open, closed, both, or neither. In particular, open and closed sets are not mutually exclusive, meaning that it is in general possible for a subset of a topological space to simultaneously be both an open subset
6457:
6160:
2157:
2682:
4804:
4649:
1930:
2338:
1058:
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3007:
1253:
1395:
A metric space is a topological space, whose topology consists of the collection of all subsets that are unions of open balls. There are, however, topological spaces that are not metric spaces.
1107:
6395:
2438:
1201:
5414:
5990:
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426:
as the only such set for "measuring distance", all points are close to 0 since there is only one possible degree of accuracy one may achieve in approximating 0: being a member of
5569:
4883:
1988:
1957:
5593:
2814:
6645:
4040:
2269:
430:. Thus, we find that in some sense, every real number is distance 0 away from 0. It may help in this case to think of the measure as being a binary condition: all things in
6851:
if and only if every regular closed subset is preopen or equivalently, if every semi-open subset is preopen. Moreover, the space is totally disconnected if and only if the
6674:
5954:
5799:
6293:
5626:
5543:
5335:
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4212:
3784:
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1655:
1579:
935:
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3456:
1553:
1471:
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3191:
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3583:
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1817:
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6255:
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3479:
804:
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1390:
1279:
1149:
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902:
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2008:
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529:
6469:
4384:{\displaystyle A~\subseteq ~\operatorname {int} _{X}\left(\operatorname {cl} _{X}A\right)~\cup ~\operatorname {cl} _{X}\left(\operatorname {int} _{X}A\right)}
2687:
6400:
7792:
414:. This innovative idea has far-reaching consequences; in particular, by defining different collections of sets containing 0 (distinct from the sets (−
6111:
2095:
1206:
2640:
227:, but on which no distance is defined in general. Less intuitive topologies are used in other branches of mathematics; for example, the
1697:
itself are always clopen. These two sets are the most well-known examples of clopen subsets and they show that clopen subsets exist in
3066:
Every regular open set (resp. regular closed set) is an open subset (resp. is a closed subset) although in general, the converses are
4743:
4606:
1063:
1866:
447:; a member of this neighborhood basis is referred to as an open set. In fact, one may generalize these notions to an arbitrary set (
2301:
1008:
3899:{\displaystyle A~\subseteq ~\operatorname {int} _{X}\left(\operatorname {cl} _{X}\left(\operatorname {int} _{X}A\right)\right)}
267:. Therefore, topological spaces may be seen as a generalization of spaces equipped with a notion of distance, which are called
4497:{\displaystyle A~\subseteq ~\operatorname {cl} _{X}\left(\operatorname {int} _{X}\left(\operatorname {cl} _{X}A\right)\right)}
422:)), one may find different results regarding the distance between 0 and other real numbers. For example, if we were to define
7362:
7332:
7234:
2971:
111:
491:
to a greater degree of accuracy. Bearing this in mind, one may define the remaining axioms that the family of sets about
503:
Several definitions are given here, in an increasing order of technicality. Each one is a special case of the next one.
7426:
360:
7780:
7775:
7305:
7121:
7089:
7045:
6361:
2411:
843:
This generalizes the
Euclidean space example, since Euclidean space with the Euclidean distance is a metric space.
1154:
7770:
5386:
1288:
Infinite intersections of open sets need not be open. For example, the intersection of all intervals of the form
5959:
6892:
3235:
4508:
7672:
7397:
410:= 0. Thus, rather than speaking of a concrete Euclidean metric, one may use sets to describe points close to
256:
3121:
and other topological structures that deal with the notions of closeness and convergence for spaces such as
2499:
1584:
1291:
2766:
940:
706:
180:. These conditions are very loose, and allow enormous flexibility in the choice of open sets. For example,
17:
2740:
7392:
3143:
contains a (possibly empty) open set; the maximum (ordered under inclusion) such open set is called the
255:, there exists an open set not containing the other (distinct) point, the two points are referred to as
7680:
5548:
4855:
467:) for otherwise we may not have a well-defined method to measure distance. For example, every point in
5864:{\displaystyle \operatorname {cl} _{X}A=\operatorname {cl} _{X}\left(\operatorname {int} _{X}A\right)}
1966:
1935:
7813:
5578:
3083:
2792:
165:
6618:
4013:
2250:
2247:
is a clopen subset. For a more advanced example reminiscent of the discrete topology, suppose that
7479:
6751:
6650:
5930:
6272:
5605:
5522:
5314:
4923:
4191:
3763:
2542:
1844:
1822:
1754:
1724:
1660:
1500:
7765:
7751:
3485:
that is open in the first topology might fail to be open in the second topology. For example, if
3090:
5019:
4888:
4134:
2473:
1625:
1558:
914:
7700:
7621:
7498:
7486:
7459:
7419:
7350:
6701:
3728:
3429:
3159:
1526:
1444:
7695:
7387:
7229:, Graduate Texts in Mathematics, vol. 2 (2nd ed.), Springer-Verlag, pp. 19–21,
4990:
3289:
3164:
7542:
7469:
7224:
7077:
7033:
6231:) if it is equal to the set of all of its θ-cluster points (resp. δ-cluster points). A point
5675:
3562:
3532:
2443:
2057:
1784:
7109:
6757:
6234:
6045:
5419:
4829:
3599:
3461:
783:
7690:
7642:
7616:
7464:
7250:
6848:
6605:{\displaystyle \operatorname {int} _{X}A~\subseteq ~\operatorname {int} _{X}B~\subseteq ~B}
5596:
4001:{\displaystyle A~\subseteq ~\operatorname {int} _{X}\left(\operatorname {cl} _{X}A\right).}
3720:
3405:
3144:
3059:{\displaystyle \operatorname {Bd} \left(\operatorname {Int} S\right)=\operatorname {Bd} S.}
1369:
1264:
1134:
1114:
988:
968:
887:
859:
852:
264:
177:
130:
5789:{\displaystyle A~\subseteq ~\operatorname {cl} _{X}\left(\operatorname {int} _{X}A\right)}
8:
7537:
6898:
6529:{\displaystyle A~\subseteq ~\operatorname {cl} _{X}A~\subseteq ~\operatorname {cl} _{X}B}
3377:
has the characteristic property that it is a countable union of disjoint open intervals.
3325:
2821:
2817:
1741:
are both open, and that they are also closed, since each is the complement of the other.
201:
193:
87:
7741:
6980:
6907: – A function that sends open (resp. closed) subsets to open (resp. closed) subsets
6338:
5080:
4720:
4697:
4541:
4237:
4111:
4065:
3351:
3266:
2562:
2278:
2159:
are open (because they cannot be written as a union of open intervals); this means that
2016:
677:
7735:
7705:
7685:
7606:
7596:
7474:
7454:
7354:
6960:
6936:
6854:
6830:
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6782:
6733:
6318:
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6091:
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6015:
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5701:
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5054:
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2210:
2186:
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1993:
1745:
1704:
1680:
1480:
1424:
1349:
653:
627:
607:
593:
557:
514:
441:
In general, one refers to the family of sets containing 0, used to approximate 0, as a
232:
161:
61:
3643:. This is because when the surrounding space is the rational numbers, for every point
7730:
7723:
7589:
7547:
7412:
7368:
7358:
7328:
7311:
7301:
7230:
7117:
7085:
7041:
7038:
A Mathematical Gift: The
Interplay Between Topology, Functions, Geometry, and Algebra
6954:
4666:
3395:
3118:
2875:
2204:
1282:
252:
185:
173:
153:
126:
7755:
2730:{\displaystyle \operatorname {Bd} \left({\overline {S}}\right)=\operatorname {Bd} S}
1701:
topological space. To see, it suffices to remark that, by definition of a topology,
311:, one can speak of the set of all points close to that real number; that is, within
7503:
7449:
7105:
6452:{\displaystyle B\cap \operatorname {int} _{X}\left(\operatorname {cl} _{X}U\right)}
3671:. On the other hand, when the surrounding space is the reals, then for every point
3093:
of an open set (relative to the space that the topology is defined on) is called a
2632:
279:
248:
228:
1841:
by definition of the
Euclidean topology. It is not closed since its complement in
455:) of that set, one may define a collection of sets "around" (that is, containing)
172:, and the whole set itself. A set in which such a collection is given is called a
7562:
7557:
6874:
6868:
6800:
5508:
3716:
3636:
3501:
can be given its own topology (called the 'subspace topology') defined by "a set
2869:
532:
224:
205:
7745:
3385:
463:. Of course, this collection would have to satisfy certain properties (known as
7652:
7584:
3151:. It can be constructed by taking the union of all the open sets contained in
3126:
133:
defined between every two points), an open set is a set that, with every point
107:
363:(−1, 1); that is, the set of all real numbers between −1 and 1. However, with
7807:
7662:
7572:
7552:
7342:
7315:
6804:
3122:
1774:
259:. In this manner, one may speak of whether two points, or more generally two
236:
137:
in it, contains all points of the metric space that are sufficiently near to
7372:
6871: – Map that satisfies a condition similar to that of being an open map.
483:
should be in this family. Once we begin to define "smaller" sets containing
282:; that is, a function which measures the distance between two real numbers:
7647:
7567:
7513:
4105:
3930:
3640:
750:
575:
375:
are precisely the points of (−0.5, 0.5). Clearly, these points approximate
268:
122:
7657:
2272:
275:
209:
99:
7601:
7532:
7491:
6886:
6880:
5572:
3105:
and the full space are examples of sets that are both open and closed.
3098:
3094:
3082:
of any number of open sets, or infinitely many open sets, is open. The
1416:
160:
of a given set, a collection that has the property of containing every
91:
7626:
6155:{\displaystyle x\in U\subseteq \operatorname {sCl} _{X}U\subseteq A.}
3374:
3102:
1749:
671:
169:
115:
6680:
Every α-open subset is semi-open, semi-preopen, preopen, and b-open.
2408:
an open subset or else a closed subset, but never both; that is, if
2152:{\displaystyle \mathbb {R} \setminus K=(-\infty ,0)\cup [1,\infty )}
215:
The most common case of a topology without any distance is given by
7611:
7579:
7435:
6904:
3391:
3320:
3240:
3114:
451:); rather than just the real numbers. In this case, given a point (
216:
3458:", despite the fact that all the topological data is contained in
2677:{\displaystyle \operatorname {Int} \left({\overline {S}}\right)=S}
1403:
263:, of a topological space are "near" without concretely defining a
192:
subset can be open except the space itself and the empty set (the
30:
6913:
339:
becomes smaller and smaller, one obtains points that approximate
7323:
Hart, Klaas Pieter; Nagata, Jun-iti; Vaughan, Jerry E. (2004).
7084:. The Sally Series. American Mathematical Society. p. 29.
4799:{\displaystyle x_{\bullet }=\left(x_{i}\right)_{i=1}^{\infty }}
4644:{\displaystyle U\subseteq A\subseteq \operatorname {cl} _{X}U.}
3133:
260:
157:
35:
1925:{\displaystyle I^{\complement }=(-\infty ,0]\cup [1,\infty ),}
247:
Intuitively, an open set provides a method to distinguish two
6977:
is both a regular open subset and a regular closed subset of
398:
to be smaller and smaller. In particular, sets of the form (−
4671:
if it satisfies any of the following equivalent conditions:
4411:
if it satisfies any of the following equivalent conditions:
3935:
if it satisfies any of the following equivalent conditions:
7404:
343:
to a higher and higher degree of accuracy. For example, if
7040:. Vol. 3. American Mathematical Society. p. 38.
3596:
is defined as the set of rational numbers in the interval
2333:{\displaystyle \tau :={\mathcal {U}}\cup \{\varnothing \}}
1053:{\displaystyle \left\{U_{i}:i\in I\right\}\subseteq \tau }
6877: – Collection of open sets used to define a topology
1932:
which is not open; indeed, an open interval contained in
6901: – Mathematical function revertible near each point
212:, which were originally defined by means of a distance.
7013:
6909:
Pages displaying short descriptions of redirect targets
2900:
is a regular open set if and only if its complement in
7206:
7204:
7202:
7200:
7198:
7196:
7194:
7192:
7190:
6916: – Collection of subsets that generate a topology
3117:. The concept is required to define and make sense of
2920:
is a regular closed set, where by definition a subset
434:
are equally close to 0, while any item that is not in
7142:
6983:
6963:
6939:
6833:
6813:
6785:
6760:
6736:
6704:
6653:
6621:
6548:
6472:
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6301:
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6114:
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3432:
3408:
3354:
3334:
3292:
3269:
3249:
3219:
3199:
3167:
3015:
3002:{\displaystyle {\overline {\operatorname {Int} S}}=S}
2974:
2946:
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2906:
2886:
2854:
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2019:
1996:
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991:
971:
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862:
786:
680:
656:
630:
610:
560:
517:
406:) give us a lot of information about points close to
394:
to higher and higher degrees of accuracy by defining
7275:
7273:
7271:
7269:
7267:
7265:
7223:
Oxtoby, John C. (1980), "4. The
Property of Baire",
7177:
7175:
7173:
7171:
7169:
7154:
7130:
7054:
6895: – Connected open subset of a topological space
5305:
The complement of a sequentially open set is called
2470:
of the following two statements is true: either (1)
1248:{\displaystyle U_{1}\cap \cdots \cap U_{n}\in \tau }
7187:
3481:If there are two topologies on the same set, a set
3386:"Open" is defined relative to a particular topology
2010:
is an example of a set that is open but not closed.
152:More generally, an open set is a member of a given
6992:
6969:
6945:
6839:
6819:
6791:
6769:
6742:
6722:
6668:
6639:
6604:
6528:
6451:
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6350:
6327:
6307:
6287:
6249:
6211:
6187:
6154:
6100:
6080:
6060:
6024:
6004:
5992:is the intersection of all semi-closed subsets of
5984:
5948:
5919:
5883:
5863:
5788:
5710:
5690:
5664:
5644:
5620:
5587:
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2151:
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2002:
1982:
1951:
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523:
86:. The red set is an open set, the blue set is its
7322:
7262:
7257:, Academic Press and Polish Scientific Publishers
7166:
7071:
7069:
3521:." This potentially introduces new open sets: if
2539:subsets that are both (i.e. that are clopen) are
90:set, and the union of the red and blue sets is a
7805:
2868:. A topological space for which there exists a
6689:Every semi-open set is b-open and semi-preopen.
6683:Every b-open set is semi-preopen (i.e. β-open).
5201: then that sequence is eventually in
3710:
3517:with an open set from the original topology on
1404:Clopen sets and non-open and/or non-closed sets
1102:{\displaystyle \bigcup _{i\in I}U_{i}\in \tau }
251:. For example, if about one of two points in a
7066:
6390:{\displaystyle B\cap \operatorname {cl} _{X}U}
5243: there does NOT exist a sequence in
2433:{\displaystyle \varnothing \neq S\subsetneq X}
7420:
6686:Every preopen set is b-open and semi-preopen.
3906:, and the complement of such a set is called
1398:
7300:. Amsterdam Boston: Elsevier/North-Holland.
6883: – Subset which is both open and closed
5283:
5220:
5207:
5139:
2872:consisting of regular open sets is called a
2327:
2321:
1990:cannot be a union of open intervals. Hence,
1379:
1373:
1196:{\displaystyle U_{1},\ldots ,U_{n}\in \tau }
386:The previous discussion shows, for the case
5409:{\displaystyle \operatorname {SeqCl} _{X}S}
4391:. The complement of a b-open set is called
3113:Open sets have a fundamental importance in
1413:a closed subset. Such subsets are known as
7788:
7761:
7427:
7413:
7249:
5985:{\displaystyle \operatorname {sCl} _{X}A,}
5630:the Baire property in the restricted sense
4260:The complement of a preopen set is called
2051:is a closed subset but not an open subset.
884:with the properties below. Each member of
379:to a greater degree of accuracy than when
4654:The complement of a β-open set is called
3086:of a finite number of open sets is open.
2100:
1849:
1827:
1759:
7114:Essentials of Topology With Applications
6199:) set, where by definition, a subset of
4531:{\displaystyle \operatorname {cl} _{X}A}
2207:(so that by definition, every subset of
29:
7341:
7160:
7148:
7136:
7060:
7019:
3555:isn't open in the original topology on
3097:. A set may be both open and closed (a
141:(that is, all points whose distance to
14:
7806:
7222:
7216:
7104:
7075:
3394:under consideration. Having opted for
2816:denote, respectively, the topological
2524:{\displaystyle X\setminus S\in \tau .}
1609:{\displaystyle X\setminus S\in \tau .}
1339:{\displaystyle \left(-1/n,1/n\right),}
223:each point, resemble an open set of a
7408:
5442:for which there exists a sequence in
3426:" rather than "the topological space
3390:Whether a set is open depends on the
3380:
2782:{\displaystyle \operatorname {Int} S}
956:{\displaystyle \varnothing \in \tau }
219:, which are topological spaces that,
145:is less than some value depending on
7295:
7279:
7210:
7181:
7031:
6889: – Complement of an open subset
4967:(that is, there exists some integer
4717:then that sequence is eventually in
3585:is open in the subspace topology on
3525:is open in the original topology on
3505:is open in the subspace topology on
3396:greater brevity over greater clarity
2756:{\displaystyle \operatorname {Bd} S}
2584:
2013:By a similar argument, the interval
1392:which is not open in the real line.
846:
5698:has the Baire property relative to
3707:contains no non-rational numbers).
1777:and every union of open intervals.
1111:Any finite intersection of sets in
27:Basic subset of a topological space
24:
6068:there exists some semiopen subset
5162: whenever a sequence in
4791:
3592:As a concrete example of this, if
2535:subset is open or closed but the
2313:
2256:
2143:
2119:
1913:
1889:
1366:is a positive integer, is the set
506:
25:
7825:
7380:
7032:Ueno, Kenji; et al. (2005).
6859:of every preopen subset is open.
5564:{\displaystyle A\bigtriangleup U}
5383:, which by definition is the set
5249:
4878:{\displaystyle x_{\bullet }\to x}
3651:, there exists a positive number
2506:
2324:
2104:
1591:
1507:
319:. In essence, points within ε of
307:. Therefore, given a real number
176:, and the collection is called a
7787:
7760:
7750:
7740:
7729:
7719:
7718:
7512:
7325:Encyclopedia of general topology
7298:Encyclopedia of general topology
6957:, in which case every subset of
1983:{\displaystyle I^{\complement }}
1952:{\displaystyle I^{\complement }}
836:has a neighborhood contained in
359:are precisely the points of the
7289:
7243:
6295:if for every open neighborhood
5891:of a semi-open set is called a
5588:{\displaystyle \bigtriangleup }
5519:if there exists an open subset
5100:which by definition is the set
4740:Explicitly, this means that if
3193:between two topological spaces
2809:{\displaystyle {\overline {S}}}
740:
7098:
7025:
6927:
6893:Domain (mathematical analysis)
6717:
6705:
6676:the following may be deduced:
6640:{\displaystyle A,B\subseteq X}
5272:
5260:
5193:
5181:
4904:
4892:
4869:
4563:There exists a preopen subset
4538:is a regular closed subset of
4035:{\displaystyle D,U\subseteq X}
3757:will be a topological space.
3744:
3732:
3615:
3603:
3445:
3433:
3302:
3177:
2264:{\displaystyle {\mathcal {U}}}
2227:is open) then every subset of
2146:
2134:
2128:
2113:
2079:
2067:
2038:
2026:
1916:
1904:
1898:
1883:
1806:
1794:
1641:
1629:
1542:
1530:
1460:
1448:
498:
390:= 0, that one may approximate
64:represents the set of points (
38:represents the set of points (
13:
1:
7006:
6669:{\displaystyle A\subseteq B,}
5949:{\displaystyle A\subseteq X,}
5257: that converges in
3489:is any topological space and
3073:
776:, there exists a real number
257:topologically distinguishable
242:
164:of its members, every finite
7434:
6288:{\displaystyle B\subseteq X}
5621:{\displaystyle A\subseteq X}
5538:{\displaystyle U\subseteq X}
5330:{\displaystyle S\subseteq X}
4940:{\displaystyle x_{\bullet }}
4207:{\displaystyle U\subseteq X}
3779:{\displaystyle A\subseteq X}
3711:Generalizations of open sets
2988:
2801:
2706:
2659:
2552:{\displaystyle \varnothing }
1856:{\displaystyle \mathbb {R} }
1834:{\displaystyle \mathbb {R} }
1766:{\displaystyle \mathbb {R} }
1734:{\displaystyle \varnothing }
1670:{\displaystyle \varnothing }
1516:{\displaystyle X\setminus S}
7:
7393:Encyclopedia of Mathematics
7116:. CRC Press. pp. 3–4.
6862:
4694:converges to some point of
2179:is neither open nor closed.
1744:The open sets of the usual
780:> 0 such that any point
10:
7830:
7681:Banach fixed-point theorem
7076:Taylor, Joseph L. (2011).
5337:is sequentially closed in
5042:{\displaystyle x_{j}\in A}
4913:{\displaystyle (X,\tau ),}
4159:{\displaystyle A=U\cap D.}
3714:
3422:as "the topological space
2489:{\displaystyle S\in \tau }
1650:{\displaystyle (X,\tau ),}
1574:{\displaystyle S\in \tau }
1399:Special types of open sets
930:{\displaystyle X\in \tau }
832:is open if every point in
699:An example of a subset of
650:is open if every point in
231:, which is fundamental in
7714:
7671:
7635:
7521:
7510:
7442:
6723:{\displaystyle (X,\tau )}
5277: to a point in
4826:and if there exists some
4168:There exists an open (in
3750:{\displaystyle (X,\tau )}
3635:is an open subset of the
3451:{\displaystyle (X,\tau )}
1548:{\displaystyle (X,\tau )}
1466:{\displaystyle (X,\tau )}
624:. Equivalently, a subset
586:) such that any point in
487:, we tend to approximate
479:degree of accuracy. Thus
367:= 0.5, the points within
327:to an accuracy of degree
7034:"The birth of manifolds"
6933:One exception if the if
6920:
6541:
6537:
5170: converges to
5009:{\displaystyle j\geq i,}
3402:endowed with a topology
3311:{\displaystyle f:X\to Y}
3186:{\displaystyle f:X\to Y}
2364:non-empty proper subset
1421:. Explicitly, a subset
705:that is not open is the
495:is required to satisfy.
184:subset can be open (the
6175:) if its complement in
5691:{\displaystyle A\cap E}
4674:Whenever a sequence in
3786:of a topological space
3691:points within distance
3659:points within distance
3578:{\displaystyle V\cap Y}
3548:{\displaystyle V\cap Y}
3513:is the intersection of
3139:of a topological space
3108:
2609:of a topological space
2459:{\displaystyle S\neq X}
2360:with the property that
2183:If a topological space
2085:{\displaystyle K=[0,1)}
1812:{\displaystyle I=(0,1)}
1773:are the empty set, the
1441:of a topological space
880:is a set of subsets of
737:, no matter how small.
578:a positive real number
351:= 1, the points within
7736:Mathematics portal
7636:Metrics and properties
7622:Second-countable space
7351:Upper Saddle River, NJ
6994:
6971:
6947:
6841:
6827:is θ-closed. A space
6821:
6793:
6771:
6770:{\displaystyle \tau .}
6744:
6724:
6670:
6641:
6606:
6530:
6453:
6391:
6352:
6329:
6309:
6289:
6251:
6250:{\displaystyle x\in X}
6213:
6189:
6156:
6102:
6082:
6062:
6061:{\displaystyle x\in A}
6026:
6006:
5986:
5950:
5921:
5885:
5865:
5790:
5712:
5692:
5666:
5646:
5622:
5589:
5565:
5539:
5496:
5476:
5456:
5436:
5435:{\displaystyle x\in X}
5410:
5371:
5351:
5331:
5294:
5094:
5065:
5043:
5010:
4981:
4961:
4941:
4914:
4879:
4846:
4845:{\displaystyle a\in A}
4820:
4800:
4734:
4711:
4688:
4645:
4597:
4577:
4555:
4532:
4498:
4385:
4251:
4228:
4208:
4182:
4160:
4125:
4098:
4079:
4056:
4036:
4002:
3900:
3800:
3780:
3751:
3625:
3624:{\displaystyle (0,1),}
3579:
3549:
3475:
3474:{\displaystyle \tau .}
3452:
3416:
3365:
3342:
3312:
3280:
3257:
3227:
3207:
3187:
3060:
3003:
2954:
2934:
2914:
2894:
2862:
2842:
2810:
2783:
2757:
2731:
2678:
2623:
2603:
2576:
2553:
2525:
2490:
2460:
2434:
2398:
2378:
2354:
2334:
2292:
2265:
2241:
2221:
2197:
2173:
2153:
2086:
2045:
2004:
1984:
1963:, and it follows that
1953:
1926:
1857:
1835:
1813:
1767:
1735:
1715:
1691:
1671:
1651:
1610:
1575:
1555:; or equivalently, if
1549:
1517:
1491:
1467:
1435:
1386:
1360:
1340:
1275:
1249:
1197:
1145:
1125:
1103:
1054:
999:
979:
957:
931:
898:
870:
800:
799:{\displaystyle y\in M}
691:
664:
638:
618:
568:
525:
459:, used to approximate
278:, one has the natural
95:
7251:Kuratowski, Kazimierz
6995:
6972:
6948:
6842:
6822:
6803:if and only if every
6794:
6772:
6745:
6725:
6671:
6642:
6615:whenever two subsets
6607:
6531:
6463:Using the fact that
6454:
6392:
6353:
6330:
6310:
6290:
6252:
6214:
6195:is a θ-closed (resp.
6190:
6157:
6103:
6083:
6063:
6027:
6007:
5987:
5951:
5922:
5886:
5866:
5791:
5713:
5693:
5667:
5647:
5623:
5590:
5566:
5540:
5497:
5477:
5457:
5437:
5411:
5372:
5352:
5332:
5295:
5095:
5066:
5044:
5011:
4982:
4962:
4942:
4915:
4880:
4847:
4821:
4801:
4735:
4712:
4689:
4646:
4598:
4578:
4556:
4533:
4499:
4386:
4252:
4234:is a dense subset of
4229:
4209:
4183:
4161:
4126:
4099:
4080:
4057:
4037:
4010:There exists subsets
4003:
3901:
3801:
3781:
3752:
3626:
3580:
3550:
3476:
3453:
3417:
3415:{\displaystyle \tau }
3366:
3343:
3328:of every open set in
3313:
3281:
3258:
3243:of every open set in
3228:
3208:
3188:
3061:
3004:
2955:
2935:
2915:
2895:
2863:
2843:
2811:
2784:
2758:
2732:
2679:
2624:
2604:
2577:
2554:
2526:
2491:
2461:
2435:
2399:
2379:
2355:
2335:
2293:
2266:
2242:
2222:
2198:
2174:
2154:
2087:
2046:
2005:
1985:
1954:
1927:
1858:
1836:
1814:
1768:
1736:
1716:
1692:
1672:
1652:
1611:
1576:
1550:
1518:
1492:
1468:
1436:
1387:
1385:{\displaystyle \{0\}}
1361:
1341:
1276:
1274:{\displaystyle \tau }
1250:
1198:
1146:
1144:{\displaystyle \tau }
1126:
1124:{\displaystyle \tau }
1104:
1055:
1000:
998:{\displaystyle \tau }
980:
978:{\displaystyle \tau }
965:Any union of sets in
958:
932:
899:
897:{\displaystyle \tau }
871:
869:{\displaystyle \tau }
801:
692:
665:
639:
619:
569:
526:
335:> 0 always but as
33:
7691:Invariance of domain
7643:Euler characteristic
7617:Bundle (mathematics)
7296:Hart, Klaas (2004).
7226:Measure and Category
7078:"Analytic functions"
6981:
6961:
6953:is endowed with the
6937:
6849:totally disconnected
6831:
6811:
6783:
6779:A topological space
6758:
6734:
6730:forms a topology on
6702:
6651:
6619:
6546:
6470:
6401:
6397:is not empty (resp.
6362:
6339:
6319:
6299:
6273:
6235:
6203:
6179:
6112:
6092:
6072:
6046:
6016:
5996:
5960:
5931:
5911:
5875:
5871:. The complement in
5800:
5732:
5702:
5676:
5656:
5636:
5632:if for every subset
5606:
5597:symmetric difference
5579:
5549:
5523:
5513:and is said to have
5486:
5466:
5446:
5420:
5387:
5361:
5341:
5315:
5308:sequentially closed
5106:
5081:
5055:
5020:
4991:
4971:
4951:
4924:
4889:
4856:
4830:
4810:
4744:
4721:
4698:
4678:
4607:
4587:
4567:
4542:
4509:
4417:
4279:
4238:
4218:
4192:
4172:
4135:
4112:
4088:
4066:
4046:
4014:
3941:
3819:
3790:
3764:
3729:
3721:Glossary of topology
3600:
3563:
3533:
3462:
3430:
3406:
3398:, we refer to a set
3352:
3332:
3290:
3267:
3247:
3217:
3197:
3165:
3013:
3009:or equivalently, if
2972:
2944:
2924:
2904:
2884:
2852:
2832:
2793:
2767:
2741:
2688:
2684:or equivalently, if
2641:
2613:
2593:
2563:
2543:
2500:
2474:
2444:
2412:
2388:
2368:
2344:
2302:
2279:
2251:
2231:
2211:
2203:is endowed with the
2187:
2163:
2096:
2058:
2017:
1994:
1967:
1936:
1867:
1845:
1823:
1785:
1755:
1725:
1705:
1681:
1661:
1626:
1585:
1559:
1527:
1523:are open subsets of
1501:
1481:
1445:
1425:
1370:
1350:
1292:
1265:
1207:
1155:
1135:
1115:
1064:
1009:
989:
969:
941:
915:
888:
860:
784:
678:
670:is the center of an
654:
628:
608:
558:
550:if, for every point
515:
168:of its members, the
7701:Tychonoff's theorem
7696:Poincaré conjecture
7450:General (point-set)
7349:(Second ed.).
6899:Local homeomorphism
5074:sequential interior
4795:
3373:An open set on the
2275:on a non-empty set
2092:nor its complement
1497:and its complement
471:should approximate
438:is not close to 0.
194:indiscrete topology
7686:De Rham cohomology
7607:Polyhedral complex
7597:Simplicial complex
7355:Prentice Hall, Inc
6993:{\displaystyle X.}
6990:
6967:
6943:
6837:
6817:
6789:
6767:
6740:
6720:
6666:
6637:
6602:
6526:
6449:
6387:
6351:{\displaystyle X,}
6348:
6325:
6305:
6285:
6247:
6209:
6185:
6152:
6098:
6078:
6058:
6022:
6002:
5982:
5946:
5917:
5881:
5861:
5796:or, equivalently,
5786:
5708:
5688:
5662:
5642:
5618:
5585:
5561:
5535:
5516:the Baire property
5492:
5472:
5462:that converges to
5452:
5432:
5416:consisting of all
5406:
5380:sequential closure
5367:
5347:
5327:
5290:
5288:
5093:{\displaystyle X,}
5090:
5061:
5039:
5006:
4977:
4957:
4937:
4910:
4875:
4842:
4816:
4796:
4760:
4733:{\displaystyle A.}
4730:
4710:{\displaystyle A,}
4707:
4684:
4641:
4593:
4573:
4554:{\displaystyle X.}
4551:
4528:
4494:
4381:
4250:{\displaystyle U.}
4247:
4224:
4204:
4178:
4156:
4124:{\displaystyle X,}
4121:
4094:
4078:{\displaystyle X,}
4075:
4052:
4032:
3998:
3896:
3796:
3776:
3747:
3621:
3575:
3545:
3471:
3448:
3412:
3381:Notes and cautions
3364:{\displaystyle Y.}
3361:
3338:
3308:
3279:{\displaystyle X.}
3276:
3253:
3223:
3203:
3183:
3056:
2999:
2964:regular closed set
2950:
2930:
2910:
2890:
2858:
2838:
2806:
2779:
2753:
2727:
2674:
2619:
2599:
2575:{\displaystyle X.}
2572:
2549:
2531:Said differently,
2521:
2486:
2456:
2430:
2394:
2374:
2350:
2330:
2291:{\displaystyle X.}
2288:
2261:
2237:
2217:
2193:
2169:
2149:
2082:
2044:{\displaystyle J=}
2041:
2000:
1980:
1949:
1922:
1853:
1831:
1809:
1763:
1746:Euclidean topology
1731:
1711:
1687:
1667:
1647:
1622:topological space
1606:
1571:
1545:
1513:
1487:
1463:
1431:
1382:
1356:
1336:
1271:
1245:
1193:
1141:
1121:
1099:
1082:
1050:
995:
975:
953:
927:
894:
866:
796:
768:if, for any point
690:{\displaystyle U.}
687:
660:
634:
614:
594:Euclidean distance
564:
521:
444:neighborhood basis
274:In the set of all
233:algebraic geometry
96:
34:Example: the blue
7801:
7800:
7590:fundamental group
7364:978-0-13-181629-9
7343:Munkres, James R.
7334:978-0-444-50355-8
7236:978-0-387-90508-2
7106:Krantz, Steven G.
7082:Complex Variables
7022:, pp. 76–77.
6970:{\displaystyle X}
6955:discrete topology
6946:{\displaystyle X}
6840:{\displaystyle X}
6820:{\displaystyle X}
6792:{\displaystyle X}
6743:{\displaystyle X}
6598:
6592:
6573:
6567:
6509:
6503:
6484:
6478:
6358:the intersection
6328:{\displaystyle x}
6308:{\displaystyle U}
6212:{\displaystyle X}
6188:{\displaystyle X}
6101:{\displaystyle X}
6081:{\displaystyle U}
6025:{\displaystyle A}
6005:{\displaystyle X}
5920:{\displaystyle X}
5884:{\displaystyle X}
5746:
5740:
5711:{\displaystyle E}
5672:the intersection
5665:{\displaystyle X}
5645:{\displaystyle E}
5495:{\displaystyle X}
5475:{\displaystyle x}
5455:{\displaystyle S}
5370:{\displaystyle S}
5350:{\displaystyle X}
5278:
5258:
5244:
5240:
5234:
5202:
5179:
5171:
5163:
5159:
5153:
5064:{\displaystyle A}
4980:{\displaystyle i}
4960:{\displaystyle A}
4947:is eventually in
4819:{\displaystyle X}
4806:is a sequence in
4687:{\displaystyle X}
4667:sequentially open
4596:{\displaystyle X}
4576:{\displaystyle U}
4431:
4425:
4341:
4335:
4293:
4287:
4227:{\displaystyle A}
4181:{\displaystyle X}
4097:{\displaystyle D}
4055:{\displaystyle U}
3955:
3949:
3833:
3827:
3799:{\displaystyle X}
3639:, but not of the
3493:is any subset of
3341:{\displaystyle X}
3256:{\displaystyle Y}
3226:{\displaystyle Y}
3206:{\displaystyle X}
3119:topological space
2991:
2953:{\displaystyle X}
2933:{\displaystyle S}
2913:{\displaystyle X}
2893:{\displaystyle X}
2876:semiregular space
2861:{\displaystyle X}
2841:{\displaystyle S}
2804:
2709:
2662:
2622:{\displaystyle X}
2602:{\displaystyle S}
2585:Regular open sets
2397:{\displaystyle X}
2377:{\displaystyle S}
2353:{\displaystyle X}
2340:is a topology on
2240:{\displaystyle X}
2220:{\displaystyle X}
2205:discrete topology
2196:{\displaystyle X}
2172:{\displaystyle K}
2054:Finally, neither
2003:{\displaystyle I}
1714:{\displaystyle X}
1690:{\displaystyle X}
1490:{\displaystyle S}
1434:{\displaystyle S}
1359:{\displaystyle n}
1283:topological space
1067:
847:Topological space
663:{\displaystyle U}
637:{\displaystyle U}
617:{\displaystyle U}
567:{\displaystyle U}
524:{\displaystyle U}
253:topological space
186:discrete topology
174:topological space
16:(Redirected from
7821:
7814:General topology
7791:
7790:
7764:
7763:
7754:
7744:
7734:
7733:
7722:
7721:
7516:
7429:
7422:
7415:
7406:
7405:
7401:
7376:
7338:
7319:
7283:
7277:
7260:
7258:
7255:Topology. Vol. 1
7247:
7241:
7239:
7220:
7214:
7208:
7185:
7179:
7164:
7158:
7152:
7146:
7140:
7134:
7128:
7127:
7102:
7096:
7095:
7073:
7064:
7058:
7052:
7051:
7029:
7023:
7017:
7000:
6999:
6997:
6996:
6991:
6976:
6974:
6973:
6968:
6952:
6950:
6949:
6944:
6931:
6910:
6846:
6844:
6843:
6838:
6826:
6824:
6823:
6818:
6805:compact subspace
6798:
6796:
6795:
6790:
6776:
6774:
6773:
6768:
6749:
6747:
6746:
6741:
6729:
6727:
6726:
6721:
6675:
6673:
6672:
6667:
6646:
6644:
6643:
6638:
6611:
6609:
6608:
6603:
6596:
6590:
6583:
6582:
6571:
6565:
6558:
6557:
6542:
6538:
6535:
6533:
6532:
6527:
6519:
6518:
6507:
6501:
6494:
6493:
6482:
6476:
6458:
6456:
6455:
6450:
6448:
6444:
6437:
6436:
6419:
6418:
6396:
6394:
6393:
6388:
6380:
6379:
6357:
6355:
6354:
6349:
6334:
6332:
6331:
6326:
6314:
6312:
6311:
6306:
6294:
6292:
6291:
6286:
6256:
6254:
6253:
6248:
6218:
6216:
6215:
6210:
6194:
6192:
6191:
6186:
6161:
6159:
6158:
6153:
6136:
6135:
6107:
6105:
6104:
6099:
6087:
6085:
6084:
6079:
6067:
6065:
6064:
6059:
6031:
6029:
6028:
6023:
6011:
6009:
6008:
6003:
5991:
5989:
5988:
5983:
5972:
5971:
5955:
5953:
5952:
5947:
5926:
5924:
5923:
5918:
5890:
5888:
5887:
5882:
5870:
5868:
5867:
5862:
5860:
5856:
5849:
5848:
5831:
5830:
5812:
5811:
5795:
5793:
5792:
5787:
5785:
5781:
5774:
5773:
5756:
5755:
5744:
5738:
5717:
5715:
5714:
5709:
5697:
5695:
5694:
5689:
5671:
5669:
5668:
5663:
5651:
5649:
5648:
5643:
5628:is said to have
5627:
5625:
5624:
5619:
5594:
5592:
5591:
5586:
5570:
5568:
5567:
5562:
5544:
5542:
5541:
5536:
5501:
5499:
5498:
5493:
5481:
5479:
5478:
5473:
5461:
5459:
5458:
5453:
5441:
5439:
5438:
5433:
5415:
5413:
5412:
5407:
5399:
5398:
5377:is equal to its
5376:
5374:
5373:
5368:
5356:
5354:
5353:
5348:
5336:
5334:
5333:
5328:
5299:
5297:
5296:
5291:
5289:
5279:
5276:
5259:
5256:
5245:
5242:
5238:
5232:
5213:
5203:
5200:
5180:
5177:
5172:
5169:
5164:
5161:
5157:
5151:
5122:
5121:
5099:
5097:
5096:
5091:
5071:is equal to its
5070:
5068:
5067:
5062:
5048:
5046:
5045:
5040:
5032:
5031:
5015:
5013:
5012:
5007:
4986:
4984:
4983:
4978:
4966:
4964:
4963:
4958:
4946:
4944:
4943:
4938:
4936:
4935:
4919:
4917:
4916:
4911:
4884:
4882:
4881:
4876:
4868:
4867:
4851:
4849:
4848:
4843:
4825:
4823:
4822:
4817:
4805:
4803:
4802:
4797:
4794:
4789:
4778:
4774:
4773:
4756:
4755:
4739:
4737:
4736:
4731:
4716:
4714:
4713:
4708:
4693:
4691:
4690:
4685:
4650:
4648:
4647:
4642:
4631:
4630:
4602:
4600:
4599:
4594:
4582:
4580:
4579:
4574:
4560:
4558:
4557:
4552:
4537:
4535:
4534:
4529:
4521:
4520:
4503:
4501:
4500:
4495:
4493:
4489:
4488:
4484:
4477:
4476:
4459:
4458:
4441:
4440:
4429:
4423:
4390:
4388:
4387:
4382:
4380:
4376:
4369:
4368:
4351:
4350:
4339:
4333:
4332:
4328:
4321:
4320:
4303:
4302:
4291:
4285:
4256:
4254:
4253:
4248:
4233:
4231:
4230:
4225:
4213:
4211:
4210:
4205:
4187:
4185:
4184:
4179:
4165:
4163:
4162:
4157:
4130:
4128:
4127:
4122:
4103:
4101:
4100:
4095:
4084:
4082:
4081:
4076:
4061:
4059:
4058:
4053:
4041:
4039:
4038:
4033:
4007:
4005:
4004:
3999:
3994:
3990:
3983:
3982:
3965:
3964:
3953:
3947:
3905:
3903:
3902:
3897:
3895:
3891:
3890:
3886:
3879:
3878:
3861:
3860:
3843:
3842:
3831:
3825:
3805:
3803:
3802:
3797:
3785:
3783:
3782:
3777:
3756:
3754:
3753:
3748:
3637:rational numbers
3630:
3628:
3627:
3622:
3584:
3582:
3581:
3576:
3554:
3552:
3551:
3546:
3480:
3478:
3477:
3472:
3457:
3455:
3454:
3449:
3421:
3419:
3418:
3413:
3370:
3368:
3367:
3362:
3347:
3345:
3344:
3339:
3317:
3315:
3314:
3309:
3285:
3283:
3282:
3277:
3262:
3260:
3259:
3254:
3232:
3230:
3229:
3224:
3212:
3210:
3209:
3204:
3192:
3190:
3189:
3184:
3065:
3063:
3062:
3057:
3040:
3036:
3008:
3006:
3005:
3000:
2992:
2987:
2976:
2959:
2957:
2956:
2951:
2939:
2937:
2936:
2931:
2919:
2917:
2916:
2911:
2899:
2897:
2896:
2891:
2867:
2865:
2864:
2859:
2847:
2845:
2844:
2839:
2815:
2813:
2812:
2807:
2805:
2797:
2788:
2786:
2785:
2780:
2762:
2760:
2759:
2754:
2736:
2734:
2733:
2728:
2714:
2710:
2702:
2683:
2681:
2680:
2675:
2667:
2663:
2655:
2633:regular open set
2628:
2626:
2625:
2620:
2608:
2606:
2605:
2600:
2581:
2579:
2578:
2573:
2558:
2556:
2555:
2550:
2530:
2528:
2527:
2522:
2495:
2493:
2492:
2487:
2465:
2463:
2462:
2457:
2439:
2437:
2436:
2431:
2403:
2401:
2400:
2395:
2383:
2381:
2380:
2375:
2359:
2357:
2356:
2351:
2339:
2337:
2336:
2331:
2317:
2316:
2297:
2295:
2294:
2289:
2270:
2268:
2267:
2262:
2260:
2259:
2246:
2244:
2243:
2238:
2226:
2224:
2223:
2218:
2202:
2200:
2199:
2194:
2178:
2176:
2175:
2170:
2158:
2156:
2155:
2150:
2103:
2091:
2089:
2088:
2083:
2050:
2048:
2047:
2042:
2009:
2007:
2006:
2001:
1989:
1987:
1986:
1981:
1979:
1978:
1962:
1958:
1956:
1955:
1950:
1948:
1947:
1931:
1929:
1928:
1923:
1879:
1878:
1862:
1860:
1859:
1854:
1852:
1840:
1838:
1837:
1832:
1830:
1818:
1816:
1815:
1810:
1772:
1770:
1769:
1764:
1762:
1740:
1738:
1737:
1732:
1720:
1718:
1717:
1712:
1696:
1694:
1693:
1688:
1676:
1674:
1673:
1668:
1656:
1654:
1653:
1648:
1615:
1613:
1612:
1607:
1580:
1578:
1577:
1572:
1554:
1552:
1551:
1546:
1522:
1520:
1519:
1514:
1496:
1494:
1493:
1488:
1472:
1470:
1469:
1464:
1440:
1438:
1437:
1432:
1391:
1389:
1388:
1383:
1365:
1363:
1362:
1357:
1345:
1343:
1342:
1337:
1332:
1328:
1324:
1310:
1280:
1278:
1277:
1272:
1260:
1254:
1252:
1251:
1246:
1238:
1237:
1219:
1218:
1202:
1200:
1199:
1194:
1186:
1185:
1167:
1166:
1150:
1148:
1147:
1142:
1130:
1128:
1127:
1122:
1108:
1106:
1105:
1100:
1092:
1091:
1081:
1059:
1057:
1056:
1051:
1043:
1039:
1026:
1025:
1004:
1002:
1001:
996:
984:
982:
981:
976:
962:
960:
959:
954:
936:
934:
933:
928:
903:
901:
900:
895:
883:
879:
875:
873:
872:
867:
828:. Equivalently,
823:
805:
803:
802:
797:
763:
736:
729:
725:
718:
712:, since neither
711:
704:
696:
694:
693:
688:
669:
667:
666:
661:
649:
643:
641:
640:
635:
623:
621:
620:
615:
603:
600:is smaller than
599:
591:
585:
581:
573:
571:
570:
565:
553:
545:
538:
530:
528:
527:
522:
306:
304:
280:Euclidean metric
229:Zariski topology
148:
144:
140:
136:
85:
59:
21:
7829:
7828:
7824:
7823:
7822:
7820:
7819:
7818:
7804:
7803:
7802:
7797:
7728:
7710:
7706:Urysohn's lemma
7667:
7631:
7517:
7508:
7480:low-dimensional
7438:
7433:
7386:
7383:
7365:
7335:
7308:
7292:
7287:
7286:
7278:
7263:
7248:
7244:
7237:
7221:
7217:
7213:, pp. 8–9.
7209:
7188:
7180:
7167:
7159:
7155:
7151:, pp. 102.
7147:
7143:
7135:
7131:
7124:
7103:
7099:
7092:
7074:
7067:
7059:
7055:
7048:
7030:
7026:
7018:
7014:
7009:
7004:
7003:
6982:
6979:
6978:
6962:
6959:
6958:
6938:
6935:
6934:
6932:
6928:
6923:
6908:
6875:Base (topology)
6869:Almost open map
6865:
6832:
6829:
6828:
6812:
6809:
6808:
6784:
6781:
6780:
6759:
6756:
6755:
6735:
6732:
6731:
6703:
6700:
6699:
6652:
6649:
6648:
6620:
6617:
6616:
6578:
6574:
6553:
6549:
6547:
6544:
6543:
6540:
6536:
6514:
6510:
6489:
6485:
6471:
6468:
6467:
6432:
6428:
6427:
6423:
6414:
6410:
6402:
6399:
6398:
6375:
6371:
6363:
6360:
6359:
6340:
6337:
6336:
6320:
6317:
6316:
6300:
6297:
6296:
6274:
6271:
6270:
6266:δ-cluster point
6260:θ-cluster point
6236:
6233:
6232:
6204:
6201:
6200:
6180:
6177:
6176:
6131:
6127:
6113:
6110:
6109:
6093:
6090:
6089:
6073:
6070:
6069:
6047:
6044:
6043:
6017:
6014:
6013:
5997:
5994:
5993:
5967:
5963:
5961:
5958:
5957:
5932:
5929:
5928:
5912:
5909:
5908:
5876:
5873:
5872:
5844:
5840:
5839:
5835:
5826:
5822:
5807:
5803:
5801:
5798:
5797:
5769:
5765:
5764:
5760:
5751:
5747:
5733:
5730:
5729:
5703:
5700:
5699:
5677:
5674:
5673:
5657:
5654:
5653:
5637:
5634:
5633:
5607:
5604:
5603:
5580:
5577:
5576:
5550:
5547:
5546:
5524:
5521:
5520:
5487:
5484:
5483:
5467:
5464:
5463:
5447:
5444:
5443:
5421:
5418:
5417:
5394:
5390:
5388:
5385:
5384:
5362:
5359:
5358:
5357:if and only if
5342:
5339:
5338:
5316:
5313:
5312:
5287:
5286:
5275:
5255:
5241:
5211:
5210:
5199:
5176:
5168:
5160:
5132:
5117:
5113:
5109:
5107:
5104:
5103:
5082:
5079:
5078:
5056:
5053:
5052:
5027:
5023:
5021:
5018:
5017:
4992:
4989:
4988:
4972:
4969:
4968:
4952:
4949:
4948:
4931:
4927:
4925:
4922:
4921:
4890:
4887:
4886:
4863:
4859:
4857:
4854:
4853:
4831:
4828:
4827:
4811:
4808:
4807:
4790:
4779:
4769:
4765:
4761:
4751:
4747:
4745:
4742:
4741:
4722:
4719:
4718:
4699:
4696:
4695:
4679:
4676:
4675:
4626:
4622:
4608:
4605:
4604:
4588:
4585:
4584:
4568:
4565:
4564:
4543:
4540:
4539:
4516:
4512:
4510:
4507:
4506:
4472:
4468:
4467:
4463:
4454:
4450:
4449:
4445:
4436:
4432:
4418:
4415:
4414:
4364:
4360:
4359:
4355:
4346:
4342:
4316:
4312:
4311:
4307:
4298:
4294:
4280:
4277:
4276:
4239:
4236:
4235:
4219:
4216:
4215:
4193:
4190:
4189:
4173:
4170:
4169:
4136:
4133:
4132:
4113:
4110:
4109:
4089:
4086:
4085:
4067:
4064:
4063:
4047:
4044:
4043:
4015:
4012:
4011:
3978:
3974:
3973:
3969:
3960:
3956:
3942:
3939:
3938:
3874:
3870:
3869:
3865:
3856:
3852:
3851:
3847:
3838:
3834:
3820:
3817:
3816:
3791:
3788:
3787:
3765:
3762:
3761:
3730:
3727:
3726:
3723:
3717:Almost open map
3713:
3601:
3598:
3597:
3564:
3561:
3560:
3534:
3531:
3530:
3509:if and only if
3463:
3460:
3459:
3431:
3428:
3427:
3407:
3404:
3403:
3388:
3383:
3353:
3350:
3349:
3333:
3330:
3329:
3291:
3288:
3287:
3268:
3265:
3264:
3248:
3245:
3244:
3218:
3215:
3214:
3198:
3195:
3194:
3166:
3163:
3162:
3111:
3076:
3026:
3022:
3014:
3011:
3010:
2977:
2975:
2973:
2970:
2969:
2945:
2942:
2941:
2925:
2922:
2921:
2905:
2902:
2901:
2885:
2882:
2881:
2880:. A subset of
2853:
2850:
2849:
2833:
2830:
2829:
2796:
2794:
2791:
2790:
2768:
2765:
2764:
2742:
2739:
2738:
2701:
2697:
2689:
2686:
2685:
2654:
2650:
2642:
2639:
2638:
2614:
2611:
2610:
2594:
2591:
2590:
2587:
2564:
2561:
2560:
2544:
2541:
2540:
2501:
2498:
2497:
2475:
2472:
2471:
2445:
2442:
2441:
2413:
2410:
2409:
2389:
2386:
2385:
2369:
2366:
2365:
2345:
2342:
2341:
2312:
2311:
2303:
2300:
2299:
2298:Then the union
2280:
2277:
2276:
2255:
2254:
2252:
2249:
2248:
2232:
2229:
2228:
2212:
2209:
2208:
2188:
2185:
2184:
2164:
2161:
2160:
2099:
2097:
2094:
2093:
2059:
2056:
2055:
2018:
2015:
2014:
1995:
1992:
1991:
1974:
1970:
1968:
1965:
1964:
1960:
1959:cannot contain
1943:
1939:
1937:
1934:
1933:
1874:
1870:
1868:
1865:
1864:
1848:
1846:
1843:
1842:
1826:
1824:
1821:
1820:
1786:
1783:
1782:
1758:
1756:
1753:
1752:
1726:
1723:
1722:
1706:
1703:
1702:
1682:
1679:
1678:
1662:
1659:
1658:
1627:
1624:
1623:
1586:
1583:
1582:
1560:
1557:
1556:
1528:
1525:
1524:
1502:
1499:
1498:
1482:
1479:
1478:
1446:
1443:
1442:
1426:
1423:
1422:
1406:
1401:
1371:
1368:
1367:
1351:
1348:
1347:
1320:
1306:
1299:
1295:
1293:
1290:
1289:
1266:
1263:
1262:
1258:
1233:
1229:
1214:
1210:
1208:
1205:
1204:
1181:
1177:
1162:
1158:
1156:
1153:
1152:
1136:
1133:
1132:
1116:
1113:
1112:
1087:
1083:
1071:
1065:
1062:
1061:
1021:
1017:
1016:
1012:
1010:
1007:
1006:
990:
987:
986:
970:
967:
966:
942:
939:
938:
916:
913:
912:
889:
886:
885:
881:
877:
861:
858:
857:
849:
807:
785:
782:
781:
753:
743:
731:
727:
720:
713:
709:
707:closed interval
700:
679:
676:
675:
655:
652:
651:
645:
629:
626:
625:
609:
606:
605:
601:
597:
587:
583:
579:
559:
556:
555:
551:
541:
534:
516:
513:
512:
509:
507:Euclidean space
501:
296:
283:
245:
225:Euclidean space
146:
142:
138:
134:
73:
47:
28:
23:
22:
15:
12:
11:
5:
7827:
7817:
7816:
7799:
7798:
7796:
7795:
7785:
7784:
7783:
7778:
7773:
7758:
7748:
7738:
7726:
7715:
7712:
7711:
7709:
7708:
7703:
7698:
7693:
7688:
7683:
7677:
7675:
7669:
7668:
7666:
7665:
7660:
7655:
7653:Winding number
7650:
7645:
7639:
7637:
7633:
7632:
7630:
7629:
7624:
7619:
7614:
7609:
7604:
7599:
7594:
7593:
7592:
7587:
7585:homotopy group
7577:
7576:
7575:
7570:
7565:
7560:
7555:
7545:
7540:
7535:
7525:
7523:
7519:
7518:
7511:
7509:
7507:
7506:
7501:
7496:
7495:
7494:
7484:
7483:
7482:
7472:
7467:
7462:
7457:
7452:
7446:
7444:
7440:
7439:
7432:
7431:
7424:
7417:
7409:
7403:
7402:
7382:
7381:External links
7379:
7378:
7377:
7363:
7339:
7333:
7320:
7306:
7291:
7288:
7285:
7284:
7261:
7242:
7235:
7215:
7186:
7165:
7163:, pp. 88.
7153:
7141:
7139:, pp. 95.
7129:
7122:
7110:"Fundamentals"
7097:
7090:
7065:
7063:, pp. 76.
7053:
7046:
7024:
7011:
7010:
7008:
7005:
7002:
7001:
6989:
6986:
6966:
6942:
6925:
6924:
6922:
6919:
6918:
6917:
6911:
6902:
6896:
6890:
6884:
6878:
6872:
6864:
6861:
6857:
6836:
6816:
6788:
6766:
6763:
6739:
6719:
6716:
6713:
6710:
6707:
6691:
6690:
6687:
6684:
6681:
6665:
6662:
6659:
6656:
6636:
6633:
6630:
6627:
6624:
6613:
6612:
6601:
6595:
6589:
6586:
6581:
6577:
6570:
6564:
6561:
6556:
6552:
6525:
6522:
6517:
6513:
6506:
6500:
6497:
6492:
6488:
6481:
6475:
6461:
6460:
6459:is not empty).
6447:
6443:
6440:
6435:
6431:
6426:
6422:
6417:
6413:
6409:
6406:
6386:
6383:
6378:
6374:
6370:
6367:
6347:
6344:
6324:
6304:
6284:
6281:
6278:
6269:) of a subset
6267:
6261:
6246:
6243:
6240:
6229:
6223:
6208:
6198:
6184:
6173:
6167:
6162:
6151:
6148:
6145:
6142:
6139:
6134:
6130:
6126:
6123:
6120:
6117:
6097:
6077:
6057:
6054:
6051:
6040:
6035:
6034:
6033:
6021:
6001:
5981:
5978:
5975:
5970:
5966:
5945:
5942:
5939:
5936:
5927:) of a subset
5916:
5905:
5895:
5880:
5859:
5855:
5852:
5847:
5843:
5838:
5834:
5829:
5825:
5821:
5818:
5815:
5810:
5806:
5784:
5780:
5777:
5772:
5768:
5763:
5759:
5754:
5750:
5743:
5737:
5726:
5721:
5720:
5719:
5707:
5687:
5684:
5681:
5661:
5641:
5617:
5614:
5611:
5584:
5560:
5557:
5554:
5534:
5531:
5528:
5517:
5511:
5504:
5491:
5471:
5451:
5431:
5428:
5425:
5405:
5402:
5397:
5393:
5381:
5366:
5346:
5326:
5323:
5320:
5309:
5303:
5302:
5301:
5300:
5285:
5282:
5274:
5271:
5268:
5265:
5262:
5254:
5251:
5248:
5237:
5231:
5228:
5225:
5222:
5219:
5216:
5214:
5212:
5209:
5206:
5198:
5195:
5192:
5189:
5186:
5183:
5178: in
5175:
5167:
5156:
5150:
5147:
5144:
5141:
5138:
5135:
5133:
5131:
5128:
5125:
5120:
5116:
5112:
5111:
5089:
5086:
5075:
5060:
5050:
5038:
5035:
5030:
5026:
5005:
5002:
4999:
4996:
4976:
4956:
4934:
4930:
4909:
4906:
4903:
4900:
4897:
4894:
4874:
4871:
4866:
4862:
4841:
4838:
4835:
4815:
4793:
4788:
4785:
4782:
4777:
4772:
4768:
4764:
4759:
4754:
4750:
4729:
4726:
4706:
4703:
4683:
4669:
4662:
4658:
4652:
4651:
4640:
4637:
4634:
4629:
4625:
4621:
4618:
4615:
4612:
4592:
4572:
4561:
4550:
4547:
4527:
4524:
4519:
4515:
4504:
4492:
4487:
4483:
4480:
4475:
4471:
4466:
4462:
4457:
4453:
4448:
4444:
4439:
4435:
4428:
4422:
4409:
4403:
4398:
4395:
4379:
4375:
4372:
4367:
4363:
4358:
4354:
4349:
4345:
4338:
4331:
4327:
4324:
4319:
4315:
4310:
4306:
4301:
4297:
4290:
4284:
4273:
4268:
4264:
4258:
4257:
4246:
4243:
4223:
4203:
4200:
4197:
4177:
4166:
4155:
4152:
4149:
4146:
4143:
4140:
4120:
4117:
4093:
4074:
4071:
4051:
4031:
4028:
4025:
4022:
4019:
4008:
3997:
3993:
3989:
3986:
3981:
3977:
3972:
3968:
3963:
3959:
3952:
3946:
3933:
3924:
3918:
3913:
3910:
3894:
3889:
3885:
3882:
3877:
3873:
3868:
3864:
3859:
3855:
3850:
3846:
3841:
3837:
3830:
3824:
3813:
3795:
3775:
3772:
3769:
3746:
3743:
3740:
3737:
3734:
3712:
3709:
3690:
3687:such that all
3682:
3658:
3655:such that all
3620:
3617:
3614:
3611:
3608:
3605:
3574:
3571:
3568:
3544:
3541:
3538:
3470:
3467:
3447:
3444:
3441:
3438:
3435:
3411:
3387:
3384:
3382:
3379:
3360:
3357:
3337:
3323:
3307:
3304:
3301:
3298:
3295:
3275:
3272:
3252:
3238:
3222:
3202:
3182:
3179:
3176:
3173:
3170:
3127:uniform spaces
3110:
3107:
3075:
3072:
3069:
3055:
3052:
3049:
3046:
3043:
3039:
3035:
3032:
3029:
3025:
3021:
3018:
2998:
2995:
2990:
2986:
2983:
2980:
2966:
2949:
2929:
2909:
2889:
2878:
2857:
2837:
2803:
2800:
2778:
2775:
2772:
2752:
2749:
2746:
2726:
2723:
2720:
2717:
2713:
2708:
2705:
2700:
2696:
2693:
2673:
2670:
2666:
2661:
2658:
2653:
2649:
2646:
2635:
2618:
2598:
2586:
2583:
2571:
2568:
2548:
2538:
2534:
2520:
2517:
2514:
2511:
2508:
2505:
2485:
2482:
2479:
2469:
2455:
2452:
2449:
2429:
2426:
2423:
2420:
2417:
2407:
2393:
2373:
2363:
2349:
2329:
2326:
2323:
2320:
2315:
2310:
2307:
2287:
2284:
2258:
2236:
2216:
2192:
2181:
2180:
2168:
2148:
2145:
2142:
2139:
2136:
2133:
2130:
2127:
2124:
2121:
2118:
2115:
2112:
2109:
2106:
2102:
2081:
2078:
2075:
2072:
2069:
2066:
2063:
2052:
2040:
2037:
2034:
2031:
2028:
2025:
2022:
2011:
1999:
1977:
1973:
1946:
1942:
1921:
1918:
1915:
1912:
1909:
1906:
1903:
1900:
1897:
1894:
1891:
1888:
1885:
1882:
1877:
1873:
1851:
1829:
1808:
1805:
1802:
1799:
1796:
1793:
1790:
1775:open intervals
1761:
1730:
1710:
1700:
1686:
1666:
1657:the empty set
1646:
1643:
1640:
1637:
1634:
1631:
1621:
1605:
1602:
1599:
1596:
1593:
1590:
1570:
1567:
1564:
1544:
1541:
1538:
1535:
1532:
1512:
1509:
1506:
1486:
1476:
1462:
1459:
1456:
1453:
1450:
1430:
1419:
1412:
1405:
1402:
1400:
1397:
1381:
1378:
1375:
1355:
1335:
1331:
1327:
1323:
1319:
1316:
1313:
1309:
1305:
1302:
1298:
1270:
1261:together with
1256:
1255:
1244:
1241:
1236:
1232:
1228:
1225:
1222:
1217:
1213:
1192:
1189:
1184:
1180:
1176:
1173:
1170:
1165:
1161:
1140:
1120:
1109:
1098:
1095:
1090:
1086:
1080:
1077:
1074:
1070:
1049:
1046:
1042:
1038:
1035:
1032:
1029:
1024:
1020:
1015:
994:
974:
963:
952:
949:
946:
926:
923:
920:
893:
865:
848:
845:
795:
792:
789:
742:
739:
686:
683:
659:
633:
613:
582:(depending on
563:
520:
508:
505:
500:
497:
244:
241:
108:generalization
26:
9:
6:
4:
3:
2:
7826:
7815:
7812:
7811:
7809:
7794:
7786:
7782:
7779:
7777:
7774:
7772:
7769:
7768:
7767:
7759:
7757:
7753:
7749:
7747:
7743:
7739:
7737:
7732:
7727:
7725:
7717:
7716:
7713:
7707:
7704:
7702:
7699:
7697:
7694:
7692:
7689:
7687:
7684:
7682:
7679:
7678:
7676:
7674:
7670:
7664:
7663:Orientability
7661:
7659:
7656:
7654:
7651:
7649:
7646:
7644:
7641:
7640:
7638:
7634:
7628:
7625:
7623:
7620:
7618:
7615:
7613:
7610:
7608:
7605:
7603:
7600:
7598:
7595:
7591:
7588:
7586:
7583:
7582:
7581:
7578:
7574:
7571:
7569:
7566:
7564:
7561:
7559:
7556:
7554:
7551:
7550:
7549:
7546:
7544:
7541:
7539:
7536:
7534:
7530:
7527:
7526:
7524:
7520:
7515:
7505:
7502:
7500:
7499:Set-theoretic
7497:
7493:
7490:
7489:
7488:
7485:
7481:
7478:
7477:
7476:
7473:
7471:
7468:
7466:
7463:
7461:
7460:Combinatorial
7458:
7456:
7453:
7451:
7448:
7447:
7445:
7441:
7437:
7430:
7425:
7423:
7418:
7416:
7411:
7410:
7407:
7399:
7395:
7394:
7389:
7385:
7384:
7374:
7370:
7366:
7360:
7356:
7352:
7348:
7344:
7340:
7336:
7330:
7326:
7321:
7317:
7313:
7309:
7307:0-444-50355-2
7303:
7299:
7294:
7293:
7281:
7276:
7274:
7272:
7270:
7268:
7266:
7256:
7252:
7246:
7238:
7232:
7228:
7227:
7219:
7212:
7207:
7205:
7203:
7201:
7199:
7197:
7195:
7193:
7191:
7183:
7178:
7176:
7174:
7172:
7170:
7162:
7157:
7150:
7145:
7138:
7133:
7125:
7123:9781420089745
7119:
7115:
7111:
7107:
7101:
7093:
7091:9780821869017
7087:
7083:
7079:
7072:
7070:
7062:
7057:
7049:
7047:9780821832844
7043:
7039:
7035:
7028:
7021:
7016:
7012:
6987:
6984:
6964:
6956:
6940:
6930:
6926:
6915:
6912:
6906:
6903:
6900:
6897:
6894:
6891:
6888:
6885:
6882:
6879:
6876:
6873:
6870:
6867:
6866:
6860:
6858:
6856:
6853:
6850:
6834:
6814:
6806:
6802:
6786:
6777:
6764:
6761:
6753:
6737:
6714:
6711:
6708:
6695:
6694:be preopen.
6688:
6685:
6682:
6679:
6678:
6677:
6663:
6660:
6657:
6654:
6634:
6631:
6628:
6625:
6622:
6599:
6593:
6587:
6584:
6579:
6575:
6568:
6562:
6559:
6554:
6550:
6523:
6520:
6515:
6511:
6504:
6498:
6495:
6490:
6486:
6479:
6473:
6466:
6465:
6464:
6445:
6441:
6438:
6433:
6429:
6424:
6420:
6415:
6411:
6407:
6404:
6384:
6381:
6376:
6372:
6368:
6365:
6345:
6342:
6322:
6302:
6282:
6279:
6276:
6268:
6265:
6262:
6259:
6244:
6241:
6238:
6230:
6227:
6224:
6221:
6206:
6196:
6182:
6174:
6171:
6168:
6165:
6163:
6149:
6146:
6143:
6140:
6137:
6132:
6128:
6124:
6121:
6118:
6115:
6095:
6075:
6055:
6052:
6049:
6041:
6038:
6036:
6019:
6012:that contain
5999:
5979:
5976:
5973:
5968:
5964:
5943:
5940:
5937:
5934:
5914:
5906:
5903:
5900:
5899:
5897:
5893:
5878:
5857:
5853:
5850:
5845:
5841:
5836:
5832:
5827:
5823:
5819:
5816:
5813:
5808:
5804:
5782:
5778:
5775:
5770:
5766:
5761:
5757:
5752:
5748:
5741:
5735:
5727:
5724:
5722:
5705:
5685:
5682:
5679:
5659:
5639:
5631:
5615:
5612:
5609:
5601:
5600:
5598:
5582:
5574:
5573:meager subset
5558:
5555:
5552:
5532:
5529:
5526:
5518:
5515:
5512:
5510:
5507:
5505:
5503:
5489:
5469:
5449:
5429:
5426:
5423:
5403:
5400:
5395:
5391:
5382:
5379:
5364:
5344:
5324:
5321:
5318:
5310:
5307:
5280:
5269:
5266:
5263:
5252:
5246:
5235:
5229:
5226:
5223:
5217:
5215:
5204:
5196:
5190:
5187:
5184:
5173:
5165:
5154:
5148:
5145:
5142:
5136:
5134:
5129:
5126:
5123:
5118:
5114:
5102:
5101:
5087:
5084:
5076:
5073:
5058:
5051:
5036:
5033:
5028:
5024:
5003:
5000:
4997:
4994:
4987:such that if
4974:
4954:
4932:
4928:
4907:
4901:
4898:
4895:
4872:
4864:
4860:
4852:is such that
4839:
4836:
4833:
4813:
4786:
4783:
4780:
4775:
4770:
4766:
4762:
4757:
4752:
4748:
4727:
4724:
4704:
4701:
4681:
4673:
4672:
4670:
4668:
4665:
4663:
4661:
4659:
4656:
4638:
4635:
4632:
4627:
4623:
4619:
4616:
4613:
4610:
4590:
4570:
4562:
4548:
4545:
4525:
4522:
4517:
4513:
4505:
4490:
4485:
4481:
4478:
4473:
4469:
4464:
4460:
4455:
4451:
4446:
4442:
4437:
4433:
4426:
4420:
4413:
4412:
4410:
4407:
4404:
4401:
4399:
4396:
4393:
4377:
4373:
4370:
4365:
4361:
4356:
4352:
4347:
4343:
4336:
4329:
4325:
4322:
4317:
4313:
4308:
4304:
4299:
4295:
4288:
4282:
4274:
4271:
4269:
4267:
4265:
4262:
4244:
4241:
4221:
4201:
4198:
4195:
4175:
4167:
4153:
4150:
4147:
4144:
4141:
4138:
4118:
4115:
4107:
4091:
4072:
4069:
4049:
4029:
4026:
4023:
4020:
4017:
4009:
3995:
3991:
3987:
3984:
3979:
3975:
3970:
3966:
3961:
3957:
3950:
3944:
3937:
3936:
3934:
3932:
3928:
3925:
3922:
3919:
3916:
3914:
3911:
3908:
3892:
3887:
3883:
3880:
3875:
3871:
3866:
3862:
3857:
3853:
3848:
3844:
3839:
3835:
3828:
3822:
3814:
3811:
3809:
3808:
3807:
3793:
3773:
3770:
3767:
3758:
3741:
3738:
3735:
3722:
3718:
3708:
3706:
3702:
3698:
3694:
3688:
3686:
3680:
3678:
3674:
3670:
3666:
3662:
3656:
3654:
3650:
3646:
3642:
3638:
3634:
3618:
3612:
3609:
3606:
3595:
3590:
3588:
3572:
3569:
3566:
3558:
3542:
3539:
3536:
3528:
3524:
3520:
3516:
3512:
3508:
3504:
3500:
3496:
3492:
3488:
3484:
3468:
3465:
3442:
3439:
3436:
3425:
3409:
3401:
3397:
3393:
3378:
3376:
3371:
3358:
3355:
3335:
3327:
3322:
3319:
3305:
3299:
3296:
3293:
3286:The function
3273:
3270:
3250:
3242:
3237:
3234:
3220:
3200:
3180:
3174:
3171:
3168:
3161:
3156:
3154:
3150:
3146:
3142:
3138:
3135:
3130:
3128:
3124:
3123:metric spaces
3120:
3116:
3106:
3104:
3100:
3096:
3092:
3087:
3085:
3081:
3071:
3067:
3053:
3050:
3047:
3044:
3041:
3037:
3033:
3030:
3027:
3023:
3019:
3016:
2996:
2993:
2984:
2981:
2978:
2967:
2965:
2962:
2947:
2927:
2907:
2887:
2879:
2877:
2874:
2871:
2855:
2835:
2827:
2823:
2819:
2798:
2776:
2773:
2770:
2750:
2747:
2744:
2724:
2721:
2718:
2715:
2711:
2703:
2698:
2694:
2691:
2671:
2668:
2664:
2656:
2651:
2647:
2644:
2636:
2634:
2631:
2616:
2596:
2582:
2569:
2566:
2546:
2536:
2532:
2518:
2515:
2512:
2509:
2503:
2496:or else, (2)
2483:
2480:
2477:
2467:
2453:
2450:
2447:
2427:
2424:
2421:
2418:
2415:
2405:
2391:
2371:
2361:
2347:
2318:
2308:
2305:
2285:
2282:
2274:
2234:
2214:
2206:
2190:
2166:
2140:
2137:
2131:
2125:
2122:
2116:
2110:
2107:
2076:
2073:
2070:
2064:
2061:
2053:
2035:
2032:
2029:
2023:
2020:
2012:
1997:
1975:
1971:
1944:
1940:
1919:
1910:
1907:
1901:
1895:
1892:
1886:
1880:
1875:
1871:
1803:
1800:
1797:
1791:
1788:
1781:The interval
1780:
1779:
1778:
1776:
1751:
1747:
1742:
1728:
1708:
1698:
1684:
1664:
1644:
1638:
1635:
1632:
1619:
1616:
1603:
1600:
1597:
1594:
1588:
1568:
1565:
1562:
1539:
1536:
1533:
1510:
1504:
1484:
1474:
1457:
1454:
1451:
1428:
1420:
1418:
1415:
1410:
1396:
1393:
1376:
1353:
1333:
1329:
1325:
1321:
1317:
1314:
1311:
1307:
1303:
1300:
1296:
1286:
1284:
1268:
1242:
1239:
1234:
1230:
1226:
1223:
1220:
1215:
1211:
1190:
1187:
1182:
1178:
1174:
1171:
1168:
1163:
1159:
1138:
1118:
1110:
1096:
1093:
1088:
1084:
1078:
1075:
1072:
1068:
1047:
1044:
1040:
1036:
1033:
1030:
1027:
1022:
1018:
1013:
992:
972:
964:
950:
947:
944:
924:
921:
918:
911:
910:
909:
907:
904:is called an
891:
863:
856:
855:
844:
841:
839:
835:
831:
827:
822:
818:
814:
810:
793:
790:
787:
779:
775:
771:
767:
761:
757:
752:
748:
738:
734:
728:[0,1]
724:
717:
710:[0,1]
708:
703:
697:
684:
681:
674:contained in
673:
657:
648:
631:
611:
595:
590:
577:
561:
549:
544:
540:
537:
518:
504:
496:
494:
490:
486:
482:
478:
474:
470:
466:
462:
458:
454:
450:
446:
445:
439:
437:
433:
429:
425:
421:
417:
413:
409:
405:
401:
397:
393:
389:
384:
382:
378:
374:
370:
366:
362:
358:
354:
350:
346:
342:
338:
334:
330:
326:
322:
318:
314:
310:
303:
299:
294:
290:
286:
281:
277:
272:
270:
269:metric spaces
266:
262:
258:
254:
250:
240:
238:
237:scheme theory
234:
230:
226:
222:
218:
213:
211:
207:
206:connectedness
203:
197:
195:
191:
187:
183:
179:
175:
171:
167:
163:
159:
155:
150:
132:
128:
124:
119:
117:
113:
112:open interval
109:
105:
101:
93:
89:
84:
80:
76:
72:) satisfying
71:
67:
63:
58:
54:
50:
46:) satisfying
45:
41:
37:
32:
19:
7793:Publications
7658:Chern number
7648:Betti number
7531: /
7528:
7522:Key concepts
7470:Differential
7391:
7346:
7327:. Elsevier.
7324:
7297:
7290:Bibliography
7282:, p. 8.
7254:
7245:
7225:
7218:
7184:, p. 9.
7161:Munkres 2000
7156:
7149:Munkres 2000
7144:
7137:Munkres 2000
7132:
7113:
7100:
7081:
7061:Munkres 2000
7056:
7037:
7027:
7020:Munkres 2000
7015:
6929:
6852:
6778:
6696:
6692:
6614:
6462:
6264:
6258:
6257:is called a
6226:
6220:
6170:
6164:
6042:if for each
6037:
6032:as a subset.
5904:semi-closure
5902:
5892:
5723:
5629:
5595:denotes the
5514:
5506:
5378:
5306:
5304:
5072:
4664:
4655:
4653:
4408:semi-preopen
4406:
4400:
4392:
4270:
4261:
4259:
4106:dense subset
3927:
3921:
3915:
3907:
3810:
3759:
3725:Throughout,
3724:
3704:
3700:
3696:
3692:
3684:
3676:
3672:
3668:
3667:are also in
3664:
3660:
3652:
3648:
3644:
3641:real numbers
3632:
3593:
3591:
3586:
3556:
3526:
3522:
3518:
3514:
3510:
3506:
3502:
3498:
3494:
3490:
3486:
3482:
3423:
3399:
3389:
3372:
3157:
3152:
3148:
3140:
3136:
3131:
3112:
3088:
3084:intersection
3077:
2961:
2960:is called a
2873:
2630:
2629:is called a
2588:
2182:
1743:
1677:and the set
1617:
1414:
1407:
1394:
1287:
1281:is called a
1257:
905:
853:
850:
842:
837:
833:
829:
825:
820:
816:
812:
808:
777:
773:
769:
765:
759:
755:
751:metric space
746:
744:
741:Metric space
732:
722:
715:
701:
698:
646:
588:
576:there exists
547:
542:
535:
510:
502:
492:
488:
484:
480:
476:
472:
468:
464:
460:
456:
452:
448:
443:
442:
440:
435:
431:
427:
423:
419:
415:
411:
407:
403:
399:
395:
391:
387:
385:
380:
376:
372:
368:
364:
356:
352:
348:
344:
340:
336:
332:
331:. Note that
328:
324:
323:approximate
320:
316:
312:
308:
301:
297:
292:
288:
284:
276:real numbers
273:
246:
220:
214:
198:
189:
181:
166:intersection
151:
123:metric space
120:
103:
97:
82:
78:
74:
69:
65:
56:
52:
48:
43:
39:
18:Open subsets
7756:Wikiversity
7673:Key results
6039:semi-θ-open
5956:denoted by
5894:semi-closed
5602:The subset
5509:almost open
5311:. A subset
4062:is open in
3923:nearly open
3806:is called:
3348:is open in
3263:is open in
2468:exactly one
2273:ultrafilter
1819:is open in
1417:clopen sets
824:belongs to
806:satisfying
726:belongs to
604:belongs to
499:Definitions
210:compactness
100:mathematics
7602:CW complex
7543:Continuity
7533:Closed set
7492:cohomology
7388:"Open set"
7007:References
6887:Closed set
6881:Clopen set
6219:is called
6108:such that
5545:such that
4603:such that
4214:such that
4042:such that
3715:See also:
3497:, the set
3318:is called
3236:continuous
3099:clopen set
3095:closed set
3091:complement
3074:Properties
1473:is called
1131:belong to
985:belong to
764:is called
533:Euclidean
295:) = |
243:Motivation
202:continuity
154:collection
92:closed set
60:. The red
7781:geometric
7776:algebraic
7627:Cobordism
7563:Hausdorff
7558:connected
7475:Geometric
7465:Continuum
7455:Algebraic
7398:EMS Press
7316:162131277
7280:Hart 2004
7211:Hart 2004
7182:Hart 2004
6801:Hausdorff
6762:τ
6715:τ
6658:⊆
6632:⊆
6594:⊆
6585:
6569:⊆
6560:
6521:
6505:⊆
6496:
6480:⊆
6439:
6421:
6408:∩
6382:
6369:∩
6280:⊆
6263:(resp. a
6242:∈
6144:⊆
6138:
6125:⊆
6119:∈
6053:∈
5974:
5938:⊆
5851:
5833:
5814:
5776:
5758:
5742:⊆
5725:semi-open
5683:∩
5613:⊆
5583:△
5556:△
5530:⊆
5427:∈
5401:
5322:⊆
5270:τ
5250:∖
5227:∈
5191:τ
5146:∈
5124:
5034:∈
4998:≥
4933:∙
4902:τ
4870:→
4865:∙
4837:∈
4792:∞
4753:∙
4633:
4620:⊆
4614:⊆
4523:
4479:
4461:
4443:
4427:⊆
4371:
4353:
4337:∪
4323:
4305:
4289:⊆
4263:preclosed
4199:⊆
4188:) subset
4148:∩
4027:⊆
3985:
3967:
3951:⊆
3881:
3863:
3845:
3829:⊆
3771:⊆
3760:A subset
3742:τ
3703:(because
3683:positive
3679:there is
3570:∩
3540:∩
3466:τ
3443:τ
3410:τ
3375:real line
3303:→
3178:→
3103:empty set
3048:
3031:
3020:
2989:¯
2982:
2802:¯
2774:
2748:
2722:
2707:¯
2695:
2660:¯
2648:
2589:A subset
2547:∅
2516:τ
2513:∈
2507:∖
2484:τ
2481:∈
2451:≠
2425:⊊
2419:≠
2416:∅
2325:∅
2319:∪
2306:τ
2144:∞
2132:∪
2120:∞
2117:−
2105:∖
1976:∁
1945:∁
1914:∞
1902:∪
1890:∞
1887:−
1876:∁
1750:real line
1729:∅
1665:∅
1639:τ
1601:τ
1598:∈
1592:∖
1569:τ
1566:∈
1540:τ
1508:∖
1458:τ
1301:−
1269:τ
1243:τ
1240:∈
1227:∩
1224:⋯
1221:∩
1191:τ
1188:∈
1172:…
1139:τ
1119:τ
1097:τ
1094:∈
1076:∈
1069:⋃
1048:τ
1045:⊆
1034:∈
993:τ
973:τ
951:τ
948:∈
945:∅
925:τ
922:∈
892:τ
876:on a set
864:τ
791:∈
745:A subset
672:open ball
511:A subset
217:manifolds
170:empty set
116:real line
7808:Category
7746:Wikibook
7724:Category
7612:Manifold
7580:Homotopy
7538:Interior
7529:Open set
7487:Homology
7436:Topology
7373:42683260
7347:Topology
7345:(2000).
7253:(1966),
7108:(2009).
6905:Open map
6863:See also
6750:that is
6647:satisfy
6228:δ-closed
6222:θ-closed
6197:δ-closed
5575:, where
4657:β-closed
4394:b-closed
3929:locally
3909:α-closed
3657:rational
3392:topology
3241:preimage
3160:function
3145:interior
3115:topology
2822:interior
2818:boundary
2737:, where
1477:if both
906:open set
854:topology
730:for any
361:interval
347:= 0 and
265:distance
178:topology
131:distance
104:open set
88:boundary
7771:general
7573:uniform
7553:compact
7504:Digital
7400:, 2001
6914:Subbase
6855:closure
6225:(resp.
6169:(resp.
3917:preopen
3699:are in
3559:, then
3324:if the
3239:if the
3101:). The
2826:closure
2466:) then
2440:(where
1748:of the
819:) <
531:of the
261:subsets
158:subsets
129:with a
114:in the
7766:Topics
7568:metric
7443:Fields
7371:
7361:
7331:
7314:
7304:
7233:
7120:
7088:
7044:
6597:
6591:
6572:
6566:
6508:
6502:
6483:
6477:
6172:δ-open
6166:θ-open
5745:
5739:
5239:
5233:
5158:
5152:
5115:SeqInt
4430:
4424:
4402:β-open
4340:
4334:
4292:
4286:
4272:b-open
3954:
3948:
3832:
3826:
3812:α-open
3529:, but
3134:subset
3132:Every
3070:true.
2824:, and
2789:, and
2406:either
2271:is an
1475:clopen
1346:where
735:> 0
592:whose
539:-space
465:axioms
305:|
249:points
208:, and
188:), or
110:of an
36:circle
7548:Space
6921:Notes
6754:than
6752:finer
5571:is a
5392:SeqCl
5016:then
4920:then
4104:is a
3931:dense
3926:, or
3631:then
3326:image
3080:union
2533:every
2362:every
1699:every
1203:then
1151:: if
1060:then
1005:: if
749:of a
596:from
383:= 1.
182:every
162:union
121:In a
106:is a
102:, an
81:<
7369:OCLC
7359:ISBN
7329:ISBN
7312:OCLC
7302:ISBN
7231:ISBN
7118:ISBN
7086:ISBN
7042:ISBN
5907:(in
5901:The
5502:).
5482:(in
4131:and
3719:and
3689:real
3321:open
3213:and
3125:and
3109:Uses
3078:The
2870:base
2559:and
2537:only
1721:and
1581:and
937:and
766:open
721:1 +
719:nor
714:0 -
548:open
477:some
235:and
221:near
62:disk
6847:is
6807:of
6799:is
6576:int
6551:int
6539:and
6412:int
6335:in
6315:of
6129:sCl
6088:of
5965:sCl
5898:.
5896:set
5842:int
5767:int
5728:if
5652:of
5077:in
4885:in
4660:.
4583:of
4452:int
4405:or
4362:int
4296:int
4275:if
4266:.
4108:of
3958:int
3872:int
3836:int
3815:if
3695:of
3675:in
3663:of
3647:in
3233:is
3147:of
3068:not
3028:Int
2979:Int
2968:if
2940:of
2848:in
2828:of
2771:Int
2645:Int
2637:if
2404:is
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