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Thus continuous functions are exactly those functions that preserve (in the forward direction) the "closeness" relationship between points and sets: a function is continuous if and only if whenever a point is close to a set then the image of that point is close to the image of that set. Similarly,
2766:). It is useful to distinguish between the interior and the surface of the sphere, so we distinguish between the open 3-ball (the interior of the sphere), and the closed 3-ball – the closure of the open 3-ball that is the open 3-ball plus the surface (the surface as the sphere itself).
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Note that these properties are also satisfied if "closure", "superset", "intersection", "contains/containing", "smallest" and "closed" are replaced by "interior", "subset", "union", "contained in", "largest", and "open". For more on this matter, see
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All properties of the closure can be derived from this definition and a few properties of the above categories. Moreover, this definition makes precise the analogy between the topological closure and other types of closures (for example
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many sets, the closure of the union and the union of the closures are equal; the union of zero sets is the empty set, and so this statement contains the earlier statement about the closure of the empty set as a special
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Therefore, the abstract theory of closure operators and the
Kuratowski closure axioms can be readily translated into the language of interior operators by replacing sets with their
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in a 3 dimensional space. Implicitly there are two regions of interest created by this sphere; the sphere itself and its interior (which is called an open 3-
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But just as a union of infinitely many closed sets is not necessarily closed, so too does closure not necessarily distribute over infinite unions: that is,
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These examples show that the closure of a set depends upon the topology of the underlying space. The last two examples are special cases of the following.
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to be the set of real numbers and define the interval in the same way then the closure of that interval is well defined and would be the set of all
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5670:{\displaystyle \operatorname {int} _{X}\left(\bigcap _{i\in \mathbb {N} }\operatorname {cl} _{X}S_{i}\right)=\operatorname {int} _{X}\left.}
5460:{\displaystyle \operatorname {cl} _{X}\left(\bigcup _{i\in \mathbb {N} }\operatorname {int} _{X}S_{i}\right)=\operatorname {cl} _{X}\left.}
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6057:{\displaystyle \operatorname {cl} _{X}\left(\bigcup _{i\in I}S_{i}\right)\neq \bigcup _{i\in I}\operatorname {cl} _{X}S_{i}}
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and then unioned together. In this way, this result can be viewed as the analogue of the well-known fact that a subset
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3492:{\displaystyle \operatorname {cl} _{X}\left(\{z\in \mathbb {C} :|z|>1\}\right)=\{z\in \mathbb {C} :|z|\geq 1\}.}
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Thus, just as the union of two closed sets is closed, so too does closure distribute over binary unions: that is,
882:. The difference between the two definitions is subtle but important – namely, in the definition of a limit point
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The closure of the union of infinitely many sets need not equal the union of the closures, but it is always a
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11533:{\displaystyle \varnothing ~=~\operatorname {cl} _{T}(S\cap T)~\neq ~T\cap \operatorname {cl} _{X}S~=~\{0\}.}
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is allowed). Note that this definition does not depend upon whether neighbourhoods are required to be open.
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5949:{\displaystyle \operatorname {cl} _{X}(S\cup T)=(\operatorname {cl} _{X}S)\cup (\operatorname {cl} _{X}T).}
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of the topological closure, which still make sense when applied to other types of closures (see below).
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8885:{\displaystyle \operatorname {cl} _{X}S=\bigcup _{U\in {\mathcal {U}}}\operatorname {cl} _{U}(U\cap S)}
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itself, we have that the closure of the empty set is the empty set, and for every non-empty subset
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The closure of a set also depends upon in which space we are taking the closure. For example, if
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9705:{\displaystyle f\left(\operatorname {cl} _{X}A\right)~\subseteq ~\operatorname {cl} _{Y}(f(A)).}
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8701:{\displaystyle C\subseteq \operatorname {cl} _{X}(T\cap S)\subseteq \operatorname {cl} _{X}S.}
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For a general topological space, this statement remains true if one replaces "sequence" by "
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10460:{\displaystyle \operatorname {cl} _{Y}f(A)\subseteq f\left(\operatorname {cl} _{X}A\right)}
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5127:{\displaystyle \operatorname {int} _{X}S=X\setminus \operatorname {cl} _{X}(X\setminus S).}
5048:{\displaystyle \operatorname {cl} _{X}S=X\setminus \operatorname {int} _{X}(X\setminus S),}
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7602:{\displaystyle \operatorname {cl} _{T}(S\cap T)~\subseteq ~T\cap \operatorname {cl} _{X}S}
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7116:{\displaystyle T\cap \operatorname {cl} _{X}S\subseteq T\cap C=\operatorname {cl} _{T}S.}
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of every closed subset of the codomain is closed in the domain; explicitly, this means:
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every limit point is a point of closure, but not every point of closure is a limit point
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The definition of a point of closure of a set is closely related to the definition of a
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On the set of real numbers one can put other topologies rather than the standard one.
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is always guaranteed, where this containment could be strict (consider for instance
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In general, the closure operator does not commute with intersections. However, in a
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97:. Intuitively, the closure can be thought of as all the points that are either in
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6613:{\displaystyle \operatorname {cl} _{T}S\subseteq T\cap \operatorname {cl} _{X}S}
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8955:{\displaystyle \operatorname {cl} _{U}(S\cap U)=U\cap \operatorname {cl} _{X}S}
7822:{\displaystyle \operatorname {cl} _{T}(S\cap T)=T\cap \operatorname {cl} _{X}S}
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3945:, since every set is closed (and also open), every set is equal to its closure.
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One may define the closure operator in terms of universal arrows, as follows.
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of (but need not be equal to) the intersection of the closures of the sets.
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6422:{\displaystyle \operatorname {cl} _{T}S~=~T\cap \operatorname {cl} _{X}S.}
6255:{\displaystyle \operatorname {cl} _{T}S\subseteq \operatorname {cl} _{X}S}
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8488:{\displaystyle s\in T\cap S\subseteq \operatorname {cl} _{T}(T\cap S)=C}
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from the definition of the subspace topology, there must exist some set
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8548:{\displaystyle \operatorname {cl} _{X}S\subseteq (X\setminus T)\cup C.}
7304:{\displaystyle \operatorname {cl} _{T}S=T\cap \operatorname {cl} _{X}S}
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3590:(For a general topological space, this property is equivalent to the
1840:.) can be defined using any of the following equivalent definitions:
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8628:{\displaystyle T\cap \operatorname {cl} _{X}S\subseteq T\cap C=C.}
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has no well defined closure due to boundary elements not being in
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In other words, every non-empty subset of an indiscrete space is
10909:
This category — also a partial order — then has initial object
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8295:{\displaystyle X\setminus (T\setminus C)=(X\setminus T)\cup C}
6905:{\displaystyle S\subseteq \operatorname {cl} _{T}S\subseteq C}
27:
All points and limit points in a subset of a topological space
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Sometimes the second or third property above is taken as the
3927:{\displaystyle \operatorname {cl} _{X}((0,1))=\mathbb {R} .}
2802:{\displaystyle \varnothing =\operatorname {cl} \varnothing }
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9144:
can be computed "locally" in the sets of any open cover of
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in which the only closed (open) sets are the empty set and
4652:{\displaystyle \operatorname {cl} _{X}:\wp (X)\to \wp (X)}
4202:{\displaystyle S=\{q\in \mathbb {Q} :q^{2}>2,q>0\},}
10578:{\displaystyle \operatorname {cl} _{Y}f(C)\subseteq f(C)}
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in the definitions. The set of all limit points of a set
522:
is allowed). Another way to express this is to say that
11248:
Pages displaying short descriptions of redirect targets
11221:
Pages displaying short descriptions of redirect targets
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in which the objects are subsets and the morphisms are
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8057:{\displaystyle T\cap \operatorname {cl} _{X}(T\cap S)}
1713:
are clear from context then it may also be denoted by
1208:. A point of closure which is not a limit point is an
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11424:{\displaystyle \operatorname {cl} _{X}S=[0,\infty ),}
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then a topological space is obtained by defining the
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3809:{\displaystyle \operatorname {cl} _{X}((0,1))=(0,1).}
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3703:{\displaystyle \operatorname {cl} _{X}((0,1))=[0,1).}
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2713:
2679:
2632:
2612:
2592:
2572:
2552:
2521:
2495:
2469:
2434:
2411:
2388:
2358:
2317:
2297:
2277:
2247:
2221:
2188:
2165:
2139:
2115:
2089:
2065:
2035:
2013:
1993:
1973:
1953:
1929:
1903:
1879:
1849:
1826:
1806:
1777:
1747:
1719:
1699:
1679:
1659:
1626:
1581:
1546:
1523:
1483:
1463:
1443:
1423:
1403:
1383:
1361:
1341:
1318:
1298:
1278:
1258:
1238:
1218:
1178:
1151:
1131:
1125:
has more strict condition than a point of closure of
1111:
1091:
1071:
1051:
1031:
1011:
991:
969:
948:
928:
908:
888:
844:
818:
798:
778:
758:
724:
701:
681:
646:
568:
548:
528:
502:
461:
435:
409:
389:
369:
347:
327:
304:
281:
258:
238:
218:
194:
174:
150:
10845:
The set of closed subsets containing a fixed subset
8006:{\displaystyle C:=\operatorname {cl} _{T}(T\cap S),}
7023:{\displaystyle \operatorname {cl} _{X}S\subseteq C.}
9098:. In words, this result shows that the closure in
2349:The closure of a set has the following properties.
11715:
11675:
11618:, p. 38 use the second property as the definition.
11532:
11423:
11369:
11337:
11299:
11182:
11155:
11117:
11094:
11056:
11036:
11008:
10973:
10950:
10930:
10901:
10863:
10837:
10802:
10778:
10758:
10738:
10715:
10695:
10666:
10640:
10606:
10577:
10520:
10488:
10459:
10385:
10353:
10330:
10301:
10281:
10257:
10212:
10180:
10151:
10122:
10102:
10076:
10055:
10023:
10003:
9984:
9955:
9931:
9889:
9859:
9839:
9816:
9787:
9759:
9730:
9704:
9616:
9587:
9553:
9530:
9510:
9490:
9443:
9393:
9360:
9340:
9314:
9294:
9274:
9250:
9226:
9202:
9182:
9156:
9136:
9110:
9086:
9058:
9038:
9014:
8984:
8954:
8884:
8800:
8774:
8750:
8719:
8700:
8627:
8567:
8547:
8487:
8418:
8398:
8372:
8352:
8314:
8294:
8232:
8209:
8183:
8163:
8143:
8120:
8094:
8056:
8005:
7951:
7928:
7908:
7861:
7841:
7821:
7751:
7731:
7711:
7673:
7629:
7601:
7525:
7505:
7485:
7451:
7410:
7370:
7339:
7303:
7245:
7209:
7187:
7167:
7134:
7115:
7042:
7022:
6980:
6947:
6924:
6904:
6859:
6811:
6791:
6771:
6745:
6722:
6689:
6669:
6645:
6612:
6550:
6530:
6491:
6468:
6421:
6354:
6334:
6314:
6294:
6274:
6254:
6202:
6178:
6158:
6134:
6114:
6076:
6056:
5948:
5818:
5795:
5764:
5722:
5698:
5669:
5508:
5488:
5459:
5298:
5278:
5248:
5222:
5156:
5126:
5047:
4971:
4933:
4899:
4879:
4842:
4812:
4789:
4767:
4740:
4713:
4677:
4651:
4594:
4555:
4523:
4492:
4448:
4417:
4397:
4375:
4355:
4331:
4311:
4291:
4267:
4247:
4221:
4201:
4127:
4098:
4070:
4029:
4006:
3986:
3966:
3926:
3864:
3838:
3808:
3732:
3702:
3626:
3582:
3540:
3513:
3491:
3367:
3324:
3302:
3273:
3251:
3222:
3200:
3178:
3155:
3123:
3103:
3071:
2998:
2976:
2948:
2926:
2901:
2867:
2841:
2821:
2801:
2722:
2691:
2647:
2618:
2598:
2578:
2558:
2536:
2507:
2481:
2452:
2417:
2394:
2370:
2338:
2303:
2283:
2259:
2233:
2206:
2171:
2151:
2124:
2101:
2074:
2047:
2019:
1999:
1979:
1959:
1935:
1915:
1888:
1861:
1832:
1812:
1792:
1763:
1734:
1705:
1685:
1665:
1645:
1612:
1567:
1529:
1489:
1469:
1449:
1429:
1409:
1389:
1370:
1347:
1324:
1304:
1284:
1264:
1244:
1224:
1184:
1157:
1137:
1117:
1097:
1077:
1057:
1037:
1017:
997:
975:
954:
934:
914:
894:
856:
830:
804:
784:
764:
730:
710:
687:
652:
632:
554:
534:
514:
488:
447:
421:
395:
375:
356:
333:
313:
287:
264:
244:
224:
200:
180:
156:
6860:{\displaystyle \operatorname {cl} _{T}S=T\cap C.}
3974:since the only closed sets are the empty set and
12238:
1613:{\displaystyle \operatorname {cl} _{(X,\tau )}S}
647:
591:
9595:is continuous if and only if for every subset
4106:is the set of rational numbers, with the usual
3072:{\displaystyle \operatorname {cl} _{X}((0,1))=}
2809:. In other words, the closure of the empty set
11228:, a set equal to the closure of their interior
9963:is continuous if and only if for every subset
9932:{\displaystyle x\in \operatorname {cl} _{X}A,}
9046:). This equality is particularly useful when
7829:will hold (no matter the relationship between
6531:{\displaystyle T\cap \operatorname {cl} _{X}S}
3368:{\displaystyle \mathbb {C} =\mathbb {R} ^{2},}
633:{\displaystyle d(x,S):=\inf _{s\in S}d(x,s)=0}
11853:
11240: – Largest open subset of some given set
11018:Similarly, since every closed set containing
10616:
8095:{\displaystyle T\cap S\subseteq T\subseteq X}
4602:, the topological closure induces a function
11713:
11583:
11524:
11518:
11246: – Cluster point in a topological space
4193:
4148:
3483:
3447:
3436:
3400:
5765:{\displaystyle \operatorname {cl} _{X}S=S.}
4071:{\displaystyle \operatorname {cl} _{X}A=X.}
3583:{\displaystyle \operatorname {cl} _{X}S=S.}
12221:
12194:
11860:
11846:
11735:
11714:Hocking, John G.; Young, Gail S. (1988) ,
11571:
11095:{\displaystyle (I\downarrow X\setminus A)}
11044:corresponds with an open set contained in
11009:{\displaystyle A\to \operatorname {cl} A.}
10528:is a (strongly) closed map if and only if
10393:is a (strongly) closed map if and only if
3744:in which every set is closed (open), then
3079:. In other words., the closure of the set
275:This definition generalizes to any subset
11762:
11695:
11639:
11607:
7623:
7361:
7340:{\displaystyle \operatorname {cl} _{X}S;}
7246:{\displaystyle \operatorname {cl} _{X}S.}
5638:
5553:
5428:
5343:
4972:{\displaystyle \operatorname {int} _{X},}
4858:
4783:
4714:{\displaystyle \operatorname {cl} _{X}S,}
4391:
4241:
4158:
4118:
3917:
3858:
3832:
3726:
3620:
3457:
3410:
3352:
3343:
3293:
3267:
3242:
3216:
3194:
2992:
2942:
2920:
671:by replacing "open ball" or "ball" with "
11770:Convex Analysis in General Vector Spaces
11753:
11559:
11234: – Set of all limit points of a set
11156:{\displaystyle \operatorname {int} (A),}
11102:as the set of open subsets contained in
9738:that belongs to the closure of a subset
9403:
6981:{\displaystyle \operatorname {cl} _{X}S}
6723:{\displaystyle \operatorname {cl} _{T}S}
6646:{\displaystyle \operatorname {cl} _{T}S}
6469:{\displaystyle \operatorname {cl} _{X}S}
1646:{\displaystyle \operatorname {cl} _{X}S}
1193:. A limit point of a set is also called
117:. The notion of closure is in many ways
6115:{\displaystyle S\subseteq T\subseteq X}
5681:
4934:{\displaystyle \operatorname {cl} _{X}}
2902:{\displaystyle X=\operatorname {cl} X.}
14:
12239:
11744:
11627:
11611:
9795:necessarily belongs to the closure of
2453:{\displaystyle S=\operatorname {cl} S}
2109:is the smallest closed set containing
11841:
11673:
11655:
11615:
11595:
10938:Thus there is a universal arrow from
10931:{\displaystyle \operatorname {cl} A.}
10084:is continuous at a fixed given point
4775:may be used instead. Conversely, if
2747:
2648:{\displaystyle \operatorname {cl} S.}
2537:{\displaystyle \operatorname {cl} T.}
1735:{\displaystyle \operatorname {cl} S,}
105:. A point which is in the closure of
11370:{\displaystyle S\cap T=\varnothing }
9394:{\displaystyle U\in {\mathcal {U}}.}
8353:{\displaystyle (X\setminus T)\cup C}
4659:that is defined by sending a subset
4299:, which would be the lower bound of
2692:{\displaystyle \operatorname {cl} S}
2508:{\displaystyle \operatorname {cl} S}
2371:{\displaystyle \operatorname {cl} S}
2234:{\displaystyle \operatorname {cl} S}
2152:{\displaystyle \operatorname {cl} S}
2102:{\displaystyle \operatorname {cl} S}
2048:{\displaystyle \operatorname {cl} S}
1916:{\displaystyle \operatorname {cl} S}
1862:{\displaystyle \operatorname {cl} S}
9939:then this terminology allows for a
9015:{\displaystyle U\in {\mathcal {U}}}
8985:{\displaystyle U\in {\mathcal {U}}}
8635:The reverse inclusion follows from
5223:{\displaystyle S_{1},S_{2},\ldots }
4459:
1833:{\displaystyle \operatorname {Cl} }
1813:{\displaystyle \operatorname {cl} }
1500:
695:be a subset of a topological space
133:
73:may equivalently be defined as the
24:
11412:
11329:
11285:
10361:In terms of the closure operator,
9712:That is to say, given any element
9563:In terms of the closure operator,
9383:
9243:
9079:
9007:
8977:
8847:
8743:
7703:
7656:
7440:
4637:
4622:
4563:, into itself which satisfies the
4539:
2738:" (as described in the article on
2189:
1292:which contains no other points of
25:
12263:
11813:
11364:
11083:
11028:
8527:
8335:
8277:
8259:
8250:
8201:
8112:
5112:
5090:
5033:
5011:
4880:{\displaystyle \mathbb {c} (S)=S}
4556:{\displaystyle {\mathcal {P}}(X)}
2796:
1025:but it also must have a point of
12220:
12193:
12183:
12173:
12162:
12152:
12151:
11945:
10902:{\displaystyle (A\downarrow I).}
6302:is equal to the intersection of
5170:the following result does hold:
4405:. However, if we instead define
1764:{\displaystyle {\overline {S}},}
1272:and there is a neighbourhood of
11747:Foundations of General Topology
11696:Gemignani, Michael C. (1990) ,
11682:, Saunders College Publishing,
11649:
11300:{\displaystyle T:=(-\infty ,0]}
9070:and the sets in the open cover
7674:{\displaystyle T=(-\infty ,0],}
7513:is not necessarily a subset of
7371:{\displaystyle X=\mathbb {R} ,}
4824:as being exactly those subsets
4797:is a closure operator on a set
4741:{\displaystyle {\overline {S}}}
4275:nor its complement can contain
4110:induced by the Euclidean space
667:This definition generalizes to
11621:
11601:
11589:
11577:
11565:
11553:
11475:
11463:
11415:
11403:
11338:{\displaystyle S:=(0,\infty )}
11332:
11320:
11294:
11279:
11260:
11147:
11141:
11089:
11077:
11071:
11064:we can interpret the category
10991:
10893:
10887:
10881:
10826:
10687:
10572:
10566:
10557:
10551:
10512:
10422:
10416:
10377:
10325:
10319:
10249:
10222:
10204:
10198:
10175:
10169:
10047:
10041:
10011:maps points that are close to
9811:
9805:
9782:
9776:
9696:
9693:
9687:
9681:
9579:
9485:
9479:
9451:between topological spaces is
9435:
9251:{\displaystyle {\mathcal {U}}}
9087:{\displaystyle {\mathcal {U}}}
8924:
8912:
8879:
8867:
8751:{\displaystyle {\mathcal {U}}}
8673:
8661:
8533:
8521:
8476:
8464:
8341:
8329:
8283:
8271:
8265:
8253:
8051:
8039:
7997:
7985:
7909:{\displaystyle S,T\subseteq X}
7791:
7779:
7706:
7694:
7665:
7650:
7630:{\displaystyle X=\mathbb {R} }
7565:
7553:
7486:{\displaystyle S,T\subseteq X}
7452:{\displaystyle T=(0,\infty ).}
7443:
7431:
7402:
7390:
5940:
5921:
5915:
5896:
5890:
5878:
5855:of the union of the closures.
5230:be a sequence of subsets of a
5118:
5106:
5039:
5027:
4949:operator, which is denoted by
4868:
4862:
4646:
4640:
4634:
4631:
4625:
4589:
4577:
4550:
4544:
3910:
3907:
3895:
3892:
3839:{\displaystyle X=\mathbb {R} }
3800:
3788:
3782:
3779:
3767:
3764:
3733:{\displaystyle X=\mathbb {R} }
3694:
3682:
3676:
3673:
3661:
3658:
3627:{\displaystyle X=\mathbb {R} }
3473:
3465:
3426:
3418:
3208:, then the closure of the set
3150:
3138:
3098:
3086:
3066:
3054:
3048:
3045:
3033:
3030:
2330:
2318:
2198:
2192:
2007:is also a point of closure of
1673:is understood), where if both
1599:
1587:
1559:
1547:
985:, i.e., each neighbourhood of
867:
621:
609:
584:
572:
477:
465:
341:as a metric space with metric
128:
13:
1:
11740:, vol. I, Academic Press
11546:
11197:), since all are examples of
10607:{\displaystyle C\subseteq X.}
10489:{\displaystyle A\subseteq X.}
10152:{\displaystyle A\subseteq X,}
9985:{\displaystyle A\subseteq X,}
9760:{\displaystyle A\subseteq X,}
9617:{\displaystyle A\subseteq X,}
9408:
8720:{\displaystyle \blacksquare }
8555:Intersecting both sides with
7739:happens to an open subset of
7712:{\displaystyle S=(0,\infty )}
7135:{\displaystyle \blacksquare }
7030:Intersecting both sides with
4128:{\displaystyle \mathbb {R} ,}
3303:{\displaystyle \mathbb {R} .}
3252:{\displaystyle \mathbb {R} .}
2207:{\displaystyle \partial (S).}
11867:
11037:{\displaystyle X\setminus A}
10864:{\displaystyle A\subseteq X}
10031:to points that are close to
9890:{\displaystyle A\subseteq X}
9183:{\displaystyle S\subseteq X}
9137:{\displaystyle S\subseteq X}
8801:{\displaystyle S\subseteq X}
8210:{\displaystyle T\setminus C}
8121:{\displaystyle T\setminus C}
7168:{\displaystyle S\subseteq T}
6772:{\displaystyle C\subseteq X}
4843:{\displaystyle S\subseteq X}
4790:{\displaystyle \mathbb {c} }
4733:
4678:{\displaystyle S\subseteq X}
4398:{\displaystyle \mathbb {Q} }
4248:{\displaystyle \mathbb {Q} }
3865:{\displaystyle \mathbb {R} }
3525:subset of a Euclidean space
3274:{\displaystyle \mathbb {Q} }
3223:{\displaystyle \mathbb {Q} }
3201:{\displaystyle \mathbb {R} }
2999:{\displaystyle \mathbb {R} }
2949:{\displaystyle \mathbb {C} }
2927:{\displaystyle \mathbb {R} }
2842:{\displaystyle \varnothing }
2822:{\displaystyle \varnothing }
2482:{\displaystyle S\subseteq T}
1820:is sometimes capitalized to
1753:
7:
11826:Encyclopedia of Mathematics
11776:World Scientific Publishing
11774:. River Edge, N.J. London:
11745:Pervin, William J. (1965),
11219: – Algebraic structure
11204:
10871:can be identified with the
9943:description of continuity:
9847:If we declare that a point
4449:{\displaystyle {\sqrt {2}}}
4356:{\displaystyle {\sqrt {2}}}
4292:{\displaystyle {\sqrt {2}}}
2753:
2055:is the intersection of all
489:{\displaystyle d(x,s)<r}
10:
12268:
12114:Banach fixed-point theorem
11660:, Wm. C. Brown Publisher,
10617:Categorical interpretation
10226:
9412:
4907:of these subsets form the
4463:
2958:standard (metric) topology
2339:{\displaystyle (X,\tau ).}
1987:, and each limit point of
1568:{\displaystyle (X,\tau ),}
1504:
1212:. In other words, a point
871:
772:if every neighbourhood of
137:
29:
12147:
12104:
12068:
11954:
11943:
11875:
11232:Derived set (mathematics)
10838:{\displaystyle I:T\to P.}
9491:{\displaystyle f^{-1}(C)}
7637:with the usual topology,
7311:to be a proper subset of
4595:{\displaystyle (X,\tau )}
4565:Kuratowski closure axioms
4470:Kuratowski closure axioms
2267:for which there exists a
1967:is a point of closure of
1397:is a point of closure of
922:, every neighbourhood of
542:is a point of closure of
383:is a point of closure of
188:is a point of closure of
11754:Schubert, Horst (1968),
11658:Introduction to Topology
11656:Baker, Crump W. (1991),
11584:Hocking & Young 1988
11253:
10746:Furthermore, a topology
10585:for every closed subset
10521:{\displaystyle f:X\to Y}
10386:{\displaystyle f:X\to Y}
10269:if and only if whenever
10258:{\displaystyle f:X\to Y}
10110:if and only if whenever
9588:{\displaystyle f:X\to Y}
9444:{\displaystyle f:X\to Y}
8380:as a subset (because if
7411:{\displaystyle S=(0,1),}
1232:is an isolated point of
942:must contain a point of
32:Closure (disambiguation)
11736:Kuratowski, K. (1966),
11700:(2nd ed.), Dover,
11674:Croom, Fred H. (1989),
10981:given by the inclusion
10810:with inclusion functor
9341:{\displaystyle S\cap U}
3186:is the Euclidean space
2984:is the Euclidean space
1945:all of its limit points
1793:{\displaystyle S{}^{-}}
1252:if it is an element of
1085:to be a limit point of
455:such that the distance
12169:Mathematics portal
12069:Metrics and properties
12055:Second-countable space
11678:Principles of Topology
11534:
11425:
11371:
11339:
11301:
11184:
11157:
11119:
11096:
11058:
11038:
11010:
10975:
10952:
10932:
10903:
10865:
10839:
10804:
10780:
10760:
10740:
10717:
10697:
10696:{\displaystyle A\to B}
10668:
10642:
10608:
10579:
10522:
10490:
10461:
10387:
10355:
10338:is a closed subset of
10332:
10303:
10289:is a closed subset of
10283:
10259:
10214:
10182:
10153:
10124:
10104:
10103:{\displaystyle x\in X}
10078:
10057:
10025:
10005:
9986:
9957:
9933:
9891:
9861:
9841:
9818:
9789:
9761:
9732:
9731:{\displaystyle x\in X}
9706:
9618:
9589:
9555:
9538:is a closed subset of
9532:
9512:
9492:
9445:
9395:
9362:
9342:
9316:
9296:
9276:
9252:
9228:
9210:if and only if it is "
9204:
9184:
9158:
9138:
9112:
9088:
9060:
9040:
9016:
8986:
8956:
8886:
8802:
8776:
8752:
8721:
8702:
8629:
8569:
8549:
8495:), which implies that
8489:
8420:
8400:
8399:{\displaystyle s\in S}
8374:
8354:
8316:
8302:is a closed subset of
8296:
8234:
8211:
8185:
8165:
8145:
8122:
8096:
8058:
8007:
7953:
7930:
7910:
7863:
7843:
7823:
7753:
7733:
7713:
7675:
7631:
7603:
7527:
7507:
7487:
7453:
7412:
7372:
7341:
7305:
7247:
7211:
7189:
7169:
7136:
7117:
7044:
7024:
6982:
6949:
6926:
6906:
6861:
6813:
6793:
6773:
6747:
6730:is a closed subset of
6724:
6691:
6671:
6647:
6614:
6562:), which implies that
6558:(by definition of the
6552:
6538:is a closed subset of
6532:
6493:
6476:is a closed subset of
6470:
6423:
6356:
6336:
6316:
6296:
6276:
6256:
6204:
6180:
6160:
6136:
6116:
6078:
6058:
5950:
5820:
5797:
5766:
5724:
5700:
5671:
5510:
5490:
5461:
5300:
5280:
5250:
5224:
5158:
5128:
5049:
4973:
4935:
4901:
4881:
4844:
4814:
4791:
4769:
4742:
4715:
4679:
4653:
4596:
4557:
4525:
4494:
4450:
4419:
4399:
4377:
4357:
4333:
4313:
4293:
4269:
4249:
4223:
4203:
4129:
4100:
4072:
4031:
4008:
3988:
3968:
3928:
3866:
3840:
3810:
3734:
3704:
3628:
3584:
3542:
3515:
3493:
3369:
3326:
3304:
3275:
3253:
3224:
3202:
3180:
3157:
3125:
3105:
3073:
3000:
2978:
2950:
2928:
2903:
2869:
2843:
2823:
2803:
2724:
2693:
2649:
2620:
2600:
2580:
2566:is a closed set, then
2560:
2538:
2509:
2483:
2454:
2419:
2396:
2372:
2340:
2305:
2285:
2261:
2260:{\displaystyle x\in X}
2235:
2208:
2173:
2153:
2126:
2103:
2076:
2049:
2021:
2001:
1981:
1961:
1937:
1917:
1890:
1863:
1834:
1814:
1794:
1765:
1736:
1707:
1687:
1667:
1647:
1614:
1569:
1531:
1491:
1471:
1451:
1431:
1411:
1391:
1372:
1349:
1326:
1306:
1286:
1266:
1246:
1226:
1186:
1159:
1139:
1119:
1099:
1079:
1059:
1039:
1019:
999:
977:
956:
936:
916:
896:
858:
857:{\displaystyle s\in S}
832:
806:
786:
766:
732:
712:
689:
654:
634:
556:
536:
516:
490:
449:
448:{\displaystyle s\in S}
423:
422:{\displaystyle r>0}
397:
377:
358:
335:
315:
289:
266:
246:
226:
202:
182:
158:
11764:Zălinescu, Constantin
11535:
11426:
11372:
11340:
11302:
11185:
11158:
11120:
11097:
11059:
11039:
11011:
10976:
10953:
10933:
10904:
10866:
10840:
10805:
10781:
10761:
10741:
10718:
10698:
10669:
10648:may be realized as a
10643:
10609:
10580:
10523:
10491:
10462:
10388:
10356:
10333:
10304:
10284:
10260:
10215:
10213:{\displaystyle f(A).}
10183:
10154:
10130:is close to a subset
10125:
10105:
10079:
10058:
10056:{\displaystyle f(A).}
10026:
10006:
9987:
9958:
9934:
9892:
9862:
9842:
9819:
9790:
9762:
9733:
9707:
9619:
9590:
9556:
9533:
9513:
9493:
9446:
9404:Functions and closure
9396:
9363:
9343:
9317:
9297:
9277:
9253:
9229:
9205:
9185:
9159:
9139:
9113:
9089:
9061:
9041:
9017:
8987:
8957:
8887:
8803:
8777:
8753:
8722:
8703:
8630:
8570:
8550:
8490:
8421:
8401:
8375:
8355:
8317:
8297:
8235:
8212:
8186:
8166:
8146:
8123:
8097:
8059:
8008:
7954:
7931:
7911:
7864:
7844:
7824:
7754:
7734:
7714:
7676:
7632:
7604:
7528:
7508:
7488:
7454:
7413:
7373:
7342:
7306:
7248:
7212:
7190:
7175:is a dense subset of
7170:
7137:
7118:
7045:
7025:
6983:
6950:
6927:
6907:
6862:
6814:
6794:
6774:
6748:
6725:
6692:
6672:
6648:
6615:
6553:
6533:
6494:
6471:
6424:
6357:
6337:
6317:
6297:
6277:
6257:
6210:induces on it), then
6205:
6181:
6161:
6137:
6117:
6079:
6059:
5951:
5821:
5798:
5767:
5725:
5701:
5672:
5511:
5491:
5489:{\displaystyle S_{i}}
5462:
5301:
5281:
5279:{\displaystyle S_{i}}
5251:
5232:complete metric space
5225:
5168:complete metric space
5159:
5129:
5050:
4974:
4936:
4914:The closure operator
4902:
4882:
4845:
4815:
4792:
4770:
4768:{\displaystyle S^{-}}
4743:
4716:
4680:
4654:
4597:
4558:
4526:
4495:
4451:
4420:
4400:
4378:
4358:
4334:
4314:
4294:
4270:
4250:
4224:
4204:
4130:
4101:
4073:
4032:
4009:
3989:
3969:
3929:
3867:
3841:
3811:
3735:
3705:
3629:
3585:
3543:
3516:
3494:
3370:
3327:
3305:
3276:
3254:
3225:
3203:
3181:
3158:
3126:
3106:
3104:{\displaystyle (0,1)}
3074:
3001:
2979:
2951:
2929:
2904:
2870:
2844:
2824:
2804:
2725:
2694:
2667:first-countable space
2650:
2621:
2601:
2581:
2561:
2539:
2510:
2484:
2455:
2420:
2397:
2373:
2341:
2306:
2286:
2262:
2236:
2209:
2174:
2154:
2127:
2104:
2077:
2050:
2022:
2002:
1982:
1962:
1938:
1918:
1891:
1864:
1835:
1815:
1795:
1766:
1737:
1708:
1706:{\displaystyle \tau }
1688:
1668:
1666:{\displaystyle \tau }
1648:
1615:
1570:
1532:
1507:Closure (mathematics)
1492:
1472:
1452:
1432:
1412:
1392:
1373:
1350:
1327:
1307:
1287:
1267:
1247:
1227:
1187:
1160:
1140:
1120:
1100:
1080:
1060:
1045:that is not equal to
1040:
1020:
1000:
978:
957:
937:
917:
897:
859:
833:
807:
787:
767:
733:
713:
690:
655:
653:{\displaystyle \inf }
635:
557:
537:
517:
491:
450:
424:
398:
378:
359:
336:
321:Fully expressed, for
316:
290:
267:
247:
227:
203:
183:
159:
12124:Invariance of domain
12076:Euler characteristic
12050:Bundle (mathematics)
11435:
11381:
11349:
11311:
11270:
11244:Limit point of a set
11171:
11132:
11106:
11068:
11048:
11022:
10985:
10962:
10942:
10913:
10878:
10849:
10814:
10794:
10770:
10750:
10727:
10707:
10681:
10658:
10632:
10589:
10532:
10500:
10471:
10397:
10365:
10342:
10331:{\displaystyle f(C)}
10313:
10293:
10273:
10237:
10229:Open and closed maps
10192:
10181:{\displaystyle f(x)}
10163:
10134:
10114:
10088:
10068:
10035:
10015:
9995:
9967:
9947:
9901:
9875:
9851:
9828:
9817:{\displaystyle f(A)}
9799:
9788:{\displaystyle f(x)}
9770:
9742:
9716:
9627:
9599:
9567:
9542:
9522:
9502:
9463:
9423:
9372:
9352:
9326:
9306:
9286:
9266:
9238:
9218:
9194:
9168:
9148:
9122:
9102:
9074:
9050:
9030:
9022:is endowed with the
8996:
8966:
8896:
8812:
8808:is any subset then:
8786:
8766:
8738:
8711:
8639:
8579:
8559:
8499:
8430:
8410:
8384:
8364:
8326:
8306:
8244:
8221:
8195:
8175:
8155:
8132:
8106:
8068:
8017:
7963:
7940:
7920:
7888:
7853:
7833:
7763:
7743:
7723:
7685:
7641:
7613:
7537:
7517:
7497:
7465:
7422:
7381:
7351:
7315:
7257:
7221:
7201:
7179:
7153:
7126:
7054:
7034:
6992:
6959:
6936:
6916:
6871:
6823:
6803:
6783:
6757:
6734:
6701:
6681:
6661:
6624:
6566:
6542:
6503:
6480:
6447:
6366:
6346:
6326:
6306:
6286:
6266:
6214:
6194:
6186:is endowed with the
6170:
6150:
6126:
6094:
6068:
5960:
5862:
5832:of sets is always a
5807:
5787:
5734:
5714:
5690:
5682:Facts about closures
5520:
5500:
5473:
5310:
5290:
5263:
5237:
5188:
5145:
5065:
4986:
4953:
4918:
4891:
4854:
4828:
4801:
4779:
4752:
4725:
4689:
4663:
4606:
4574:
4534:
4512:
4484:
4436:
4409:
4387:
4367:
4343:
4323:
4303:
4279:
4259:
4237:
4231:both closed and open
4213:
4139:
4114:
4090:
4040:
4018:
3998:
3978:
3955:
3876:
3854:
3822:
3818:If one considers on
3748:
3716:
3712:If one considers on
3642:
3636:lower limit topology
3634:is endowed with the
3610:
3552:
3529:
3505:
3379:
3339:
3316:
3289:
3263:
3238:
3212:
3190:
3170:
3135:
3115:
3083:
3014:
2988:
2968:
2938:
2916:
2878:
2856:
2833:
2813:
2781:
2711:
2677:
2630:
2610:
2590:
2570:
2550:
2519:
2493:
2467:
2432:
2409:
2386:
2356:
2315:
2295:
2275:
2245:
2219:
2186:
2163:
2137:
2113:
2087:
2063:
2033:
2011:
1991:
1971:
1951:
1927:
1901:
1877:
1847:
1824:
1804:
1775:
1745:
1717:
1697:
1677:
1657:
1624:
1579:
1544:
1521:
1481:
1477:is a limit point of
1461:
1441:
1421:
1401:
1381:
1359:
1339:
1316:
1296:
1276:
1256:
1236:
1216:
1176:
1149:
1129:
1109:
1089:
1069:
1049:
1029:
1009:
989:
967:
946:
926:
906:
886:
880:limit point of a set
874:Limit point of a set
842:
816:
796:
792:contains a point of
776:
756:
722:
699:
679:
644:
566:
546:
526:
500:
459:
433:
407:
387:
367:
345:
325:
302:
279:
256:
236:
232:contains a point of
216:
192:
172:
148:
30:For other uses, see
12134:Tychonoff's theorem
12129:Poincaré conjecture
11883:General (point-set)
11698:Elementary Topology
11238:Interior (topology)
9455:if and only if the
9415:Continuous function
9234:", meaning that if
7253:It is possible for
6322:and the closure of
6262:and the closure of
5776:The closure of the
5182: —
4887:(so complements in
4721:where the notation
4363:is irrational. So,
4319:, but cannot be in
3234:is the whole space
2740:filters in topology
1105:. A limit point of
831:{\displaystyle x=s}
515:{\displaystyle x=s}
252:(this point can be
12119:De Rham cohomology
12040:Polyhedral complex
12030:Simplicial complex
11821:"Closure of a set"
11530:
11421:
11367:
11335:
11297:
11226:Closed regular set
11183:{\displaystyle A.}
11180:
11153:
11118:{\displaystyle A,}
11115:
11092:
11054:
11034:
11006:
10974:{\displaystyle I,}
10971:
10948:
10928:
10899:
10861:
10835:
10800:
10776:
10756:
10739:{\displaystyle B.}
10736:
10713:
10693:
10664:
10638:
10604:
10575:
10518:
10486:
10457:
10383:
10354:{\displaystyle Y.}
10351:
10328:
10299:
10279:
10255:
10210:
10178:
10149:
10120:
10100:
10074:
10053:
10021:
10001:
9982:
9953:
9929:
9887:
9857:
9840:{\displaystyle Y.}
9837:
9814:
9785:
9757:
9728:
9702:
9614:
9585:
9554:{\displaystyle Y.}
9551:
9528:
9508:
9488:
9441:
9391:
9358:
9338:
9312:
9292:
9272:
9248:
9224:
9200:
9180:
9154:
9134:
9108:
9084:
9056:
9036:
9012:
8982:
8952:
8882:
8853:
8798:
8772:
8748:
8717:
8698:
8625:
8565:
8545:
8485:
8416:
8396:
8370:
8350:
8312:
8292:
8233:{\displaystyle X.}
8230:
8207:
8181:
8161:
8144:{\displaystyle T,}
8141:
8118:
8102:). The complement
8092:
8054:
8013:which is equal to
8003:
7952:{\displaystyle X.}
7949:
7926:
7906:
7859:
7839:
7819:
7759:then the equality
7749:
7729:
7709:
7671:
7627:
7599:
7523:
7503:
7483:
7449:
7408:
7368:
7347:for example, take
7337:
7301:
7243:
7207:
7185:
7165:
7132:
7113:
7040:
7020:
6978:
6955:the minimality of
6948:{\displaystyle X,}
6945:
6922:
6902:
6857:
6809:
6789:
6769:
6746:{\displaystyle T,}
6743:
6720:
6687:
6667:
6643:
6610:
6548:
6528:
6492:{\displaystyle X,}
6489:
6466:
6419:
6352:
6332:
6312:
6292:
6272:
6252:
6200:
6176:
6156:
6132:
6112:
6074:
6054:
6030:
5996:
5946:
5828:The closure of an
5819:{\displaystyle X.}
5816:
5793:
5762:
5720:
5696:
5667:
5643:
5558:
5506:
5486:
5457:
5433:
5348:
5296:
5276:
5249:{\displaystyle X.}
5246:
5220:
5176:
5157:{\displaystyle X.}
5154:
5124:
5045:
4979:in the sense that
4969:
4931:
4911:of the topology).
4897:
4877:
4840:
4813:{\displaystyle X,}
4810:
4787:
4765:
4738:
4711:
4675:
4649:
4592:
4553:
4524:{\displaystyle X,}
4521:
4490:
4446:
4415:
4395:
4373:
4353:
4329:
4309:
4289:
4265:
4245:
4219:
4199:
4125:
4096:
4068:
4030:{\displaystyle X,}
4027:
4004:
3984:
3967:{\displaystyle X,}
3964:
3924:
3862:
3836:
3806:
3730:
3700:
3624:
3580:
3541:{\displaystyle X,}
3538:
3511:
3489:
3365:
3322:
3300:
3271:
3249:
3220:
3198:
3176:
3153:
3121:
3101:
3069:
2996:
2974:
2946:
2924:
2899:
2868:{\displaystyle X,}
2865:
2839:
2819:
2799:
2723:{\displaystyle S.}
2720:
2703:of all convergent
2699:is the set of all
2689:
2645:
2616:
2596:
2576:
2556:
2534:
2505:
2479:
2450:
2415:
2392:
2368:
2336:
2301:
2291:that converges to
2281:
2257:
2241:is the set of all
2231:
2204:
2169:
2149:
2125:{\displaystyle S.}
2122:
2099:
2075:{\displaystyle S.}
2072:
2045:
2017:
1997:
1977:
1957:
1933:
1913:
1889:{\displaystyle S.}
1886:
1869:is the set of all
1859:
1830:
1810:
1790:
1761:
1732:
1703:
1683:
1663:
1643:
1610:
1565:
1527:
1487:
1467:
1447:
1427:
1407:
1387:
1371:{\displaystyle x,}
1368:
1345:
1322:
1302:
1282:
1262:
1242:
1222:
1199:accumulation point
1182:
1155:
1135:
1115:
1095:
1075:
1055:
1035:
1015:
995:
973:
952:
932:
912:
892:
854:
828:
802:
782:
762:
728:
711:{\displaystyle X.}
708:
685:
669:topological spaces
650:
630:
605:
552:
532:
512:
486:
445:
429:there exists some
419:
393:
373:
357:{\displaystyle d,}
354:
331:
314:{\displaystyle X.}
311:
285:
262:
242:
222:
198:
178:
154:
85:, and also as the
61:together with all
12252:Closure operators
12234:
12233:
12023:fundamental group
11785:978-981-4488-15-0
11758:, Allyn and Bacon
11517:
11511:
11486:
11480:
11449:
11443:
11195:algebraic closure
11057:{\displaystyle A}
10951:{\displaystyle A}
10803:{\displaystyle P}
10779:{\displaystyle X}
10759:{\displaystyle T}
10716:{\displaystyle A}
10667:{\displaystyle P}
10641:{\displaystyle X}
10467:for every subset
10302:{\displaystyle X}
10282:{\displaystyle C}
10123:{\displaystyle x}
10077:{\displaystyle f}
10024:{\displaystyle A}
10004:{\displaystyle f}
9956:{\displaystyle f}
9860:{\displaystyle x}
9667:
9661:
9531:{\displaystyle C}
9511:{\displaystyle X}
9361:{\displaystyle U}
9315:{\displaystyle X}
9295:{\displaystyle S}
9275:{\displaystyle X}
9227:{\displaystyle X}
9203:{\displaystyle X}
9157:{\displaystyle X}
9111:{\displaystyle X}
9096:coordinate charts
9059:{\displaystyle X}
9039:{\displaystyle X}
9026:induced on it by
9024:subspace topology
8834:
8775:{\displaystyle X}
8734:Consequently, if
8731:
8730:
8568:{\displaystyle T}
8419:{\displaystyle T}
8373:{\displaystyle S}
8315:{\displaystyle X}
8191:now implies that
8184:{\displaystyle X}
8164:{\displaystyle T}
7929:{\displaystyle T}
7862:{\displaystyle T}
7842:{\displaystyle S}
7752:{\displaystyle X}
7732:{\displaystyle T}
7576:
7570:
7526:{\displaystyle T}
7506:{\displaystyle S}
7210:{\displaystyle T}
7188:{\displaystyle T}
7146:
7145:
7043:{\displaystyle T}
6925:{\displaystyle C}
6812:{\displaystyle X}
6792:{\displaystyle C}
6690:{\displaystyle S}
6670:{\displaystyle T}
6657:closed subset of
6560:subspace topology
6551:{\displaystyle T}
6499:the intersection
6393:
6387:
6355:{\displaystyle X}
6335:{\displaystyle S}
6315:{\displaystyle T}
6295:{\displaystyle T}
6275:{\displaystyle S}
6203:{\displaystyle X}
6188:subspace topology
6179:{\displaystyle T}
6159:{\displaystyle X}
6135:{\displaystyle T}
6077:{\displaystyle I}
6064:is possible when
6015:
5981:
5796:{\displaystyle X}
5780:is the empty set;
5723:{\displaystyle X}
5699:{\displaystyle S}
5626:
5541:
5509:{\displaystyle X}
5416:
5331:
5299:{\displaystyle X}
5174:
4900:{\displaystyle X}
4736:
4569:topological space
4493:{\displaystyle X}
4444:
4418:{\displaystyle X}
4376:{\displaystyle S}
4351:
4332:{\displaystyle S}
4312:{\displaystyle S}
4287:
4268:{\displaystyle S}
4222:{\displaystyle S}
4108:relative topology
4099:{\displaystyle X}
4007:{\displaystyle A}
3987:{\displaystyle X}
3742:discrete topology
3514:{\displaystyle S}
3325:{\displaystyle X}
3179:{\displaystyle X}
3124:{\displaystyle X}
2977:{\displaystyle X}
2771:topological space
2619:{\displaystyle A}
2599:{\displaystyle S}
2579:{\displaystyle A}
2559:{\displaystyle A}
2418:{\displaystyle S}
2395:{\displaystyle S}
2304:{\displaystyle x}
2284:{\displaystyle S}
2172:{\displaystyle S}
2020:{\displaystyle S}
2000:{\displaystyle S}
1980:{\displaystyle S}
1960:{\displaystyle S}
1947:. (Each point of
1936:{\displaystyle S}
1871:points of closure
1756:
1686:{\displaystyle X}
1539:topological space
1530:{\displaystyle S}
1490:{\displaystyle S}
1470:{\displaystyle x}
1450:{\displaystyle S}
1437:is an element of
1430:{\displaystyle x}
1410:{\displaystyle S}
1390:{\displaystyle x}
1348:{\displaystyle S}
1325:{\displaystyle x}
1305:{\displaystyle S}
1285:{\displaystyle x}
1265:{\displaystyle S}
1245:{\displaystyle S}
1225:{\displaystyle x}
1185:{\displaystyle S}
1158:{\displaystyle S}
1138:{\displaystyle S}
1118:{\displaystyle S}
1098:{\displaystyle S}
1078:{\displaystyle x}
1058:{\displaystyle x}
1038:{\displaystyle S}
1018:{\displaystyle x}
998:{\displaystyle x}
976:{\displaystyle x}
955:{\displaystyle S}
935:{\displaystyle x}
915:{\displaystyle S}
895:{\displaystyle x}
805:{\displaystyle S}
785:{\displaystyle x}
765:{\displaystyle S}
731:{\displaystyle x}
688:{\displaystyle S}
590:
555:{\displaystyle S}
535:{\displaystyle x}
396:{\displaystyle S}
376:{\displaystyle x}
334:{\displaystyle X}
288:{\displaystyle S}
265:{\displaystyle x}
245:{\displaystyle S}
225:{\displaystyle x}
201:{\displaystyle S}
181:{\displaystyle x}
164:as a subset of a
157:{\displaystyle S}
121:to the notion of
69:. The closure of
51:topological space
16:(Redirected from
12259:
12247:General topology
12224:
12223:
12197:
12196:
12187:
12177:
12167:
12166:
12155:
12154:
11949:
11862:
11855:
11848:
11839:
11838:
11834:
11809:
11806:Internet Archive
11773:
11766:(30 July 2002).
11759:
11750:
11749:, Academic Press
11741:
11732:
11721:
11710:
11692:
11681:
11670:
11643:
11637:
11631:
11625:
11619:
11605:
11599:
11593:
11587:
11581:
11575:
11569:
11563:
11557:
11540:
11539:
11537:
11536:
11531:
11515:
11509:
11502:
11501:
11484:
11478:
11459:
11458:
11447:
11441:
11430:
11428:
11427:
11422:
11393:
11392:
11376:
11374:
11373:
11368:
11345:it follows that
11344:
11342:
11341:
11336:
11306:
11304:
11303:
11298:
11264:
11249:
11222:
11199:universal arrows
11189:
11187:
11186:
11181:
11162:
11160:
11159:
11154:
11124:
11122:
11121:
11116:
11101:
11099:
11098:
11093:
11063:
11061:
11060:
11055:
11043:
11041:
11040:
11035:
11015:
11013:
11012:
11007:
10980:
10978:
10977:
10972:
10957:
10955:
10954:
10949:
10937:
10935:
10934:
10929:
10908:
10906:
10905:
10900:
10870:
10868:
10867:
10862:
10844:
10842:
10841:
10836:
10809:
10807:
10806:
10801:
10785:
10783:
10782:
10777:
10765:
10763:
10762:
10757:
10745:
10743:
10742:
10737:
10722:
10720:
10719:
10714:
10702:
10700:
10699:
10694:
10673:
10671:
10670:
10665:
10647:
10645:
10644:
10639:
10613:
10611:
10610:
10605:
10584:
10582:
10581:
10576:
10544:
10543:
10527:
10525:
10524:
10519:
10495:
10493:
10492:
10487:
10466:
10464:
10463:
10458:
10456:
10452:
10445:
10444:
10409:
10408:
10392:
10390:
10389:
10384:
10360:
10358:
10357:
10352:
10337:
10335:
10334:
10329:
10308:
10306:
10305:
10300:
10288:
10286:
10285:
10280:
10265:is a (strongly)
10264:
10262:
10261:
10256:
10219:
10217:
10216:
10211:
10187:
10185:
10184:
10179:
10158:
10156:
10155:
10150:
10129:
10127:
10126:
10121:
10109:
10107:
10106:
10101:
10083:
10081:
10080:
10075:
10062:
10060:
10059:
10054:
10030:
10028:
10027:
10022:
10010:
10008:
10007:
10002:
9991:
9989:
9988:
9983:
9962:
9960:
9959:
9954:
9938:
9936:
9935:
9930:
9919:
9918:
9896:
9894:
9893:
9888:
9866:
9864:
9863:
9858:
9846:
9844:
9843:
9838:
9823:
9821:
9820:
9815:
9794:
9792:
9791:
9786:
9766:
9764:
9763:
9758:
9737:
9735:
9734:
9729:
9711:
9709:
9708:
9703:
9677:
9676:
9665:
9659:
9658:
9654:
9647:
9646:
9623:
9621:
9620:
9615:
9594:
9592:
9591:
9586:
9560:
9558:
9557:
9552:
9537:
9535:
9534:
9529:
9517:
9515:
9514:
9509:
9497:
9495:
9494:
9489:
9478:
9477:
9450:
9448:
9447:
9442:
9400:
9398:
9397:
9392:
9387:
9386:
9367:
9365:
9364:
9359:
9347:
9345:
9344:
9339:
9321:
9319:
9318:
9313:
9301:
9299:
9298:
9293:
9281:
9279:
9278:
9273:
9257:
9255:
9254:
9249:
9247:
9246:
9233:
9231:
9230:
9225:
9209:
9207:
9206:
9201:
9189:
9187:
9186:
9181:
9163:
9161:
9160:
9155:
9143:
9141:
9140:
9135:
9117:
9115:
9114:
9109:
9093:
9091:
9090:
9085:
9083:
9082:
9065:
9063:
9062:
9057:
9045:
9043:
9042:
9037:
9021:
9019:
9018:
9013:
9011:
9010:
8991:
8989:
8988:
8983:
8981:
8980:
8961:
8959:
8958:
8953:
8945:
8944:
8908:
8907:
8891:
8889:
8888:
8883:
8863:
8862:
8852:
8851:
8850:
8824:
8823:
8807:
8805:
8804:
8799:
8781:
8779:
8778:
8773:
8757:
8755:
8754:
8749:
8747:
8746:
8726:
8724:
8723:
8718:
8707:
8705:
8704:
8699:
8688:
8687:
8657:
8656:
8634:
8632:
8631:
8626:
8597:
8596:
8574:
8572:
8571:
8566:
8554:
8552:
8551:
8546:
8511:
8510:
8494:
8492:
8491:
8486:
8460:
8459:
8425:
8423:
8422:
8417:
8405:
8403:
8402:
8397:
8379:
8377:
8376:
8371:
8359:
8357:
8356:
8351:
8321:
8319:
8318:
8313:
8301:
8299:
8298:
8293:
8239:
8237:
8236:
8231:
8217:is also open in
8216:
8214:
8213:
8208:
8190:
8188:
8187:
8182:
8170:
8168:
8167:
8162:
8150:
8148:
8147:
8142:
8127:
8125:
8124:
8119:
8101:
8099:
8098:
8093:
8063:
8061:
8060:
8055:
8035:
8034:
8012:
8010:
8009:
8004:
7981:
7980:
7958:
7956:
7955:
7950:
7935:
7933:
7932:
7927:
7916:and assume that
7915:
7913:
7912:
7907:
7873:
7872:
7868:
7866:
7865:
7860:
7848:
7846:
7845:
7840:
7828:
7826:
7825:
7820:
7812:
7811:
7775:
7774:
7758:
7756:
7755:
7750:
7738:
7736:
7735:
7730:
7718:
7716:
7715:
7710:
7680:
7678:
7677:
7672:
7636:
7634:
7633:
7628:
7626:
7608:
7606:
7605:
7600:
7592:
7591:
7574:
7568:
7549:
7548:
7532:
7530:
7529:
7524:
7512:
7510:
7509:
7504:
7492:
7490:
7489:
7484:
7458:
7456:
7455:
7450:
7417:
7415:
7414:
7409:
7377:
7375:
7374:
7369:
7364:
7346:
7344:
7343:
7338:
7327:
7326:
7310:
7308:
7307:
7302:
7294:
7293:
7269:
7268:
7252:
7250:
7249:
7244:
7233:
7232:
7216:
7214:
7213:
7208:
7194:
7192:
7191:
7186:
7174:
7172:
7171:
7166:
7149:It follows that
7141:
7139:
7138:
7133:
7122:
7120:
7119:
7114:
7103:
7102:
7072:
7071:
7049:
7047:
7046:
7041:
7029:
7027:
7026:
7021:
7004:
7003:
6987:
6985:
6984:
6979:
6971:
6970:
6954:
6952:
6951:
6946:
6931:
6929:
6928:
6923:
6911:
6909:
6908:
6903:
6889:
6888:
6866:
6864:
6863:
6858:
6835:
6834:
6818:
6816:
6815:
6810:
6798:
6796:
6795:
6790:
6778:
6776:
6775:
6770:
6752:
6750:
6749:
6744:
6729:
6727:
6726:
6721:
6713:
6712:
6696:
6694:
6693:
6688:
6676:
6674:
6673:
6668:
6652:
6650:
6649:
6644:
6636:
6635:
6619:
6617:
6616:
6611:
6603:
6602:
6578:
6577:
6557:
6555:
6554:
6549:
6537:
6535:
6534:
6529:
6521:
6520:
6498:
6496:
6495:
6490:
6475:
6473:
6472:
6467:
6459:
6458:
6432:
6431:
6428:
6426:
6425:
6420:
6409:
6408:
6391:
6385:
6378:
6377:
6361:
6359:
6358:
6353:
6341:
6339:
6338:
6333:
6321:
6319:
6318:
6313:
6301:
6299:
6298:
6293:
6281:
6279:
6278:
6273:
6261:
6259:
6258:
6253:
6245:
6244:
6226:
6225:
6209:
6207:
6206:
6201:
6185:
6183:
6182:
6177:
6165:
6163:
6162:
6157:
6141:
6139:
6138:
6133:
6121:
6119:
6118:
6113:
6083:
6081:
6080:
6075:
6063:
6061:
6060:
6055:
6053:
6052:
6040:
6039:
6029:
6011:
6007:
6006:
6005:
5995:
5972:
5971:
5955:
5953:
5952:
5947:
5933:
5932:
5908:
5907:
5874:
5873:
5825:
5823:
5822:
5817:
5802:
5800:
5799:
5794:
5771:
5769:
5768:
5763:
5746:
5745:
5729:
5727:
5726:
5721:
5705:
5703:
5702:
5697:
5676:
5674:
5673:
5668:
5663:
5659:
5658:
5654:
5653:
5652:
5642:
5641:
5617:
5616:
5599:
5598:
5586:
5582:
5581:
5580:
5568:
5567:
5557:
5556:
5532:
5531:
5515:
5513:
5512:
5507:
5495:
5493:
5492:
5487:
5485:
5484:
5466:
5464:
5463:
5458:
5453:
5449:
5448:
5444:
5443:
5442:
5432:
5431:
5407:
5406:
5389:
5388:
5376:
5372:
5371:
5370:
5358:
5357:
5347:
5346:
5322:
5321:
5305:
5303:
5302:
5297:
5285:
5283:
5282:
5277:
5275:
5274:
5255:
5253:
5252:
5247:
5229:
5227:
5226:
5221:
5213:
5212:
5200:
5199:
5183:
5180:
5163:
5161:
5160:
5155:
5133:
5131:
5130:
5125:
5102:
5101:
5077:
5076:
5054:
5052:
5051:
5046:
5023:
5022:
4998:
4997:
4978:
4976:
4975:
4970:
4965:
4964:
4940:
4938:
4937:
4932:
4930:
4929:
4906:
4904:
4903:
4898:
4886:
4884:
4883:
4878:
4861:
4849:
4847:
4846:
4841:
4819:
4817:
4816:
4811:
4796:
4794:
4793:
4788:
4786:
4774:
4772:
4771:
4766:
4764:
4763:
4747:
4745:
4744:
4739:
4737:
4729:
4720:
4718:
4717:
4712:
4701:
4700:
4684:
4682:
4681:
4676:
4658:
4656:
4655:
4650:
4618:
4617:
4601:
4599:
4598:
4593:
4562:
4560:
4559:
4554:
4543:
4542:
4530:
4528:
4527:
4522:
4499:
4497:
4496:
4491:
4477:closure operator
4466:Closure operator
4460:Closure operator
4455:
4453:
4452:
4447:
4445:
4440:
4424:
4422:
4421:
4416:
4404:
4402:
4401:
4396:
4394:
4382:
4380:
4379:
4374:
4362:
4360:
4359:
4354:
4352:
4347:
4338:
4336:
4335:
4330:
4318:
4316:
4315:
4310:
4298:
4296:
4295:
4290:
4288:
4283:
4274:
4272:
4271:
4266:
4255:because neither
4254:
4252:
4251:
4246:
4244:
4228:
4226:
4225:
4220:
4208:
4206:
4205:
4200:
4174:
4173:
4161:
4134:
4132:
4131:
4126:
4121:
4105:
4103:
4102:
4097:
4077:
4075:
4074:
4069:
4052:
4051:
4036:
4034:
4033:
4028:
4013:
4011:
4010:
4005:
3993:
3991:
3990:
3985:
3973:
3971:
3970:
3965:
3950:indiscrete space
3933:
3931:
3930:
3925:
3920:
3888:
3887:
3871:
3869:
3868:
3863:
3861:
3848:trivial topology
3845:
3843:
3842:
3837:
3835:
3815:
3813:
3812:
3807:
3760:
3759:
3739:
3737:
3736:
3731:
3729:
3709:
3707:
3706:
3701:
3654:
3653:
3633:
3631:
3630:
3625:
3623:
3589:
3587:
3586:
3581:
3564:
3563:
3547:
3545:
3544:
3539:
3520:
3518:
3517:
3512:
3498:
3496:
3495:
3490:
3476:
3468:
3460:
3443:
3439:
3429:
3421:
3413:
3391:
3390:
3374:
3372:
3371:
3366:
3361:
3360:
3355:
3346:
3331:
3329:
3328:
3323:
3309:
3307:
3306:
3301:
3296:
3280:
3278:
3277:
3272:
3270:
3258:
3256:
3255:
3250:
3245:
3232:rational numbers
3229:
3227:
3226:
3221:
3219:
3207:
3205:
3204:
3199:
3197:
3185:
3183:
3182:
3177:
3162:
3160:
3159:
3156:{\displaystyle }
3154:
3130:
3128:
3127:
3122:
3110:
3108:
3107:
3102:
3078:
3076:
3075:
3070:
3026:
3025:
3005:
3003:
3002:
2997:
2995:
2983:
2981:
2980:
2975:
2955:
2953:
2952:
2947:
2945:
2933:
2931:
2930:
2925:
2923:
2908:
2906:
2905:
2900:
2874:
2872:
2871:
2866:
2848:
2846:
2845:
2840:
2828:
2826:
2825:
2820:
2808:
2806:
2805:
2800:
2748:closure operator
2729:
2727:
2726:
2721:
2698:
2696:
2695:
2690:
2654:
2652:
2651:
2646:
2625:
2623:
2622:
2617:
2605:
2603:
2602:
2597:
2585:
2583:
2582:
2577:
2565:
2563:
2562:
2557:
2543:
2541:
2540:
2535:
2514:
2512:
2511:
2506:
2488:
2486:
2485:
2480:
2459:
2457:
2456:
2451:
2424:
2422:
2421:
2416:
2401:
2399:
2398:
2393:
2377:
2375:
2374:
2369:
2345:
2343:
2342:
2337:
2310:
2308:
2307:
2302:
2290:
2288:
2287:
2282:
2266:
2264:
2263:
2258:
2240:
2238:
2237:
2232:
2213:
2211:
2210:
2205:
2178:
2176:
2175:
2170:
2159:is the union of
2158:
2156:
2155:
2150:
2131:
2129:
2128:
2123:
2108:
2106:
2105:
2100:
2081:
2079:
2078:
2073:
2054:
2052:
2051:
2046:
2026:
2024:
2023:
2018:
2006:
2004:
2003:
1998:
1986:
1984:
1983:
1978:
1966:
1964:
1963:
1958:
1942:
1940:
1939:
1934:
1922:
1920:
1919:
1914:
1895:
1893:
1892:
1887:
1868:
1866:
1865:
1860:
1839:
1837:
1836:
1831:
1819:
1817:
1816:
1811:
1799:
1797:
1796:
1791:
1789:
1788:
1783:
1770:
1768:
1767:
1762:
1757:
1749:
1741:
1739:
1738:
1733:
1712:
1710:
1709:
1704:
1692:
1690:
1689:
1684:
1672:
1670:
1669:
1664:
1652:
1650:
1649:
1644:
1636:
1635:
1619:
1617:
1616:
1611:
1603:
1602:
1574:
1572:
1571:
1566:
1536:
1534:
1533:
1528:
1501:Closure of a set
1496:
1494:
1493:
1488:
1476:
1474:
1473:
1468:
1456:
1454:
1453:
1448:
1436:
1434:
1433:
1428:
1416:
1414:
1413:
1408:
1396:
1394:
1393:
1388:
1377:
1375:
1374:
1369:
1354:
1352:
1351:
1346:
1335:For a given set
1331:
1329:
1328:
1323:
1311:
1309:
1308:
1303:
1291:
1289:
1288:
1283:
1271:
1269:
1268:
1263:
1251:
1249:
1248:
1243:
1231:
1229:
1228:
1223:
1191:
1189:
1188:
1183:
1164:
1162:
1161:
1156:
1144:
1142:
1141:
1136:
1124:
1122:
1121:
1116:
1104:
1102:
1101:
1096:
1084:
1082:
1081:
1076:
1064:
1062:
1061:
1056:
1044:
1042:
1041:
1036:
1024:
1022:
1021:
1016:
1004:
1002:
1001:
996:
982:
980:
979:
974:
961:
959:
958:
953:
941:
939:
938:
933:
921:
919:
918:
913:
901:
899:
898:
893:
863:
861:
860:
855:
837:
835:
834:
829:
811:
809:
808:
803:
791:
789:
788:
783:
771:
769:
768:
763:
742:point of closure
737:
735:
734:
729:
717:
715:
714:
709:
694:
692:
691:
686:
659:
657:
656:
651:
639:
637:
636:
631:
604:
562:if the distance
561:
559:
558:
553:
541:
539:
538:
533:
521:
519:
518:
513:
495:
493:
492:
487:
454:
452:
451:
446:
428:
426:
425:
420:
402:
400:
399:
394:
382:
380:
379:
374:
363:
361:
360:
355:
340:
338:
337:
332:
320:
318:
317:
312:
294:
292:
291:
286:
271:
269:
268:
263:
251:
249:
248:
243:
231:
229:
228:
223:
207:
205:
204:
199:
187:
185:
184:
179:
163:
161:
160:
155:
134:Point of closure
116:
111:point of closure
108:
104:
100:
96:
80:
72:
68:
60:
53:consists of all
48:
21:
12267:
12266:
12262:
12261:
12260:
12258:
12257:
12256:
12237:
12236:
12235:
12230:
12161:
12143:
12139:Urysohn's lemma
12100:
12064:
11950:
11941:
11913:low-dimensional
11871:
11866:
11819:
11816:
11786:
11730:
11708:
11690:
11668:
11652:
11647:
11646:
11638:
11634:
11626:
11622:
11606:
11602:
11594:
11590:
11582:
11578:
11572:Kuratowski 1966
11570:
11566:
11558:
11554:
11549:
11544:
11543:
11497:
11493:
11454:
11450:
11436:
11433:
11432:
11431:which implies
11388:
11384:
11382:
11379:
11378:
11350:
11347:
11346:
11312:
11309:
11308:
11271:
11268:
11267:
11265:
11261:
11256:
11247:
11220:
11217:Closure algebra
11207:
11172:
11169:
11168:
11133:
11130:
11129:
11127:terminal object
11107:
11104:
11103:
11069:
11066:
11065:
11049:
11046:
11045:
11023:
11020:
11019:
10986:
10983:
10982:
10963:
10960:
10959:
10943:
10940:
10939:
10914:
10911:
10910:
10879:
10876:
10875:
10850:
10847:
10846:
10815:
10812:
10811:
10795:
10792:
10791:
10771:
10768:
10767:
10751:
10748:
10747:
10728:
10725:
10724:
10723:is a subset of
10708:
10705:
10704:
10682:
10679:
10678:
10659:
10656:
10655:
10633:
10630:
10629:
10619:
10590:
10587:
10586:
10539:
10535:
10533:
10530:
10529:
10501:
10498:
10497:
10472:
10469:
10468:
10440:
10436:
10435:
10431:
10404:
10400:
10398:
10395:
10394:
10366:
10363:
10362:
10343:
10340:
10339:
10314:
10311:
10310:
10294:
10291:
10290:
10274:
10271:
10270:
10238:
10235:
10234:
10231:
10225:
10193:
10190:
10189:
10164:
10161:
10160:
10135:
10132:
10131:
10115:
10112:
10111:
10089:
10086:
10085:
10069:
10066:
10065:
10036:
10033:
10032:
10016:
10013:
10012:
9996:
9993:
9992:
9968:
9965:
9964:
9948:
9945:
9944:
9914:
9910:
9902:
9899:
9898:
9876:
9873:
9872:
9852:
9849:
9848:
9829:
9826:
9825:
9800:
9797:
9796:
9771:
9768:
9767:
9743:
9740:
9739:
9717:
9714:
9713:
9672:
9668:
9642:
9638:
9637:
9633:
9628:
9625:
9624:
9600:
9597:
9596:
9568:
9565:
9564:
9543:
9540:
9539:
9523:
9520:
9519:
9503:
9500:
9499:
9470:
9466:
9464:
9461:
9460:
9424:
9421:
9420:
9417:
9411:
9406:
9382:
9381:
9373:
9370:
9369:
9353:
9350:
9349:
9327:
9324:
9323:
9322:if and only if
9307:
9304:
9303:
9287:
9284:
9283:
9267:
9264:
9263:
9242:
9241:
9239:
9236:
9235:
9219:
9216:
9215:
9195:
9192:
9191:
9169:
9166:
9165:
9149:
9146:
9145:
9123:
9120:
9119:
9103:
9100:
9099:
9094:are domains of
9078:
9077:
9075:
9072:
9071:
9051:
9048:
9047:
9031:
9028:
9027:
9006:
9005:
8997:
8994:
8993:
8976:
8975:
8967:
8964:
8963:
8940:
8936:
8903:
8899:
8897:
8894:
8893:
8858:
8854:
8846:
8845:
8838:
8819:
8815:
8813:
8810:
8809:
8787:
8784:
8783:
8767:
8764:
8763:
8742:
8741:
8739:
8736:
8735:
8732:
8712:
8709:
8708:
8683:
8679:
8652:
8648:
8640:
8637:
8636:
8592:
8588:
8580:
8577:
8576:
8560:
8557:
8556:
8506:
8502:
8500:
8497:
8496:
8455:
8451:
8431:
8428:
8427:
8411:
8408:
8407:
8385:
8382:
8381:
8365:
8362:
8361:
8327:
8324:
8323:
8307:
8304:
8303:
8245:
8242:
8241:
8222:
8219:
8218:
8196:
8193:
8192:
8176:
8173:
8172:
8156:
8153:
8152:
8133:
8130:
8129:
8107:
8104:
8103:
8069:
8066:
8065:
8030:
8026:
8018:
8015:
8014:
7976:
7972:
7964:
7961:
7960:
7941:
7938:
7937:
7921:
7918:
7917:
7889:
7886:
7885:
7878:
7854:
7851:
7850:
7834:
7831:
7830:
7807:
7803:
7770:
7766:
7764:
7761:
7760:
7744:
7741:
7740:
7724:
7721:
7720:
7719:), although if
7686:
7683:
7682:
7642:
7639:
7638:
7622:
7614:
7611:
7610:
7587:
7583:
7544:
7540:
7538:
7535:
7534:
7518:
7515:
7514:
7498:
7495:
7494:
7466:
7463:
7462:
7423:
7420:
7419:
7382:
7379:
7378:
7360:
7352:
7349:
7348:
7322:
7318:
7316:
7313:
7312:
7289:
7285:
7264:
7260:
7258:
7255:
7254:
7228:
7224:
7222:
7219:
7218:
7217:is a subset of
7202:
7199:
7198:
7180:
7177:
7176:
7154:
7151:
7150:
7147:
7127:
7124:
7123:
7098:
7094:
7067:
7063:
7055:
7052:
7051:
7035:
7032:
7031:
6999:
6995:
6993:
6990:
6989:
6966:
6962:
6960:
6957:
6956:
6937:
6934:
6933:
6917:
6914:
6913:
6884:
6880:
6872:
6869:
6868:
6830:
6826:
6824:
6821:
6820:
6804:
6801:
6800:
6784:
6781:
6780:
6758:
6755:
6754:
6735:
6732:
6731:
6708:
6704:
6702:
6699:
6698:
6682:
6679:
6678:
6662:
6659:
6658:
6631:
6627:
6625:
6622:
6621:
6598:
6594:
6573:
6569:
6567:
6564:
6563:
6543:
6540:
6539:
6516:
6512:
6504:
6501:
6500:
6481:
6478:
6477:
6454:
6450:
6448:
6445:
6444:
6437:
6404:
6400:
6373:
6369:
6367:
6364:
6363:
6347:
6344:
6343:
6327:
6324:
6323:
6307:
6304:
6303:
6287:
6284:
6283:
6267:
6264:
6263:
6240:
6236:
6221:
6217:
6215:
6212:
6211:
6195:
6192:
6191:
6171:
6168:
6167:
6151:
6148:
6147:
6127:
6124:
6123:
6095:
6092:
6091:
6069:
6066:
6065:
6048:
6044:
6035:
6031:
6019:
6001:
5997:
5985:
5980:
5976:
5967:
5963:
5961:
5958:
5957:
5928:
5924:
5903:
5899:
5869:
5865:
5863:
5860:
5859:
5808:
5805:
5804:
5788:
5785:
5784:
5783:The closure of
5772:In particular:
5741:
5737:
5735:
5732:
5731:
5730:if and only if
5715:
5712:
5711:
5691:
5688:
5687:
5684:
5679:
5648:
5644:
5637:
5630:
5625:
5621:
5612:
5608:
5607:
5603:
5594:
5590:
5576:
5572:
5563:
5559:
5552:
5545:
5540:
5536:
5527:
5523:
5521:
5518:
5517:
5501:
5498:
5497:
5480:
5476:
5474:
5471:
5470:
5438:
5434:
5427:
5420:
5415:
5411:
5402:
5398:
5397:
5393:
5384:
5380:
5366:
5362:
5353:
5349:
5342:
5335:
5330:
5326:
5317:
5313:
5311:
5308:
5307:
5291:
5288:
5287:
5270:
5266:
5264:
5261:
5260:
5238:
5235:
5234:
5208:
5204:
5195:
5191:
5189:
5186:
5185:
5181:
5178:
5146:
5143:
5142:
5097:
5093:
5072:
5068:
5066:
5063:
5062:
5018:
5014:
4993:
4989:
4987:
4984:
4983:
4960:
4956:
4954:
4951:
4950:
4925:
4921:
4919:
4916:
4915:
4892:
4889:
4888:
4857:
4855:
4852:
4851:
4829:
4826:
4825:
4802:
4799:
4798:
4782:
4780:
4777:
4776:
4759:
4755:
4753:
4750:
4749:
4728:
4726:
4723:
4722:
4696:
4692:
4690:
4687:
4686:
4664:
4661:
4660:
4613:
4609:
4607:
4604:
4603:
4575:
4572:
4571:
4538:
4537:
4535:
4532:
4531:
4513:
4510:
4509:
4485:
4482:
4481:
4472:
4462:
4439:
4437:
4434:
4433:
4410:
4407:
4406:
4390:
4388:
4385:
4384:
4368:
4365:
4364:
4346:
4344:
4341:
4340:
4324:
4321:
4320:
4304:
4301:
4300:
4282:
4280:
4277:
4276:
4260:
4257:
4256:
4240:
4238:
4235:
4234:
4214:
4211:
4210:
4169:
4165:
4157:
4140:
4137:
4136:
4117:
4115:
4112:
4111:
4091:
4088:
4087:
4047:
4043:
4041:
4038:
4037:
4019:
4016:
4015:
3999:
3996:
3995:
3979:
3976:
3975:
3956:
3953:
3952:
3916:
3883:
3879:
3877:
3874:
3873:
3857:
3855:
3852:
3851:
3831:
3823:
3820:
3819:
3755:
3751:
3749:
3746:
3745:
3725:
3717:
3714:
3713:
3649:
3645:
3643:
3640:
3639:
3619:
3611:
3608:
3607:
3595:
3559:
3555:
3553:
3550:
3549:
3530:
3527:
3526:
3506:
3503:
3502:
3472:
3464:
3456:
3425:
3417:
3409:
3399:
3395:
3386:
3382:
3380:
3377:
3376:
3356:
3351:
3350:
3342:
3340:
3337:
3336:
3317:
3314:
3313:
3292:
3290:
3287:
3286:
3266:
3264:
3261:
3260:
3241:
3239:
3236:
3235:
3215:
3213:
3210:
3209:
3193:
3191:
3188:
3187:
3171:
3168:
3167:
3136:
3133:
3132:
3116:
3113:
3112:
3111:as a subset of
3084:
3081:
3080:
3021:
3017:
3015:
3012:
3011:
2991:
2989:
2986:
2985:
2969:
2966:
2965:
2941:
2939:
2936:
2935:
2919:
2917:
2914:
2913:
2879:
2876:
2875:
2857:
2854:
2853:
2834:
2831:
2830:
2814:
2811:
2810:
2782:
2779:
2778:
2756:
2712:
2709:
2708:
2678:
2675:
2674:
2631:
2628:
2627:
2611:
2608:
2607:
2606:if and only if
2591:
2588:
2587:
2571:
2568:
2567:
2551:
2548:
2547:
2520:
2517:
2516:
2515:is a subset of
2494:
2491:
2490:
2468:
2465:
2464:
2433:
2430:
2429:
2410:
2407:
2406:
2387:
2384:
2383:
2357:
2354:
2353:
2316:
2313:
2312:
2296:
2293:
2292:
2276:
2273:
2272:
2246:
2243:
2242:
2220:
2217:
2216:
2187:
2184:
2183:
2164:
2161:
2160:
2138:
2135:
2134:
2114:
2111:
2110:
2088:
2085:
2084:
2064:
2061:
2060:
2034:
2031:
2030:
2012:
2009:
2008:
1992:
1989:
1988:
1972:
1969:
1968:
1952:
1949:
1948:
1928:
1925:
1924:
1902:
1899:
1898:
1878:
1875:
1874:
1848:
1845:
1844:
1825:
1822:
1821:
1805:
1802:
1801:
1784:
1782:
1781:
1776:
1773:
1772:
1748:
1746:
1743:
1742:
1718:
1715:
1714:
1698:
1695:
1694:
1678:
1675:
1674:
1658:
1655:
1654:
1631:
1627:
1625:
1622:
1621:
1620:or possibly by
1586:
1582:
1580:
1577:
1576:
1545:
1542:
1541:
1522:
1519:
1518:
1509:
1503:
1482:
1479:
1478:
1462:
1459:
1458:
1442:
1439:
1438:
1422:
1419:
1418:
1417:if and only if
1402:
1399:
1398:
1382:
1379:
1378:
1360:
1357:
1356:
1340:
1337:
1336:
1317:
1314:
1313:
1297:
1294:
1293:
1277:
1274:
1273:
1257:
1254:
1253:
1237:
1234:
1233:
1217:
1214:
1213:
1177:
1174:
1173:
1150:
1147:
1146:
1130:
1127:
1126:
1110:
1107:
1106:
1090:
1087:
1086:
1070:
1067:
1066:
1050:
1047:
1046:
1030:
1027:
1026:
1010:
1007:
1006:
990:
987:
986:
968:
965:
964:
947:
944:
943:
927:
924:
923:
907:
904:
903:
887:
884:
883:
876:
870:
843:
840:
839:
817:
814:
813:
797:
794:
793:
777:
774:
773:
757:
754:
753:
723:
720:
719:
700:
697:
696:
680:
677:
676:
645:
642:
641:
594:
567:
564:
563:
547:
544:
543:
527:
524:
523:
501:
498:
497:
460:
457:
456:
434:
431:
430:
408:
405:
404:
388:
385:
384:
368:
365:
364:
346:
343:
342:
326:
323:
322:
303:
300:
299:
280:
277:
276:
257:
254:
253:
237:
234:
233:
217:
214:
213:
193:
190:
189:
173:
170:
169:
166:Euclidean space
149:
146:
145:
142:
136:
131:
114:
106:
102:
101:or "very near"
98:
94:
78:
70:
66:
58:
49:of points in a
46:
35:
28:
23:
22:
15:
12:
11:
5:
12265:
12255:
12254:
12249:
12232:
12231:
12229:
12228:
12218:
12217:
12216:
12211:
12206:
12191:
12181:
12171:
12159:
12148:
12145:
12144:
12142:
12141:
12136:
12131:
12126:
12121:
12116:
12110:
12108:
12102:
12101:
12099:
12098:
12093:
12088:
12086:Winding number
12083:
12078:
12072:
12070:
12066:
12065:
12063:
12062:
12057:
12052:
12047:
12042:
12037:
12032:
12027:
12026:
12025:
12020:
12018:homotopy group
12010:
12009:
12008:
12003:
11998:
11993:
11988:
11978:
11973:
11968:
11958:
11956:
11952:
11951:
11944:
11942:
11940:
11939:
11934:
11929:
11928:
11927:
11917:
11916:
11915:
11905:
11900:
11895:
11890:
11885:
11879:
11877:
11873:
11872:
11865:
11864:
11857:
11850:
11842:
11836:
11835:
11815:
11814:External links
11812:
11811:
11810:
11784:
11760:
11751:
11742:
11733:
11728:
11711:
11706:
11693:
11688:
11671:
11666:
11651:
11648:
11645:
11644:
11640:Zălinescu 2002
11632:
11620:
11608:Gemignani 1990
11600:
11588:
11576:
11564:
11551:
11550:
11548:
11545:
11542:
11541:
11529:
11526:
11523:
11520:
11514:
11508:
11505:
11500:
11496:
11492:
11489:
11483:
11477:
11474:
11471:
11468:
11465:
11462:
11457:
11453:
11446:
11440:
11420:
11417:
11414:
11411:
11408:
11405:
11402:
11399:
11396:
11391:
11387:
11366:
11363:
11360:
11357:
11354:
11334:
11331:
11328:
11325:
11322:
11319:
11316:
11296:
11293:
11290:
11287:
11284:
11281:
11278:
11275:
11258:
11257:
11255:
11252:
11251:
11250:
11241:
11235:
11229:
11223:
11214:
11211:Adherent point
11206:
11203:
11179:
11176:
11152:
11149:
11146:
11143:
11140:
11137:
11114:
11111:
11091:
11088:
11085:
11082:
11079:
11076:
11073:
11053:
11033:
11030:
11027:
11005:
11002:
10999:
10996:
10993:
10990:
10970:
10967:
10947:
10927:
10924:
10921:
10918:
10898:
10895:
10892:
10889:
10886:
10883:
10873:comma category
10860:
10857:
10854:
10834:
10831:
10828:
10825:
10822:
10819:
10799:
10775:
10755:
10735:
10732:
10712:
10692:
10689:
10686:
10676:inclusion maps
10663:
10637:
10618:
10615:
10603:
10600:
10597:
10594:
10574:
10571:
10568:
10565:
10562:
10559:
10556:
10553:
10550:
10547:
10542:
10538:
10517:
10514:
10511:
10508:
10505:
10496:Equivalently,
10485:
10482:
10479:
10476:
10455:
10451:
10448:
10443:
10439:
10434:
10430:
10427:
10424:
10421:
10418:
10415:
10412:
10407:
10403:
10382:
10379:
10376:
10373:
10370:
10350:
10347:
10327:
10324:
10321:
10318:
10298:
10278:
10254:
10251:
10248:
10245:
10242:
10227:Main article:
10224:
10221:
10209:
10206:
10203:
10200:
10197:
10177:
10174:
10171:
10168:
10148:
10145:
10142:
10139:
10119:
10099:
10096:
10093:
10073:
10052:
10049:
10046:
10043:
10040:
10020:
10000:
9981:
9978:
9975:
9972:
9952:
9928:
9925:
9922:
9917:
9913:
9909:
9906:
9886:
9883:
9880:
9870:
9856:
9836:
9833:
9813:
9810:
9807:
9804:
9784:
9781:
9778:
9775:
9756:
9753:
9750:
9747:
9727:
9724:
9721:
9701:
9698:
9695:
9692:
9689:
9686:
9683:
9680:
9675:
9671:
9664:
9657:
9653:
9650:
9645:
9641:
9636:
9632:
9613:
9610:
9607:
9604:
9584:
9581:
9578:
9575:
9572:
9550:
9547:
9527:
9507:
9487:
9484:
9481:
9476:
9473:
9469:
9440:
9437:
9434:
9431:
9428:
9413:Main article:
9410:
9407:
9405:
9402:
9390:
9385:
9380:
9377:
9357:
9337:
9334:
9331:
9311:
9291:
9271:
9245:
9223:
9212:locally closed
9199:
9179:
9176:
9173:
9153:
9133:
9130:
9127:
9118:of any subset
9107:
9081:
9055:
9035:
9009:
9004:
9001:
8979:
8974:
8971:
8951:
8948:
8943:
8939:
8935:
8932:
8929:
8926:
8923:
8920:
8917:
8914:
8911:
8906:
8902:
8881:
8878:
8875:
8872:
8869:
8866:
8861:
8857:
8849:
8844:
8841:
8837:
8833:
8830:
8827:
8822:
8818:
8797:
8794:
8791:
8771:
8745:
8729:
8728:
8716:
8697:
8694:
8691:
8686:
8682:
8678:
8675:
8672:
8669:
8666:
8663:
8660:
8655:
8651:
8647:
8644:
8624:
8621:
8618:
8615:
8612:
8609:
8606:
8603:
8600:
8595:
8591:
8587:
8584:
8564:
8544:
8541:
8538:
8535:
8532:
8529:
8526:
8523:
8520:
8517:
8514:
8509:
8505:
8484:
8481:
8478:
8475:
8472:
8469:
8466:
8463:
8458:
8454:
8450:
8447:
8444:
8441:
8438:
8435:
8415:
8395:
8392:
8389:
8369:
8349:
8346:
8343:
8340:
8337:
8334:
8331:
8311:
8291:
8288:
8285:
8282:
8279:
8276:
8273:
8270:
8267:
8264:
8261:
8258:
8255:
8252:
8249:
8229:
8226:
8206:
8203:
8200:
8180:
8171:being open in
8160:
8140:
8137:
8117:
8114:
8111:
8091:
8088:
8085:
8082:
8079:
8076:
8073:
8053:
8050:
8047:
8044:
8041:
8038:
8033:
8029:
8025:
8022:
8002:
7999:
7996:
7993:
7990:
7987:
7984:
7979:
7975:
7971:
7968:
7948:
7945:
7925:
7905:
7902:
7899:
7896:
7893:
7880:
7879:
7876:
7871:
7858:
7838:
7818:
7815:
7810:
7806:
7802:
7799:
7796:
7793:
7790:
7787:
7784:
7781:
7778:
7773:
7769:
7748:
7728:
7708:
7705:
7702:
7699:
7696:
7693:
7690:
7670:
7667:
7664:
7661:
7658:
7655:
7652:
7649:
7646:
7625:
7621:
7618:
7598:
7595:
7590:
7586:
7582:
7579:
7573:
7567:
7564:
7561:
7558:
7555:
7552:
7547:
7543:
7522:
7502:
7482:
7479:
7476:
7473:
7470:
7448:
7445:
7442:
7439:
7436:
7433:
7430:
7427:
7407:
7404:
7401:
7398:
7395:
7392:
7389:
7386:
7367:
7363:
7359:
7356:
7336:
7333:
7330:
7325:
7321:
7300:
7297:
7292:
7288:
7284:
7281:
7278:
7275:
7272:
7267:
7263:
7242:
7239:
7236:
7231:
7227:
7206:
7196:if and only if
7184:
7164:
7161:
7158:
7144:
7143:
7131:
7112:
7109:
7106:
7101:
7097:
7093:
7090:
7087:
7084:
7081:
7078:
7075:
7070:
7066:
7062:
7059:
7039:
7019:
7016:
7013:
7010:
7007:
7002:
6998:
6977:
6974:
6969:
6965:
6944:
6941:
6921:
6901:
6898:
6895:
6892:
6887:
6883:
6879:
6876:
6856:
6853:
6850:
6847:
6844:
6841:
6838:
6833:
6829:
6808:
6788:
6768:
6765:
6762:
6742:
6739:
6719:
6716:
6711:
6707:
6686:
6666:
6656:
6642:
6639:
6634:
6630:
6609:
6606:
6601:
6597:
6593:
6590:
6587:
6584:
6581:
6576:
6572:
6547:
6527:
6524:
6519:
6515:
6511:
6508:
6488:
6485:
6465:
6462:
6457:
6453:
6439:
6438:
6435:
6430:
6418:
6415:
6412:
6407:
6403:
6399:
6396:
6390:
6384:
6381:
6376:
6372:
6351:
6331:
6311:
6291:
6271:
6251:
6248:
6243:
6239:
6235:
6232:
6229:
6224:
6220:
6199:
6175:
6166:(meaning that
6155:
6131:
6111:
6108:
6105:
6102:
6099:
6088:
6087:
6086:
6085:
6073:
6051:
6047:
6043:
6038:
6034:
6028:
6025:
6022:
6018:
6014:
6010:
6004:
6000:
5994:
5991:
5988:
5984:
5979:
5975:
5970:
5966:
5945:
5942:
5939:
5936:
5931:
5927:
5923:
5920:
5917:
5914:
5911:
5906:
5902:
5898:
5895:
5892:
5889:
5886:
5883:
5880:
5877:
5872:
5868:
5849:
5837:
5826:
5815:
5812:
5792:
5781:
5761:
5758:
5755:
5752:
5749:
5744:
5740:
5719:
5695:
5683:
5680:
5678:
5677:
5666:
5662:
5657:
5651:
5647:
5640:
5636:
5633:
5629:
5624:
5620:
5615:
5611:
5606:
5602:
5597:
5593:
5589:
5585:
5579:
5575:
5571:
5566:
5562:
5555:
5551:
5548:
5544:
5539:
5535:
5530:
5526:
5505:
5483:
5479:
5467:
5456:
5452:
5447:
5441:
5437:
5430:
5426:
5423:
5419:
5414:
5410:
5405:
5401:
5396:
5392:
5387:
5383:
5379:
5375:
5369:
5365:
5361:
5356:
5352:
5345:
5341:
5338:
5334:
5329:
5325:
5320:
5316:
5295:
5273:
5269:
5245:
5242:
5219:
5216:
5211:
5207:
5203:
5198:
5194:
5172:
5153:
5150:
5135:
5134:
5123:
5120:
5117:
5114:
5111:
5108:
5105:
5100:
5096:
5092:
5089:
5086:
5083:
5080:
5075:
5071:
5056:
5055:
5044:
5041:
5038:
5035:
5032:
5029:
5026:
5021:
5017:
5013:
5010:
5007:
5004:
5001:
4996:
4992:
4968:
4963:
4959:
4928:
4924:
4896:
4876:
4873:
4870:
4867:
4864:
4860:
4839:
4836:
4833:
4809:
4806:
4785:
4762:
4758:
4735:
4732:
4710:
4707:
4704:
4699:
4695:
4674:
4671:
4668:
4648:
4645:
4642:
4639:
4636:
4633:
4630:
4627:
4624:
4621:
4616:
4612:
4591:
4588:
4585:
4582:
4579:
4552:
4549:
4546:
4541:
4520:
4517:
4489:
4479:
4461:
4458:
4443:
4432:
4428:
4414:
4393:
4372:
4350:
4328:
4308:
4286:
4264:
4243:
4218:
4198:
4195:
4192:
4189:
4186:
4183:
4180:
4177:
4172:
4168:
4164:
4160:
4156:
4153:
4150:
4147:
4144:
4124:
4120:
4095:
4084:
4083:
4067:
4064:
4061:
4058:
4055:
4050:
4046:
4026:
4023:
4003:
3983:
3963:
3960:
3946:
3943:discrete space
3935:
3934:
3923:
3919:
3915:
3912:
3909:
3906:
3903:
3900:
3897:
3894:
3891:
3886:
3882:
3860:
3834:
3830:
3827:
3816:
3805:
3802:
3799:
3796:
3793:
3790:
3787:
3784:
3781:
3778:
3775:
3772:
3769:
3766:
3763:
3758:
3754:
3728:
3724:
3721:
3710:
3699:
3696:
3693:
3690:
3687:
3684:
3681:
3678:
3675:
3672:
3669:
3666:
3663:
3660:
3657:
3652:
3648:
3622:
3618:
3615:
3600:
3599:
3593:
3579:
3576:
3573:
3570:
3567:
3562:
3558:
3537:
3534:
3510:
3499:
3488:
3485:
3482:
3479:
3475:
3471:
3467:
3463:
3459:
3455:
3452:
3449:
3446:
3442:
3438:
3435:
3432:
3428:
3424:
3420:
3416:
3412:
3408:
3405:
3402:
3398:
3394:
3389:
3385:
3364:
3359:
3354:
3349:
3345:
3321:
3310:
3299:
3295:
3269:
3248:
3244:
3218:
3196:
3175:
3164:
3152:
3149:
3146:
3143:
3140:
3120:
3100:
3097:
3094:
3091:
3088:
3068:
3065:
3062:
3059:
3056:
3053:
3050:
3047:
3044:
3041:
3038:
3035:
3032:
3029:
3024:
3020:
2994:
2973:
2944:
2922:
2910:
2909:
2898:
2895:
2892:
2889:
2886:
2883:
2864:
2861:
2850:
2838:
2818:
2798:
2795:
2792:
2789:
2786:
2777:In any space,
2755:
2752:
2719:
2716:
2688:
2685:
2682:
2661:
2656:
2655:
2644:
2641:
2638:
2635:
2615:
2595:
2575:
2555:
2544:
2533:
2530:
2527:
2524:
2504:
2501:
2498:
2478:
2475:
2472:
2461:
2449:
2446:
2443:
2440:
2437:
2427:if and only if
2414:
2403:
2391:
2367:
2364:
2361:
2347:
2346:
2335:
2332:
2329:
2326:
2323:
2320:
2300:
2280:
2256:
2253:
2250:
2230:
2227:
2224:
2214:
2203:
2200:
2197:
2194:
2191:
2168:
2148:
2145:
2142:
2132:
2121:
2118:
2098:
2095:
2092:
2082:
2071:
2068:
2044:
2041:
2038:
2028:
2016:
1996:
1976:
1956:
1943:together with
1932:
1912:
1909:
1906:
1896:
1885:
1882:
1858:
1855:
1852:
1829:
1809:
1787:
1780:
1760:
1755:
1752:
1731:
1728:
1725:
1722:
1702:
1682:
1662:
1642:
1639:
1634:
1630:
1609:
1606:
1601:
1598:
1595:
1592:
1589:
1585:
1564:
1561:
1558:
1555:
1552:
1549:
1526:
1516:
1502:
1499:
1486:
1466:
1446:
1426:
1406:
1386:
1367:
1364:
1344:
1321:
1301:
1281:
1261:
1241:
1221:
1210:isolated point
1192:
1181:
1165:is called the
1154:
1134:
1114:
1094:
1074:
1054:
1034:
1014:
1005:obviously has
994:
984:
972:
951:
931:
911:
891:
872:Main article:
869:
866:
853:
850:
847:
827:
824:
821:
801:
781:
761:
751:
749:adherent point
745:
727:
707:
704:
684:
649:
629:
626:
623:
620:
617:
614:
611:
608:
603:
600:
597:
593:
589:
586:
583:
580:
577:
574:
571:
551:
531:
511:
508:
505:
485:
482:
479:
476:
473:
470:
467:
464:
444:
441:
438:
418:
415:
412:
392:
372:
353:
350:
330:
310:
307:
284:
261:
241:
221:
197:
177:
153:
140:Adherent point
138:Main article:
135:
132:
130:
127:
26:
9:
6:
4:
3:
2:
12264:
12253:
12250:
12248:
12245:
12244:
12242:
12227:
12219:
12215:
12212:
12210:
12207:
12205:
12202:
12201:
12200:
12192:
12190:
12186:
12182:
12180:
12176:
12172:
12170:
12165:
12160:
12158:
12150:
12149:
12146:
12140:
12137:
12135:
12132:
12130:
12127:
12125:
12122:
12120:
12117:
12115:
12112:
12111:
12109:
12107:
12103:
12097:
12096:Orientability
12094:
12092:
12089:
12087:
12084:
12082:
12079:
12077:
12074:
12073:
12071:
12067:
12061:
12058:
12056:
12053:
12051:
12048:
12046:
12043:
12041:
12038:
12036:
12033:
12031:
12028:
12024:
12021:
12019:
12016:
12015:
12014:
12011:
12007:
12004:
12002:
11999:
11997:
11994:
11992:
11989:
11987:
11984:
11983:
11982:
11979:
11977:
11974:
11972:
11969:
11967:
11963:
11960:
11959:
11957:
11953:
11948:
11938:
11935:
11933:
11932:Set-theoretic
11930:
11926:
11923:
11922:
11921:
11918:
11914:
11911:
11910:
11909:
11906:
11904:
11901:
11899:
11896:
11894:
11893:Combinatorial
11891:
11889:
11886:
11884:
11881:
11880:
11878:
11874:
11870:
11863:
11858:
11856:
11851:
11849:
11844:
11843:
11840:
11832:
11828:
11827:
11822:
11818:
11817:
11807:
11803:
11799:
11795:
11791:
11787:
11781:
11777:
11772:
11771:
11765:
11761:
11757:
11752:
11748:
11743:
11739:
11734:
11731:
11729:0-486-65676-4
11725:
11720:
11719:
11712:
11709:
11707:0-486-66522-4
11703:
11699:
11694:
11691:
11689:0-03-012813-7
11685:
11680:
11679:
11672:
11669:
11667:0-697-05972-3
11663:
11659:
11654:
11653:
11642:, p. 33.
11641:
11636:
11629:
11624:
11617:
11613:
11609:
11604:
11597:
11592:
11585:
11580:
11573:
11568:
11561:
11560:Schubert 1968
11556:
11552:
11527:
11521:
11512:
11506:
11503:
11498:
11494:
11490:
11487:
11481:
11472:
11469:
11466:
11460:
11455:
11451:
11444:
11438:
11418:
11409:
11406:
11400:
11397:
11394:
11389:
11385:
11361:
11358:
11355:
11352:
11326:
11323:
11317:
11314:
11291:
11288:
11282:
11276:
11273:
11263:
11259:
11245:
11242:
11239:
11236:
11233:
11230:
11227:
11224:
11218:
11215:
11212:
11209:
11208:
11202:
11200:
11196:
11190:
11177:
11174:
11166:
11150:
11144:
11138:
11135:
11128:
11112:
11109:
11086:
11080:
11074:
11051:
11031:
11025:
11016:
11003:
11000:
10997:
10994:
10988:
10968:
10965:
10945:
10925:
10922:
10919:
10916:
10896:
10890:
10884:
10874:
10858:
10855:
10852:
10832:
10829:
10823:
10820:
10817:
10797:
10789:
10773:
10753:
10733:
10730:
10710:
10690:
10684:
10677:
10661:
10654:
10651:
10650:partial order
10635:
10627:
10622:
10614:
10601:
10598:
10595:
10592:
10569:
10563:
10560:
10554:
10548:
10545:
10540:
10536:
10515:
10509:
10506:
10503:
10483:
10480:
10477:
10474:
10453:
10449:
10446:
10441:
10437:
10432:
10428:
10425:
10419:
10413:
10410:
10405:
10401:
10380:
10374:
10371:
10368:
10348:
10345:
10322:
10316:
10296:
10276:
10268:
10252:
10246:
10243:
10240:
10230:
10220:
10207:
10201:
10195:
10172:
10166:
10146:
10143:
10140:
10137:
10117:
10097:
10094:
10091:
10071:
10050:
10044:
10038:
10018:
9998:
9979:
9976:
9973:
9970:
9950:
9942:
9941:plain English
9926:
9923:
9920:
9915:
9911:
9907:
9904:
9884:
9881:
9878:
9868:
9854:
9834:
9831:
9808:
9802:
9779:
9773:
9754:
9751:
9748:
9745:
9725:
9722:
9719:
9699:
9690:
9684:
9678:
9673:
9669:
9662:
9655:
9651:
9648:
9643:
9639:
9634:
9630:
9611:
9608:
9605:
9602:
9582:
9576:
9573:
9570:
9561:
9548:
9545:
9525:
9505:
9498:is closed in
9482:
9474:
9471:
9467:
9458:
9454:
9438:
9432:
9429:
9426:
9416:
9401:
9388:
9378:
9375:
9355:
9348:is closed in
9335:
9332:
9329:
9309:
9302:is closed in
9289:
9269:
9261:
9221:
9213:
9197:
9190:is closed in
9177:
9174:
9171:
9151:
9131:
9128:
9125:
9105:
9097:
9069:
9053:
9033:
9025:
9002:
8999:
8992:(where every
8972:
8969:
8949:
8946:
8941:
8937:
8933:
8930:
8927:
8921:
8918:
8915:
8909:
8904:
8900:
8876:
8873:
8870:
8864:
8859:
8855:
8842:
8839:
8835:
8831:
8828:
8825:
8820:
8816:
8795:
8792:
8789:
8769:
8761:
8727:
8714:
8695:
8692:
8689:
8684:
8680:
8676:
8670:
8667:
8664:
8658:
8653:
8649:
8645:
8642:
8622:
8619:
8616:
8613:
8610:
8607:
8604:
8601:
8598:
8593:
8589:
8585:
8582:
8562:
8542:
8539:
8536:
8530:
8524:
8518:
8515:
8512:
8507:
8503:
8482:
8479:
8473:
8470:
8467:
8461:
8456:
8452:
8448:
8445:
8442:
8439:
8436:
8433:
8413:
8393:
8390:
8387:
8367:
8347:
8344:
8338:
8332:
8309:
8289:
8286:
8280:
8274:
8268:
8262:
8256:
8247:
8240:Consequently
8227:
8224:
8204:
8198:
8178:
8158:
8138:
8135:
8115:
8109:
8089:
8086:
8083:
8080:
8077:
8074:
8071:
8048:
8045:
8042:
8036:
8031:
8027:
8023:
8020:
8000:
7994:
7991:
7988:
7982:
7977:
7973:
7969:
7966:
7946:
7943:
7923:
7903:
7900:
7897:
7894:
7891:
7882:
7881:
7875:
7874:
7870:
7856:
7836:
7816:
7813:
7808:
7804:
7800:
7797:
7794:
7788:
7785:
7782:
7776:
7771:
7767:
7746:
7726:
7700:
7697:
7691:
7688:
7668:
7662:
7659:
7653:
7647:
7644:
7619:
7616:
7596:
7593:
7588:
7584:
7580:
7577:
7571:
7562:
7559:
7556:
7550:
7545:
7541:
7520:
7500:
7480:
7477:
7474:
7471:
7468:
7459:
7446:
7437:
7434:
7428:
7425:
7405:
7399:
7396:
7393:
7387:
7384:
7365:
7357:
7354:
7334:
7331:
7328:
7323:
7319:
7298:
7295:
7290:
7286:
7282:
7279:
7276:
7273:
7270:
7265:
7261:
7240:
7237:
7234:
7229:
7225:
7204:
7197:
7182:
7162:
7159:
7156:
7142:
7129:
7110:
7107:
7104:
7099:
7095:
7091:
7088:
7085:
7082:
7079:
7076:
7073:
7068:
7064:
7060:
7057:
7037:
7017:
7014:
7011:
7008:
7005:
7000:
6996:
6988:implies that
6975:
6972:
6967:
6963:
6942:
6939:
6932:is closed in
6919:
6899:
6896:
6893:
6890:
6885:
6881:
6877:
6874:
6854:
6851:
6848:
6845:
6842:
6839:
6836:
6831:
6827:
6806:
6799:is closed in
6786:
6766:
6763:
6760:
6740:
6737:
6717:
6714:
6709:
6705:
6684:
6664:
6654:
6640:
6637:
6632:
6628:
6607:
6604:
6599:
6595:
6591:
6588:
6585:
6582:
6579:
6574:
6570:
6561:
6545:
6525:
6522:
6517:
6513:
6509:
6506:
6486:
6483:
6463:
6460:
6455:
6451:
6441:
6440:
6434:
6433:
6429:
6416:
6413:
6410:
6405:
6401:
6397:
6394:
6388:
6382:
6379:
6374:
6370:
6349:
6329:
6309:
6289:
6269:
6249:
6246:
6241:
6237:
6233:
6230:
6227:
6222:
6218:
6197:
6189:
6173:
6153:
6145:
6129:
6109:
6106:
6103:
6100:
6097:
6071:
6049:
6045:
6041:
6036:
6032:
6026:
6023:
6020:
6016:
6012:
6008:
6002:
5998:
5992:
5989:
5986:
5982:
5977:
5973:
5968:
5964:
5943:
5937:
5934:
5929:
5925:
5918:
5912:
5909:
5904:
5900:
5893:
5887:
5884:
5881:
5875:
5870:
5866:
5857:
5856:
5854:
5850:
5846:
5842:
5838:
5835:
5831:
5827:
5813:
5810:
5790:
5782:
5779:
5775:
5774:
5773:
5759:
5756:
5753:
5750:
5747:
5742:
5738:
5717:
5709:
5693:
5664:
5660:
5655:
5649:
5645:
5634:
5631:
5627:
5622:
5618:
5613:
5609:
5604:
5600:
5595:
5591:
5587:
5583:
5577:
5573:
5569:
5564:
5560:
5549:
5546:
5542:
5537:
5533:
5528:
5524:
5503:
5481:
5477:
5468:
5454:
5450:
5445:
5439:
5435:
5424:
5421:
5417:
5412:
5408:
5403:
5399:
5394:
5390:
5385:
5381:
5377:
5373:
5367:
5363:
5359:
5354:
5350:
5339:
5336:
5332:
5327:
5323:
5318:
5314:
5293:
5286:is closed in
5271:
5267:
5258:
5257:
5256:
5243:
5240:
5233:
5217:
5214:
5209:
5205:
5201:
5196:
5192:
5171:
5169:
5164:
5151:
5148:
5140:
5121:
5115:
5109:
5103:
5098:
5094:
5087:
5084:
5081:
5078:
5073:
5069:
5061:
5060:
5059:
5042:
5036:
5030:
5024:
5019:
5015:
5008:
5005:
5002:
4999:
4994:
4990:
4982:
4981:
4980:
4966:
4961:
4957:
4948:
4944:
4926:
4922:
4912:
4910:
4894:
4874:
4871:
4865:
4850:that satisfy
4837:
4834:
4831:
4823:
4807:
4804:
4760:
4756:
4730:
4708:
4705:
4702:
4697:
4693:
4672:
4669:
4666:
4643:
4628:
4619:
4614:
4610:
4586:
4583:
4580:
4570:
4566:
4547:
4518:
4515:
4507:
4503:
4487:
4478:
4475:
4471:
4467:
4457:
4441:
4430:
4429:greater than
4426:
4412:
4370:
4348:
4326:
4306:
4284:
4262:
4232:
4216:
4196:
4190:
4187:
4184:
4181:
4178:
4175:
4170:
4166:
4162:
4154:
4151:
4145:
4142:
4122:
4109:
4093:
4081:
4065:
4062:
4059:
4056:
4053:
4048:
4044:
4024:
4021:
4001:
3981:
3961:
3958:
3951:
3947:
3944:
3940:
3939:
3938:
3921:
3913:
3904:
3901:
3898:
3889:
3884:
3880:
3872:itself, then
3849:
3828:
3825:
3817:
3803:
3797:
3794:
3791:
3785:
3776:
3773:
3770:
3761:
3756:
3752:
3743:
3722:
3719:
3711:
3697:
3691:
3688:
3685:
3679:
3670:
3667:
3664:
3655:
3650:
3646:
3637:
3616:
3613:
3605:
3604:
3603:
3597:
3577:
3574:
3571:
3568:
3565:
3560:
3556:
3535:
3532:
3524:
3508:
3500:
3486:
3480:
3477:
3469:
3461:
3453:
3450:
3444:
3440:
3433:
3430:
3422:
3414:
3406:
3403:
3396:
3392:
3387:
3383:
3362:
3357:
3347:
3335:
3334:complex plane
3319:
3311:
3297:
3284:
3246:
3233:
3173:
3165:
3147:
3144:
3141:
3118:
3095:
3092:
3089:
3063:
3060:
3057:
3051:
3042:
3039:
3036:
3027:
3022:
3018:
3009:
2971:
2963:
2962:
2961:
2959:
2896:
2893:
2890:
2887:
2884:
2881:
2862:
2859:
2852:In any space
2851:
2836:
2816:
2793:
2790:
2787:
2784:
2776:
2775:
2774:
2772:
2767:
2765:
2761:
2751:
2749:
2743:
2741:
2737:
2733:
2717:
2714:
2707:of points in
2706:
2702:
2686:
2683:
2680:
2672:
2668:
2663:
2659:
2642:
2639:
2636:
2633:
2613:
2593:
2573:
2553:
2545:
2531:
2528:
2525:
2522:
2502:
2499:
2496:
2476:
2473:
2470:
2462:
2447:
2444:
2441:
2438:
2435:
2428:
2412:
2404:
2389:
2381:
2365:
2362:
2359:
2352:
2351:
2350:
2333:
2327:
2324:
2321:
2298:
2278:
2270:
2254:
2251:
2248:
2228:
2225:
2222:
2215:
2201:
2195:
2182:
2166:
2146:
2143:
2140:
2133:
2119:
2116:
2096:
2093:
2090:
2083:
2069:
2066:
2058:
2042:
2039:
2036:
2029:
2014:
1994:
1974:
1954:
1946:
1930:
1910:
1907:
1904:
1897:
1883:
1880:
1872:
1856:
1853:
1850:
1843:
1842:
1841:
1827:
1807:
1785:
1778:
1758:
1750:
1729:
1726:
1723:
1720:
1700:
1680:
1660:
1640:
1637:
1632:
1628:
1607:
1604:
1596:
1593:
1590:
1583:
1562:
1556:
1553:
1550:
1540:
1524:
1515:
1512:
1508:
1498:
1484:
1464:
1444:
1424:
1404:
1384:
1365:
1362:
1342:
1333:
1319:
1299:
1279:
1259:
1239:
1219:
1211:
1207:
1202:
1201:of the set.
1200:
1196:
1195:cluster point
1179:
1171:
1170:
1166:
1152:
1132:
1112:
1092:
1072:
1065:in order for
1052:
1032:
1012:
992:
970:
962:
949:
929:
909:
889:
881:
875:
865:
851:
848:
845:
825:
822:
819:
799:
779:
759:
750:
747:
744:
743:
739:
725:
705:
702:
682:
674:
673:neighbourhood
670:
665:
663:
627:
624:
618:
615:
612:
606:
601:
598:
595:
587:
581:
578:
575:
569:
549:
529:
509:
506:
503:
483:
480:
474:
471:
468:
462:
442:
439:
436:
416:
413:
410:
403:if for every
390:
370:
351:
348:
328:
308:
305:
298:
282:
273:
259:
239:
219:
211:
195:
175:
167:
151:
141:
126:
124:
120:
112:
92:
88:
84:
76:
64:
56:
52:
44:
40:
33:
19:
12226:Publications
12091:Chern number
12081:Betti number
11964: /
11955:Key concepts
11903:Differential
11824:
11804:– via
11769:
11755:
11746:
11737:
11717:
11697:
11677:
11657:
11650:Bibliography
11635:
11623:
11614:, p. 40 and
11603:
11591:
11579:
11567:
11555:
11262:
11191:
11017:
10623:
10620:
10232:
10188:is close to
9562:
9418:
8733:
8575:proves that
7883:
7460:
7148:
6442:
6342:computed in
6282:computed in
6089:
6084:is infinite.
5830:intersection
5685:
5179:(C. Ursescu)
5173:
5165:
5136:
5057:
4913:
4476:
4473:
4427:real numbers
4085:
3936:
3601:
3259:We say that
3008:real numbers
2911:
2768:
2757:
2744:
2671:metric space
2664:
2657:
2382:superset of
2348:
2271:(valued) in
1517:of a subset
1513:
1510:
1334:
1205:
1203:
1198:
1194:
1167:
877:
748:
740:
666:
297:metric space
274:
212:centered at
143:
87:intersection
63:limit points
45:of a subset
42:
36:
12189:Wikiversity
12106:Key results
11628:Pervin 1965
11612:Pervin 1965
10788:subcategory
10233:A function
10223:Closed maps
9419:A function
8128:is open in
7936:is open in
7533:then only
7050:shows that
6697:). Because
6677:containing
5496:is open in
5139:complements
4822:closed sets
4567:. Given a
4431:or equal to
2758:Consider a
2669:(such as a
2059:containing
2057:closed sets
1923:is the set
1800:(Moreover,
1575:denoted by
1497:(or both).
1169:derived set
963:other than
868:Limit point
129:Definitions
93:containing
91:closed sets
18:Set closure
12241:Categories
12035:CW complex
11976:Continuity
11966:Closed set
11925:cohomology
11616:Baker 1991
11596:Croom 1989
11547:References
10267:closed map
9453:continuous
9409:Continuity
9368:for every
9260:open cover
8962:for every
8760:open cover
6779:such that
5803:itself is
4464:See also:
2660:definition
2425:is closed
1505:See also:
1355:and point
12214:geometric
12209:algebraic
12060:Cobordism
11996:Hausdorff
11991:connected
11908:Geometric
11898:Continuum
11888:Algebraic
11831:EMS Press
11802:285163112
11722:, Dover,
11610:, p. 55,
11504:
11491:∩
11482:≠
11470:∩
11461:
11439:∅
11413:∞
11395:
11365:∅
11356:∩
11330:∞
11286:∞
11283:−
11139:
11084:∖
11078:↓
11029:∖
10998:
10992:→
10920:
10888:↓
10856:⊆
10827:→
10703:whenever
10688:→
10628:of a set
10596:⊆
10561:⊆
10546:
10513:→
10478:⊆
10447:
10426:⊆
10411:
10378:→
10250:→
10141:⊆
10095:∈
9974:⊆
9921:
9908:∈
9882:⊆
9871:a subset
9749:⊆
9723:∈
9679:
9663:⊆
9649:
9606:⊆
9580:→
9518:whenever
9472:−
9436:→
9379:∈
9333:∩
9175:⊆
9129:⊆
9003:∈
8973:∈
8947:
8934:∩
8919:∩
8910:
8874:∩
8865:
8843:∈
8836:⋃
8826:
8793:⊆
8715:◼
8690:
8677:⊆
8668:∩
8659:
8646:⊆
8611:∩
8605:⊆
8599:
8586:∩
8537:∪
8528:∖
8519:⊆
8513:
8471:∩
8462:
8449:⊆
8443:∩
8437:∈
8391:∈
8360:contains
8345:∪
8336:∖
8287:∪
8278:∖
8260:∖
8251:∖
8202:∖
8113:∖
8087:⊆
8081:⊆
8075:∩
8064:(because
8046:∩
8037:
8024:∩
7992:∩
7983:
7901:⊆
7814:
7801:∩
7786:∩
7777:
7704:∞
7657:∞
7654:−
7594:
7581:∩
7572:⊆
7560:∩
7551:
7478:⊆
7441:∞
7329:
7296:
7283:∩
7271:
7235:
7160:⊆
7130:◼
7105:
7086:∩
7080:⊆
7074:
7061:∩
7012:⊆
7006:
6973:
6897:⊆
6891:
6878:⊆
6849:∩
6837:
6764:⊆
6715:
6638:
6620:(because
6605:
6592:∩
6586:⊆
6580:
6523:
6510:∩
6461:
6411:
6398:∩
6380:
6247:
6234:⊆
6228:
6107:⊆
6101:⊆
6042:
6024:∈
6017:⋃
6013:≠
5990:∈
5983:⋃
5974:
5935:
5919:∪
5910:
5885:∪
5876:
5778:empty set
5748:
5686:A subset
5635:∈
5628:⋂
5619:
5601:
5570:
5550:∈
5543:⋂
5534:
5425:∈
5418:⋃
5409:
5391:
5360:
5340:∈
5333:⋃
5324:
5218:…
5113:∖
5104:
5091:∖
5079:
5058:and also
5034:∖
5025:
5012:∖
5000:
4909:open sets
4835:⊆
4761:−
4734:¯
4703:
4670:⊆
4638:℘
4635:→
4623:℘
4587:τ
4506:power set
4480:on a set
4155:∈
4054:
3890:
3762:
3656:
3566:
3478:≥
3454:∈
3407:∈
3393:
3028:
2891:
2837:∅
2817:∅
2797:∅
2794:
2785:∅
2705:sequences
2684:
2637:
2626:contains
2586:contains
2526:
2500:
2474:⊆
2445:
2363:
2328:τ
2252:∈
2226:
2190:∂
2144:
2094:
2040:
1908:
1854:
1786:−
1754:¯
1724:
1701:τ
1661:τ
1638:
1605:
1597:τ
1557:τ
902:of a set
849:∈
599:∈
440:∈
272:itself).
210:open ball
208:if every
12179:Wikibook
12157:Category
12045:Manifold
12013:Homotopy
11971:Interior
11962:Open set
11920:Homology
11869:Topology
11756:Topology
11738:Topology
11718:Topology
11598:, p. 104
11205:See also
11165:interior
10653:category
10626:powerset
9869:close to
9457:preimage
9068:manifold
8892:because
6867:Because
6655:smallest
6443:Because
6144:subspace
5853:superset
5845:finitely
5469:If each
5259:If each
4947:interior
4339:because
2754:Examples
2405:The set
2181:boundary
2179:and its
1332:itself.
812:(again,
123:interior
83:boundary
81:and its
39:topology
12204:general
12006:uniform
11986:compact
11937:Digital
11833:, 2001
11794:1921556
11630:, p. 41
11574:, p. 75
11562:, p. 20
9258:is any
8782:and if
8758:is any
6653:is the
6122:and if
5175:Theorem
4945:to the
4504:of the
4502:mapping
4135:and if
3948:In any
3941:In any
3638:, then
3332:is the
3010:, then
2912:Giving
2849:itself.
2750:below.
1514:closure
675:". Let
662:infimum
660:is the
89:of all
43:closure
12199:Topics
12001:metric
11876:Fields
11800:
11792:
11782:
11726:
11704:
11686:
11664:
11586:, p. 4
11516:
11510:
11485:
11479:
11448:
11442:
9666:
9660:
8406:is in
8322:where
8151:where
7575:
7569:
6392:
6386:
5834:subset
5708:closed
5177:
3523:finite
2760:sphere
2736:filter
2734:" or "
2701:limits
2380:closed
1204:Thus,
983:itself
640:where
55:points
41:, the
11981:Space
11266:From
11254:Notes
11125:with
10786:is a
10309:then
10159:then
9282:then
9066:is a
8426:then
7877:Proof
6436:Proof
6190:that
6142:is a
5848:case.
5841:union
5839:In a
5516:then
5306:then
4500:is a
4209:then
4080:dense
3596:axiom
3548:then
3521:is a
3375:then
3283:dense
2665:In a
2489:then
2378:is a
1537:of a
1312:than
738:is a
718:Then
295:of a
109:is a
75:union
11798:OCLC
11780:ISBN
11724:ISBN
11702:ISBN
11684:ISBN
11662:ISBN
11377:and
11307:and
11163:the
10624:The
7959:Let
7884:Let
7869:).
7849:and
7681:and
7493:but
7418:and
6912:and
6819:and
5184:Let
4943:dual
4468:and
4188:>
4176:>
3846:the
3740:the
3431:>
2956:the
2934:and
2764:ball
1693:and
1653:(if
1511:The
838:for
481:<
414:>
144:For
119:dual
11167:of
11136:int
10958:to
10790:of
10766:on
9897:if
9867:is
9824:in
9262:of
9214:in
8762:of
7461:If
6362::
6146:of
6090:If
5843:of
5710:in
5706:is
5592:int
5525:int
5400:int
5351:int
5141:in
5070:int
5016:int
4958:int
4941:is
4748:or
4685:to
4508:of
4233:in
4229:is
4014:of
3606:If
3501:If
3312:If
3285:in
3281:is
3230:of
3166:If
3131:is
3006:of
2964:If
2829:is
2769:In
2742:).
2732:net
2673:),
2546:If
2463:If
2311:in
2269:net
1873:of
1771:or
1457:or
1197:or
1172:of
752:of
746:or
664:.
648:inf
592:inf
113:of
77:of
65:of
57:in
37:In
12243::
11829:,
11823:,
11796:.
11790:MR
11788:.
11778:.
11495:cl
11452:cl
11386:cl
11318::=
11277::=
11201:.
10995:cl
10917:cl
10537:cl
10438:cl
10402:cl
9912:cl
9670:cl
9640:cl
8938:cl
8901:cl
8856:cl
8817:cl
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8650:cl
8590:cl
8504:cl
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8028:cl
7974:cl
7970::=
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7768:cl
7585:cl
7542:cl
7320:cl
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4474:A
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3598:.)
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2960::
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2037:cl
2027:.)
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1851:cl
1828:Cl
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1721:cl
1629:cl
1584:cl
588::=
168:,
125:.
11861:e
11854:t
11847:v
11808:.
11528:.
11525:}
11522:0
11519:{
11513:=
11507:S
11499:X
11488:T
11476:)
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11467:S
11464:(
11456:T
11445:=
11419:,
11416:)
11410:,
11407:0
11404:[
11401:=
11398:S
11390:X
11362:=
11359:T
11353:S
11333:)
11327:,
11324:0
11321:(
11315:S
11295:]
11292:0
11289:,
11280:(
11274:T
11178:.
11175:A
11151:,
11148:)
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11142:(
11113:,
11110:A
11090:)
11087:A
11081:X
11075:I
11072:(
11052:A
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11026:X
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10926:.
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10897:.
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10891:I
10885:A
10882:(
10859:X
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10833:.
10830:P
10824:T
10821::
10818:I
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10734:.
10731:B
10711:A
10691:B
10685:A
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10636:X
10602:.
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10593:C
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10570:C
10567:(
10564:f
10558:)
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10552:(
10549:f
10541:Y
10516:Y
10510:X
10507::
10504:f
10484:.
10481:X
10475:A
10454:)
10450:A
10442:X
10433:(
10429:f
10423:)
10420:A
10417:(
10414:f
10406:Y
10381:Y
10375:X
10372::
10369:f
10349:.
10346:Y
10326:)
10323:C
10320:(
10317:f
10297:X
10277:C
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10247:X
10244::
10241:f
10208:.
10205:)
10202:A
10199:(
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10176:)
10173:x
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10072:f
10051:.
10048:)
10045:A
10042:(
10039:f
10019:A
9999:f
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9927:,
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9835:.
9832:Y
9812:)
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9806:(
9803:f
9783:)
9780:x
9777:(
9774:f
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9746:A
9726:X
9720:x
9700:.
9697:)
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9691:A
9688:(
9685:f
9682:(
9674:Y
9656:)
9652:A
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9635:(
9631:f
9612:,
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9583:Y
9577:X
9574::
9571:f
9549:.
9546:Y
9526:C
9506:X
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9483:C
9480:(
9475:1
9468:f
9439:Y
9433:X
9430::
9427:f
9389:.
9384:U
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9330:S
9310:X
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9080:U
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9034:X
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8978:U
8970:U
8950:S
8942:X
8931:U
8928:=
8925:)
8922:U
8916:S
8913:(
8905:U
8880:)
8877:S
8871:U
8868:(
8860:U
8848:U
8840:U
8832:=
8829:S
8821:X
8796:X
8790:S
8770:X
8744:U
8696:.
8693:S
8685:X
8674:)
8671:S
8665:T
8662:(
8654:X
8643:C
8623:.
8620:C
8617:=
8614:C
8608:T
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8583:T
8563:T
8543:.
8540:C
8534:)
8531:T
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8516:S
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8480:=
8477:)
8474:S
8468:T
8465:(
8457:T
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8440:T
8434:s
8414:T
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8388:s
8368:S
8348:C
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8330:(
8310:X
8290:C
8284:)
8281:T
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8272:(
8269:=
8266:)
8263:C
8257:T
8254:(
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8228:.
8225:X
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8179:X
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8052:)
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7792:)
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7701:,
7698:0
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7692:=
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7669:,
7666:]
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7648:=
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7624:R
7620:=
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7557:S
7554:(
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7438:,
7435:0
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7429:=
7426:T
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7400:1
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7092:=
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