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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,
2755:). 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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5659:{\displaystyle \operatorname {int} _{X}\left(\bigcap _{i\in \mathbb {N} }\operatorname {cl} _{X}S_{i}\right)=\operatorname {int} _{X}\left.}
5449:{\displaystyle \operatorname {cl} _{X}\left(\bigcup _{i\in \mathbb {N} }\operatorname {int} _{X}S_{i}\right)=\operatorname {cl} _{X}\left.}
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6046:{\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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3481:{\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,
871:. 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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11522:{\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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5938:{\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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8874:{\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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9694:{\displaystyle f\left(\operatorname {cl} _{X}A\right)~\subseteq ~\operatorname {cl} _{Y}(f(A)).}
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8690:{\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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10449:{\displaystyle \operatorname {cl} _{Y}f(A)\subseteq f\left(\operatorname {cl} _{X}A\right)}
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5116:{\displaystyle \operatorname {int} _{X}S=X\setminus \operatorname {cl} _{X}(X\setminus S).}
5037:{\displaystyle \operatorname {cl} _{X}S=X\setminus \operatorname {int} _{X}(X\setminus S),}
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7591:{\displaystyle \operatorname {cl} _{T}(S\cap T)~\subseteq ~T\cap \operatorname {cl} _{X}S}
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7105:{\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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86:. Intuitively, the closure can be thought of as all the points that are either in
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6602:{\displaystyle \operatorname {cl} _{T}S\subseteq T\cap \operatorname {cl} _{X}S}
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8944:{\displaystyle \operatorname {cl} _{U}(S\cap U)=U\cap \operatorname {cl} _{X}S}
7811:{\displaystyle \operatorname {cl} _{T}(S\cap T)=T\cap \operatorname {cl} _{X}S}
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3934:, 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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6411:{\displaystyle \operatorname {cl} _{T}S~=~T\cap \operatorname {cl} _{X}S.}
6244:{\displaystyle \operatorname {cl} _{T}S\subseteq \operatorname {cl} _{X}S}
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8477:{\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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8537:{\displaystyle \operatorname {cl} _{X}S\subseteq (X\setminus T)\cup C.}
7293:{\displaystyle \operatorname {cl} _{T}S=T\cap \operatorname {cl} _{X}S}
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3579:(For a general topological space, this property is equivalent to the
1829:.) can be defined using any of the following equivalent definitions:
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8617:{\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
10898:
This category — also a partial order — then has initial object
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8284:{\displaystyle X\setminus (T\setminus C)=(X\setminus T)\cup C}
6894:{\displaystyle S\subseteq \operatorname {cl} _{T}S\subseteq C}
16:
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
3916:{\displaystyle \operatorname {cl} _{X}((0,1))=\mathbb {R} .}
2791:{\displaystyle \varnothing =\operatorname {cl} \varnothing }
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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
4641:{\displaystyle \operatorname {cl} _{X}:\wp (X)\to \wp (X)}
4191:{\displaystyle S=\{q\in \mathbb {Q} :q^{2}>2,q>0\},}
10567:{\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
511:
is allowed). Another way to express this is to say that
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Pages displaying short descriptions of redirect targets
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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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8046:{\displaystyle T\cap \operatorname {cl} _{X}(T\cap S)}
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are clear from context then it may also be denoted by
1197:. A point of closure which is not a limit point is an
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11413:{\displaystyle \operatorname {cl} _{X}S=[0,\infty ),}
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then a topological space is obtained by defining the
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3798:{\displaystyle \operatorname {cl} _{X}((0,1))=(0,1).}
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3692:{\displaystyle \operatorname {cl} _{X}((0,1))=[0,1).}
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2702:
2668:
2621:
2601:
2581:
2561:
2541:
2510:
2484:
2458:
2423:
2400:
2377:
2347:
2306:
2286:
2266:
2236:
2210:
2177:
2154:
2128:
2104:
2078:
2054:
2024:
2002:
1982:
1962:
1942:
1918:
1892:
1868:
1838:
1815:
1795:
1766:
1736:
1708:
1688:
1668:
1648:
1615:
1570:
1535:
1512:
1472:
1452:
1432:
1412:
1392:
1372:
1350:
1330:
1307:
1287:
1267:
1247:
1227:
1207:
1167:
1140:
1120:
1114:
has more strict condition than a point of closure of
1100:
1080:
1060:
1040:
1020:
1000:
980:
958:
937:
917:
897:
877:
833:
807:
787:
767:
747:
713:
690:
670:
635:
557:
537:
517:
491:
450:
424:
398:
378:
358:
336:
316:
293:
270:
247:
227:
207:
183:
163:
139:
10834:
The set of closed subsets containing a fixed subset
7995:{\displaystyle C:=\operatorname {cl} _{T}(T\cap S),}
7012:{\displaystyle \operatorname {cl} _{X}S\subseteq C.}
9087:. In words, this result shows that the closure in
2338:The closure of a set has the following properties.
11704:
11664:
11607:, p. 38 use the second property as the definition.
11521:
11412:
11358:
11326:
11288:
11171:
11144:
11106:
11083:
11045:
11025:
10997:
10962:
10939:
10919:
10890:
10852:
10826:
10791:
10767:
10747:
10727:
10704:
10684:
10655:
10629:
10595:
10566:
10509:
10477:
10448:
10374:
10342:
10319:
10290:
10270:
10246:
10201:
10169:
10140:
10111:
10091:
10065:
10044:
10012:
9992:
9973:
9944:
9920:
9878:
9848:
9828:
9805:
9776:
9748:
9719:
9693:
9605:
9576:
9542:
9519:
9499:
9479:
9432:
9382:
9349:
9329:
9303:
9283:
9263:
9239:
9215:
9191:
9171:
9145:
9125:
9099:
9075:
9047:
9027:
9003:
8973:
8943:
8873:
8789:
8763:
8739:
8708:
8689:
8616:
8556:
8536:
8476:
8407:
8387:
8361:
8341:
8303:
8283:
8221:
8198:
8172:
8152:
8132:
8109:
8083:
8045:
7994:
7940:
7917:
7897:
7850:
7830:
7810:
7740:
7720:
7700:
7662:
7618:
7590:
7514:
7494:
7474:
7440:
7399:
7359:
7328:
7292:
7234:
7198:
7176:
7156:
7123:
7104:
7031:
7011:
6969:
6936:
6913:
6893:
6848:
6800:
6780:
6760:
6734:
6711:
6678:
6658:
6634:
6601:
6539:
6519:
6480:
6457:
6410:
6343:
6323:
6303:
6283:
6263:
6243:
6191:
6167:
6147:
6123:
6103:
6065:
6045:
5937:
5807:
5784:
5753:
5711:
5687:
5658:
5497:
5477:
5448:
5287:
5267:
5237:
5211:
5145:
5115:
5036:
4960:
4922:
4888:
4868:
4831:
4801:
4778:
4756:
4729:
4702:
4666:
4640:
4583:
4544:
4512:
4481:
4437:
4406:
4386:
4364:
4344:
4320:
4300:
4280:
4256:
4236:
4210:
4190:
4116:
4087:
4059:
4018:
3995:
3975:
3955:
3915:
3853:
3827:
3797:
3721:
3691:
3615:
3571:
3529:
3502:
3480:
3356:
3313:
3291:
3262:
3240:
3211:
3189:
3167:
3144:
3112:
3092:
3060:
2987:
2965:
2937:
2915:
2890:
2856:
2830:
2810:
2790:
2711:
2680:
2636:
2607:
2587:
2567:
2547:
2525:
2496:
2470:
2441:
2406:
2383:
2359:
2327:
2292:
2272:
2248:
2222:
2195:
2160:
2140:
2113:
2090:
2063:
2036:
2008:
1988:
1968:
1948:
1924:
1904:
1877:
1850:
1821:
1801:
1781:
1752:
1723:
1694:
1674:
1654:
1634:
1601:
1556:
1518:
1478:
1458:
1438:
1418:
1398:
1378:
1359:
1336:
1313:
1293:
1273:
1253:
1233:
1213:
1173:
1146:
1126:
1106:
1086:
1066:
1046:
1026:
1006:
986:
964:
943:
923:
903:
883:
845:
819:
793:
773:
753:
719:
699:
676:
641:
621:
543:
523:
503:
477:
436:
410:
384:
364:
345:
322:
302:
276:
253:
233:
213:
189:
169:
145:
6849:{\displaystyle \operatorname {cl} _{T}S=T\cap C.}
3963:since the only closed sets are the empty set and
12227:
1602:{\displaystyle \operatorname {cl} _{(X,\tau )}S}
636:
580:
9584:is continuous if and only if for every subset
4095:is the set of rational numbers, with the usual
3061:{\displaystyle \operatorname {cl} _{X}((0,1))=}
2798:. In other words, the closure of the empty set
11217:, a set equal to the closure of their interior
9952:is continuous if and only if for every subset
9921:{\displaystyle x\in \operatorname {cl} _{X}A,}
9035:). This equality is particularly useful when
7818:will hold (no matter the relationship between
6520:{\displaystyle T\cap \operatorname {cl} _{X}S}
3357:{\displaystyle \mathbb {C} =\mathbb {R} ^{2},}
622:{\displaystyle d(x,S):=\inf _{s\in S}d(x,s)=0}
11842:
11229: – Largest open subset of some given set
11007:Similarly, since every closed set containing
10605:
8084:{\displaystyle T\cap S\subseteq T\subseteq X}
4591:, the topological closure induces a function
11702:
11572:
11513:
11507:
11235: – Cluster point in a topological space
4182:
4137:
3472:
3436:
3425:
3389:
5754:{\displaystyle \operatorname {cl} _{X}S=S.}
4060:{\displaystyle \operatorname {cl} _{X}A=X.}
3572:{\displaystyle \operatorname {cl} _{X}S=S.}
12210:
12183:
11849:
11835:
11724:
11703:Hocking, John G.; Young, Gail S. (1988) ,
11560:
11084:{\displaystyle (I\downarrow X\setminus A)}
11033:corresponds with an open set contained in
10998:{\displaystyle A\to \operatorname {cl} A.}
10517:is a (strongly) closed map if and only if
10382:is a (strongly) closed map if and only if
3733:in which every set is closed (open), then
3068:. In other words., the closure of the set
264:This definition generalizes to any subset
11751:
11684:
11628:
11596:
7612:
7350:
7329:{\displaystyle \operatorname {cl} _{X}S;}
7235:{\displaystyle \operatorname {cl} _{X}S.}
5627:
5542:
5417:
5332:
4961:{\displaystyle \operatorname {int} _{X},}
4847:
4772:
4703:{\displaystyle \operatorname {cl} _{X}S,}
4380:
4230:
4147:
4107:
3906:
3847:
3821:
3715:
3609:
3446:
3399:
3341:
3332:
3282:
3256:
3231:
3205:
3183:
2981:
2931:
2909:
660:by replacing "open ball" or "ball" with "
11759:Convex Analysis in General Vector Spaces
11742:
11548:
11223: – Set of all limit points of a set
11145:{\displaystyle \operatorname {int} (A),}
11091:as the set of open subsets contained in
9727:that belongs to the closure of a subset
9392:
6970:{\displaystyle \operatorname {cl} _{X}S}
6712:{\displaystyle \operatorname {cl} _{T}S}
6635:{\displaystyle \operatorname {cl} _{T}S}
6458:{\displaystyle \operatorname {cl} _{X}S}
1635:{\displaystyle \operatorname {cl} _{X}S}
1182:. A limit point of a set is also called
106:. The notion of closure is in many ways
6104:{\displaystyle S\subseteq T\subseteq X}
5670:
4923:{\displaystyle \operatorname {cl} _{X}}
2891:{\displaystyle X=\operatorname {cl} X.}
12228:
11733:
11616:
11600:
9784:necessarily belongs to the closure of
2442:{\displaystyle S=\operatorname {cl} S}
2098:is the smallest closed set containing
11830:
11662:
11644:
11604:
11584:
10927:Thus there is a universal arrow from
10920:{\displaystyle \operatorname {cl} A.}
10073:is continuous at a fixed given point
4764:may be used instead. Conversely, if
2736:
2637:{\displaystyle \operatorname {cl} S.}
2526:{\displaystyle \operatorname {cl} T.}
1724:{\displaystyle \operatorname {cl} S,}
94:. A point which is in the closure of
11359:{\displaystyle S\cap T=\varnothing }
9383:{\displaystyle U\in {\mathcal {U}}.}
8342:{\displaystyle (X\setminus T)\cup C}
4648:that is defined by sending a subset
4288:, which would be the lower bound of
2681:{\displaystyle \operatorname {cl} S}
2497:{\displaystyle \operatorname {cl} S}
2360:{\displaystyle \operatorname {cl} S}
2223:{\displaystyle \operatorname {cl} S}
2141:{\displaystyle \operatorname {cl} S}
2091:{\displaystyle \operatorname {cl} S}
2037:{\displaystyle \operatorname {cl} S}
1905:{\displaystyle \operatorname {cl} S}
1851:{\displaystyle \operatorname {cl} S}
9928:then this terminology allows for a
9004:{\displaystyle U\in {\mathcal {U}}}
8974:{\displaystyle U\in {\mathcal {U}}}
8624:The reverse inclusion follows from
5212:{\displaystyle S_{1},S_{2},\ldots }
4448:
1822:{\displaystyle \operatorname {Cl} }
1802:{\displaystyle \operatorname {cl} }
1489:
684:be a subset of a topological space
122:
62:may equivalently be defined as the
13:
11401:
11318:
11274:
10350:In terms of the closure operator,
9701:That is to say, given any element
9552:In terms of the closure operator,
9372:
9232:
9068:
8996:
8966:
8836:
8732:
7692:
7645:
7429:
4626:
4611:
4552:, into itself which satisfies the
4528:
2727:" (as described in the article on
2178:
1281:which contains no other points of
14:
12252:
11802:
11353:
11072:
11017:
8516:
8324:
8266:
8248:
8239:
8190:
8101:
5101:
5079:
5022:
5000:
4869:{\displaystyle \mathbb {c} (S)=S}
4545:{\displaystyle {\mathcal {P}}(X)}
2785:
1014:but it also must have a point of
12209:
12182:
12172:
12162:
12151:
12141:
12140:
11934:
10891:{\displaystyle (A\downarrow I).}
6291:is equal to the intersection of
5159:the following result does hold:
4394:. However, if we instead define
1753:{\displaystyle {\overline {S}},}
1261:and there is a neighbourhood of
11736:Foundations of General Topology
11685:Gemignani, Michael C. (1990) ,
11671:, Saunders College Publishing,
11638:
11289:{\displaystyle T:=(-\infty ,0]}
9059:and the sets in the open cover
7663:{\displaystyle T=(-\infty ,0],}
7502:is not necessarily a subset of
7360:{\displaystyle X=\mathbb {R} ,}
4813:as being exactly those subsets
4786:is a closure operator on a set
4730:{\displaystyle {\overline {S}}}
4264:nor its complement can contain
4099:induced by the Euclidean space
656:This definition generalizes to
11610:
11590:
11578:
11566:
11554:
11542:
11464:
11452:
11404:
11392:
11327:{\displaystyle S:=(0,\infty )}
11321:
11309:
11283:
11268:
11249:
11136:
11130:
11078:
11066:
11060:
11053:we can interpret the category
10980:
10882:
10876:
10870:
10815:
10676:
10561:
10555:
10546:
10540:
10501:
10411:
10405:
10366:
10314:
10308:
10238:
10211:
10193:
10187:
10164:
10158:
10036:
10030:
10000:maps points that are close to
9800:
9794:
9771:
9765:
9685:
9682:
9676:
9670:
9568:
9474:
9468:
9440:between topological spaces is
9424:
9240:{\displaystyle {\mathcal {U}}}
9076:{\displaystyle {\mathcal {U}}}
8913:
8901:
8868:
8856:
8740:{\displaystyle {\mathcal {U}}}
8662:
8650:
8522:
8510:
8465:
8453:
8330:
8318:
8272:
8260:
8254:
8242:
8040:
8028:
7986:
7974:
7898:{\displaystyle S,T\subseteq X}
7780:
7768:
7695:
7683:
7654:
7639:
7619:{\displaystyle X=\mathbb {R} }
7554:
7542:
7475:{\displaystyle S,T\subseteq X}
7441:{\displaystyle T=(0,\infty ).}
7432:
7420:
7391:
7379:
5929:
5910:
5904:
5885:
5879:
5867:
5844:of the union of the closures.
5219:be a sequence of subsets of a
5107:
5095:
5028:
5016:
4938:operator, which is denoted by
4857:
4851:
4635:
4629:
4623:
4620:
4614:
4578:
4566:
4539:
4533:
3899:
3896:
3884:
3881:
3828:{\displaystyle X=\mathbb {R} }
3789:
3777:
3771:
3768:
3756:
3753:
3722:{\displaystyle X=\mathbb {R} }
3683:
3671:
3665:
3662:
3650:
3647:
3616:{\displaystyle X=\mathbb {R} }
3462:
3454:
3415:
3407:
3197:, then the closure of the set
3139:
3127:
3087:
3075:
3055:
3043:
3037:
3034:
3022:
3019:
2319:
2307:
2187:
2181:
1996:is also a point of closure of
1662:is understood), where if both
1588:
1576:
1548:
1536:
974:, i.e., each neighbourhood of
856:
610:
598:
573:
561:
466:
454:
330:as a metric space with metric
117:
1:
11729:, vol. I, Academic Press
11535:
11186:), since all are examples of
10596:{\displaystyle C\subseteq X.}
10478:{\displaystyle A\subseteq X.}
10141:{\displaystyle A\subseteq X,}
9974:{\displaystyle A\subseteq X,}
9749:{\displaystyle A\subseteq X,}
9606:{\displaystyle A\subseteq X,}
9397:
8709:{\displaystyle \blacksquare }
8544:Intersecting both sides with
7728:happens to an open subset of
7701:{\displaystyle S=(0,\infty )}
7124:{\displaystyle \blacksquare }
7019:Intersecting both sides with
4117:{\displaystyle \mathbb {R} ,}
3292:{\displaystyle \mathbb {R} .}
3241:{\displaystyle \mathbb {R} .}
2196:{\displaystyle \partial (S).}
11856:
11026:{\displaystyle X\setminus A}
10853:{\displaystyle A\subseteq X}
10020:to points that are close to
9879:{\displaystyle A\subseteq X}
9172:{\displaystyle S\subseteq X}
9126:{\displaystyle S\subseteq X}
8790:{\displaystyle S\subseteq X}
8199:{\displaystyle T\setminus C}
8110:{\displaystyle T\setminus C}
7157:{\displaystyle S\subseteq T}
6761:{\displaystyle C\subseteq X}
4832:{\displaystyle S\subseteq X}
4779:{\displaystyle \mathbb {c} }
4722:
4667:{\displaystyle S\subseteq X}
4387:{\displaystyle \mathbb {Q} }
4237:{\displaystyle \mathbb {Q} }
3854:{\displaystyle \mathbb {R} }
3514:subset of a Euclidean space
3263:{\displaystyle \mathbb {Q} }
3212:{\displaystyle \mathbb {Q} }
3190:{\displaystyle \mathbb {R} }
2988:{\displaystyle \mathbb {R} }
2938:{\displaystyle \mathbb {C} }
2916:{\displaystyle \mathbb {R} }
2831:{\displaystyle \varnothing }
2811:{\displaystyle \varnothing }
2471:{\displaystyle S\subseteq T}
1809:is sometimes capitalized to
1742:
7:
11815:Encyclopedia of Mathematics
11765:World Scientific Publishing
11763:. River Edge, N.J. London:
11734:Pervin, William J. (1965),
11208: – Algebraic structure
11193:
10860:can be identified with the
9932:description of continuity:
9836:If we declare that a point
4438:{\displaystyle {\sqrt {2}}}
4345:{\displaystyle {\sqrt {2}}}
4281:{\displaystyle {\sqrt {2}}}
2742:
2044:is the intersection of all
478:{\displaystyle d(x,s)<r}
10:
12257:
12103:Banach fixed-point theorem
11649:, Wm. C. Brown Publisher,
10606:Categorical interpretation
10215:
9401:
4896:of these subsets form the
4452:
2947:standard (metric) topology
2328:{\displaystyle (X,\tau ).}
1976:, and each limit point of
1557:{\displaystyle (X,\tau ),}
1493:
1201:. In other words, a point
860:
761:if every neighbourhood of
126:
18:
12136:
12093:
12057:
11943:
11932:
11864:
11221:Derived set (mathematics)
10827:{\displaystyle I:T\to P.}
9480:{\displaystyle f^{-1}(C)}
7626:with the usual topology,
7300:to be a proper subset of
4584:{\displaystyle (X,\tau )}
4554:Kuratowski closure axioms
4459:Kuratowski closure axioms
2256:for which there exists a
1956:is a point of closure of
1386:is a point of closure of
911:, every neighbourhood of
531:is a point of closure of
372:is a point of closure of
177:is a point of closure of
11743:Schubert, Horst (1968),
11647:Introduction to Topology
11645:Baker, Crump W. (1991),
11573:Hocking & Young 1988
11242:
10735:Furthermore, a topology
10574:for every closed subset
10510:{\displaystyle f:X\to Y}
10375:{\displaystyle f:X\to Y}
10258:if and only if whenever
10247:{\displaystyle f:X\to Y}
10099:if and only if whenever
9577:{\displaystyle f:X\to Y}
9433:{\displaystyle f:X\to Y}
8369:as a subset (because if
7400:{\displaystyle S=(0,1),}
1221:is an isolated point of
931:must contain a point of
21:Closure (disambiguation)
11725:Kuratowski, K. (1966),
11689:(2nd ed.), Dover,
11663:Croom, Fred H. (1989),
10970:given by the inclusion
10799:with inclusion functor
9330:{\displaystyle S\cap U}
3175:is the Euclidean space
2973:is the Euclidean space
1934:all of its limit points
1782:{\displaystyle S{}^{-}}
1241:if it is an element of
1074:to be a limit point of
444:such that the distance
12158:Mathematics portal
12058:Metrics and properties
12044:Second-countable space
11667:Principles of Topology
11523:
11414:
11360:
11328:
11290:
11173:
11146:
11108:
11085:
11047:
11027:
10999:
10964:
10941:
10921:
10892:
10854:
10828:
10793:
10769:
10749:
10729:
10706:
10686:
10685:{\displaystyle A\to B}
10657:
10631:
10597:
10568:
10511:
10479:
10450:
10376:
10344:
10327:is a closed subset of
10321:
10292:
10278:is a closed subset of
10272:
10248:
10203:
10171:
10142:
10113:
10093:
10092:{\displaystyle x\in X}
10067:
10046:
10014:
9994:
9975:
9946:
9922:
9880:
9850:
9830:
9807:
9778:
9750:
9721:
9720:{\displaystyle x\in X}
9695:
9607:
9578:
9544:
9527:is a closed subset of
9521:
9501:
9481:
9434:
9384:
9351:
9331:
9305:
9285:
9265:
9241:
9217:
9199:if and only if it is "
9193:
9173:
9147:
9127:
9101:
9077:
9049:
9029:
9005:
8975:
8945:
8875:
8791:
8765:
8741:
8710:
8691:
8618:
8558:
8538:
8484:), which implies that
8478:
8409:
8389:
8388:{\displaystyle s\in S}
8363:
8343:
8305:
8291:is a closed subset of
8285:
8223:
8200:
8174:
8154:
8134:
8111:
8085:
8047:
7996:
7942:
7919:
7899:
7852:
7832:
7812:
7742:
7722:
7702:
7664:
7620:
7592:
7516:
7496:
7476:
7442:
7401:
7361:
7330:
7294:
7236:
7200:
7178:
7158:
7125:
7106:
7033:
7013:
6971:
6938:
6915:
6895:
6850:
6802:
6782:
6762:
6736:
6719:is a closed subset of
6713:
6680:
6660:
6636:
6603:
6551:), which implies that
6547:(by definition of the
6541:
6527:is a closed subset of
6521:
6482:
6465:is a closed subset of
6459:
6412:
6345:
6325:
6305:
6285:
6265:
6245:
6193:
6169:
6149:
6125:
6105:
6067:
6047:
5939:
5809:
5786:
5755:
5713:
5689:
5660:
5499:
5479:
5450:
5289:
5269:
5239:
5213:
5147:
5117:
5038:
4962:
4924:
4890:
4870:
4833:
4803:
4780:
4758:
4731:
4704:
4668:
4642:
4585:
4546:
4514:
4483:
4439:
4408:
4388:
4366:
4346:
4322:
4302:
4282:
4258:
4238:
4212:
4192:
4118:
4089:
4061:
4020:
3997:
3977:
3957:
3917:
3855:
3829:
3799:
3723:
3693:
3617:
3573:
3531:
3504:
3482:
3358:
3315:
3293:
3264:
3242:
3213:
3191:
3169:
3146:
3114:
3094:
3062:
2989:
2967:
2939:
2917:
2892:
2858:
2832:
2812:
2792:
2713:
2682:
2638:
2609:
2589:
2569:
2555:is a closed set, then
2549:
2527:
2498:
2472:
2443:
2408:
2385:
2361:
2329:
2294:
2274:
2250:
2249:{\displaystyle x\in X}
2224:
2197:
2162:
2142:
2115:
2092:
2065:
2038:
2010:
1990:
1970:
1950:
1926:
1906:
1879:
1852:
1823:
1803:
1783:
1754:
1725:
1696:
1676:
1656:
1636:
1603:
1558:
1520:
1480:
1460:
1440:
1420:
1400:
1380:
1361:
1338:
1315:
1295:
1275:
1255:
1235:
1215:
1175:
1148:
1128:
1108:
1088:
1068:
1048:
1028:
1008:
988:
966:
945:
925:
905:
885:
847:
846:{\displaystyle s\in S}
821:
795:
775:
755:
721:
701:
678:
643:
623:
545:
525:
505:
479:
438:
437:{\displaystyle s\in S}
412:
411:{\displaystyle r>0}
386:
366:
347:
324:
304:
278:
255:
235:
215:
191:
171:
147:
11753:Zălinescu, Constantin
11524:
11415:
11361:
11329:
11291:
11174:
11147:
11109:
11086:
11048:
11028:
11000:
10965:
10942:
10922:
10893:
10855:
10829:
10794:
10770:
10750:
10730:
10707:
10687:
10658:
10637:may be realized as a
10632:
10598:
10569:
10512:
10480:
10451:
10377:
10345:
10322:
10293:
10273:
10249:
10204:
10202:{\displaystyle f(A).}
10172:
10143:
10119:is close to a subset
10114:
10094:
10068:
10047:
10045:{\displaystyle f(A).}
10015:
9995:
9976:
9947:
9923:
9881:
9851:
9831:
9808:
9779:
9751:
9722:
9696:
9608:
9579:
9545:
9522:
9502:
9482:
9435:
9393:Functions and closure
9385:
9352:
9332:
9306:
9286:
9266:
9242:
9218:
9194:
9174:
9148:
9128:
9102:
9078:
9050:
9030:
9006:
8976:
8946:
8876:
8792:
8766:
8742:
8711:
8692:
8619:
8559:
8539:
8479:
8410:
8390:
8364:
8344:
8306:
8286:
8224:
8201:
8175:
8155:
8135:
8112:
8086:
8048:
7997:
7943:
7920:
7900:
7853:
7833:
7813:
7743:
7723:
7703:
7665:
7621:
7593:
7517:
7497:
7477:
7443:
7402:
7362:
7331:
7295:
7237:
7201:
7179:
7164:is a dense subset of
7159:
7126:
7107:
7034:
7014:
6972:
6939:
6916:
6896:
6851:
6803:
6783:
6763:
6737:
6714:
6681:
6661:
6637:
6604:
6542:
6522:
6483:
6460:
6413:
6346:
6326:
6306:
6286:
6266:
6246:
6199:induces on it), then
6194:
6170:
6150:
6126:
6106:
6068:
6048:
5940:
5810:
5787:
5756:
5714:
5690:
5661:
5500:
5480:
5478:{\displaystyle S_{i}}
5451:
5290:
5270:
5268:{\displaystyle S_{i}}
5240:
5221:complete metric space
5214:
5157:complete metric space
5148:
5118:
5039:
4963:
4925:
4903:The closure operator
4891:
4871:
4834:
4804:
4781:
4759:
4757:{\displaystyle S^{-}}
4732:
4705:
4669:
4643:
4586:
4547:
4515:
4484:
4440:
4409:
4389:
4367:
4347:
4323:
4303:
4283:
4259:
4239:
4213:
4193:
4119:
4090:
4062:
4021:
3998:
3978:
3958:
3918:
3856:
3830:
3800:
3724:
3694:
3618:
3574:
3532:
3505:
3483:
3359:
3316:
3294:
3265:
3243:
3214:
3192:
3170:
3147:
3115:
3095:
3093:{\displaystyle (0,1)}
3063:
2990:
2968:
2940:
2918:
2893:
2859:
2833:
2813:
2793:
2714:
2683:
2656:first-countable space
2639:
2610:
2590:
2570:
2550:
2528:
2499:
2473:
2444:
2409:
2386:
2362:
2330:
2295:
2275:
2251:
2225:
2198:
2163:
2143:
2116:
2093:
2066:
2039:
2011:
1991:
1971:
1951:
1927:
1907:
1880:
1853:
1824:
1804:
1784:
1755:
1726:
1697:
1695:{\displaystyle \tau }
1677:
1657:
1655:{\displaystyle \tau }
1637:
1604:
1559:
1521:
1496:Closure (mathematics)
1481:
1461:
1441:
1421:
1401:
1381:
1362:
1339:
1316:
1296:
1276:
1256:
1236:
1216:
1176:
1149:
1129:
1109:
1089:
1069:
1049:
1034:that is not equal to
1029:
1009:
989:
967:
946:
926:
906:
886:
848:
822:
796:
776:
756:
722:
702:
679:
644:
642:{\displaystyle \inf }
624:
546:
526:
506:
480:
439:
413:
387:
367:
348:
325:
310:Fully expressed, for
305:
279:
256:
236:
216:
192:
172:
148:
12113:Invariance of domain
12065:Euler characteristic
12039:Bundle (mathematics)
11424:
11370:
11338:
11300:
11259:
11233:Limit point of a set
11160:
11121:
11095:
11057:
11037:
11011:
10974:
10951:
10931:
10902:
10867:
10838:
10803:
10783:
10759:
10739:
10716:
10696:
10670:
10647:
10621:
10578:
10521:
10489:
10460:
10386:
10354:
10331:
10320:{\displaystyle f(C)}
10302:
10282:
10262:
10226:
10218:Open and closed maps
10181:
10170:{\displaystyle f(x)}
10152:
10123:
10103:
10077:
10057:
10024:
10004:
9984:
9956:
9936:
9890:
9864:
9840:
9817:
9806:{\displaystyle f(A)}
9788:
9777:{\displaystyle f(x)}
9759:
9731:
9705:
9616:
9588:
9556:
9531:
9511:
9491:
9452:
9412:
9361:
9341:
9315:
9295:
9275:
9255:
9227:
9207:
9183:
9157:
9137:
9111:
9091:
9063:
9039:
9019:
9011:is endowed with the
8985:
8955:
8885:
8801:
8797:is any subset then:
8775:
8755:
8727:
8700:
8628:
8568:
8548:
8488:
8419:
8399:
8373:
8353:
8315:
8295:
8233:
8210:
8184:
8164:
8144:
8121:
8095:
8057:
8006:
7952:
7929:
7909:
7877:
7842:
7822:
7752:
7732:
7712:
7674:
7630:
7602:
7526:
7506:
7486:
7454:
7411:
7370:
7340:
7304:
7246:
7210:
7190:
7168:
7142:
7115:
7043:
7023:
6981:
6948:
6925:
6905:
6860:
6812:
6792:
6772:
6746:
6723:
6690:
6670:
6650:
6613:
6555:
6531:
6492:
6469:
6436:
6355:
6335:
6315:
6295:
6275:
6255:
6203:
6183:
6175:is endowed with the
6159:
6139:
6115:
6083:
6057:
5949:
5851:
5821:of sets is always a
5796:
5776:
5723:
5703:
5679:
5671:Facts about closures
5509:
5489:
5462:
5299:
5279:
5252:
5226:
5177:
5134:
5054:
4975:
4942:
4907:
4880:
4843:
4817:
4790:
4768:
4741:
4714:
4678:
4652:
4595:
4563:
4523:
4501:
4473:
4425:
4398:
4376:
4356:
4332:
4312:
4292:
4268:
4248:
4226:
4220:both closed and open
4202:
4128:
4103:
4079:
4029:
4007:
3987:
3967:
3944:
3865:
3843:
3811:
3807:If one considers on
3737:
3705:
3701:If one considers on
3631:
3625:lower limit topology
3623:is endowed with the
3599:
3541:
3518:
3494:
3368:
3328:
3305:
3278:
3252:
3227:
3201:
3179:
3159:
3124:
3104:
3072:
3003:
2977:
2957:
2927:
2905:
2867:
2845:
2822:
2802:
2770:
2700:
2666:
2619:
2599:
2579:
2559:
2539:
2508:
2482:
2456:
2421:
2398:
2375:
2345:
2304:
2284:
2264:
2234:
2208:
2175:
2152:
2126:
2102:
2076:
2052:
2022:
2000:
1980:
1960:
1940:
1916:
1890:
1866:
1836:
1813:
1793:
1764:
1734:
1706:
1686:
1666:
1646:
1613:
1568:
1533:
1510:
1470:
1466:is a limit point of
1450:
1430:
1410:
1390:
1370:
1348:
1328:
1305:
1285:
1265:
1245:
1225:
1205:
1165:
1138:
1118:
1098:
1078:
1058:
1038:
1018:
998:
978:
956:
935:
915:
895:
875:
869:limit point of a set
863:Limit point of a set
831:
805:
785:
781:contains a point of
765:
745:
711:
688:
668:
633:
555:
535:
515:
489:
448:
422:
396:
376:
356:
334:
314:
291:
268:
245:
225:
221:contains a point of
205:
181:
161:
137:
19:For other uses, see
12123:Tychonoff's theorem
12118:Poincaré conjecture
11872:General (point-set)
11687:Elementary Topology
11227:Interior (topology)
9444:if and only if the
9404:Continuous function
9223:", meaning that if
7242:It is possible for
6311:and the closure of
6251:and the closure of
5765:The closure of the
5171: —
4876:(so complements in
4710:where the notation
4352:is irrational. So,
4308:, but cannot be in
3223:is the whole space
2729:filters in topology
1094:. A limit point of
820:{\displaystyle x=s}
504:{\displaystyle x=s}
241:(this point can be
12108:De Rham cohomology
12029:Polyhedral complex
12019:Simplicial complex
11810:"Closure of a set"
11519:
11410:
11356:
11324:
11286:
11215:Closed regular set
11172:{\displaystyle A.}
11169:
11142:
11107:{\displaystyle A,}
11104:
11081:
11043:
11023:
10995:
10963:{\displaystyle I,}
10960:
10937:
10917:
10888:
10850:
10824:
10789:
10765:
10745:
10728:{\displaystyle B.}
10725:
10702:
10682:
10653:
10627:
10593:
10564:
10507:
10475:
10446:
10372:
10343:{\displaystyle Y.}
10340:
10317:
10288:
10268:
10244:
10199:
10167:
10138:
10109:
10089:
10063:
10042:
10010:
9990:
9971:
9942:
9918:
9876:
9846:
9829:{\displaystyle Y.}
9826:
9803:
9774:
9746:
9717:
9691:
9603:
9574:
9543:{\displaystyle Y.}
9540:
9517:
9497:
9477:
9430:
9380:
9347:
9327:
9301:
9281:
9261:
9237:
9213:
9189:
9169:
9143:
9123:
9097:
9073:
9045:
9025:
9001:
8971:
8941:
8871:
8842:
8787:
8761:
8737:
8706:
8687:
8614:
8554:
8534:
8474:
8405:
8385:
8359:
8339:
8301:
8281:
8222:{\displaystyle X.}
8219:
8196:
8170:
8150:
8133:{\displaystyle T,}
8130:
8107:
8091:). The complement
8081:
8043:
8002:which is equal to
7992:
7941:{\displaystyle X.}
7938:
7915:
7895:
7848:
7828:
7808:
7748:then the equality
7738:
7718:
7698:
7660:
7616:
7588:
7512:
7492:
7472:
7438:
7397:
7357:
7336:for example, take
7326:
7290:
7232:
7196:
7174:
7154:
7121:
7102:
7029:
7009:
6967:
6944:the minimality of
6937:{\displaystyle X,}
6934:
6911:
6891:
6846:
6798:
6778:
6758:
6735:{\displaystyle T,}
6732:
6709:
6676:
6656:
6632:
6599:
6537:
6517:
6481:{\displaystyle X,}
6478:
6455:
6408:
6341:
6321:
6301:
6281:
6261:
6241:
6189:
6165:
6145:
6121:
6101:
6063:
6043:
6019:
5985:
5935:
5817:The closure of an
5808:{\displaystyle X.}
5805:
5782:
5751:
5709:
5685:
5656:
5632:
5547:
5495:
5475:
5446:
5422:
5337:
5285:
5265:
5238:{\displaystyle X.}
5235:
5209:
5165:
5146:{\displaystyle X.}
5143:
5113:
5034:
4968:in the sense that
4958:
4920:
4900:of the topology).
4886:
4866:
4829:
4802:{\displaystyle X,}
4799:
4776:
4754:
4727:
4700:
4664:
4638:
4581:
4542:
4513:{\displaystyle X,}
4510:
4479:
4435:
4404:
4384:
4362:
4342:
4318:
4298:
4278:
4254:
4234:
4208:
4188:
4114:
4085:
4057:
4019:{\displaystyle X,}
4016:
3993:
3973:
3956:{\displaystyle X,}
3953:
3913:
3851:
3825:
3795:
3719:
3689:
3613:
3569:
3530:{\displaystyle X,}
3527:
3500:
3478:
3354:
3311:
3289:
3260:
3238:
3209:
3187:
3165:
3142:
3110:
3090:
3058:
2985:
2963:
2935:
2913:
2888:
2857:{\displaystyle X,}
2854:
2828:
2808:
2788:
2712:{\displaystyle S.}
2709:
2692:of all convergent
2688:is the set of all
2678:
2634:
2605:
2585:
2565:
2545:
2523:
2494:
2468:
2439:
2404:
2381:
2357:
2325:
2290:
2280:that converges to
2270:
2246:
2230:is the set of all
2220:
2193:
2158:
2138:
2114:{\displaystyle S.}
2111:
2088:
2064:{\displaystyle S.}
2061:
2034:
2006:
1986:
1966:
1946:
1922:
1902:
1878:{\displaystyle S.}
1875:
1858:is the set of all
1848:
1819:
1799:
1779:
1750:
1721:
1692:
1672:
1652:
1632:
1599:
1554:
1516:
1476:
1456:
1436:
1416:
1396:
1376:
1360:{\displaystyle x,}
1357:
1334:
1311:
1291:
1271:
1251:
1231:
1211:
1188:accumulation point
1171:
1144:
1124:
1104:
1084:
1064:
1044:
1024:
1004:
984:
962:
941:
921:
901:
881:
843:
817:
791:
771:
751:
717:
700:{\displaystyle X.}
697:
674:
658:topological spaces
639:
619:
594:
541:
521:
501:
475:
434:
418:there exists some
408:
382:
362:
346:{\displaystyle d,}
343:
320:
303:{\displaystyle X.}
300:
274:
251:
231:
211:
187:
167:
143:
74:, and also as the
50:together with all
12241:Closure operators
12223:
12222:
12012:fundamental group
11774:978-981-4488-15-0
11747:, Allyn and Bacon
11506:
11500:
11475:
11469:
11438:
11432:
11184:algebraic closure
11046:{\displaystyle A}
10940:{\displaystyle A}
10792:{\displaystyle P}
10768:{\displaystyle X}
10748:{\displaystyle T}
10705:{\displaystyle A}
10656:{\displaystyle P}
10630:{\displaystyle X}
10456:for every subset
10291:{\displaystyle X}
10271:{\displaystyle C}
10112:{\displaystyle x}
10066:{\displaystyle f}
10013:{\displaystyle A}
9993:{\displaystyle f}
9945:{\displaystyle f}
9849:{\displaystyle x}
9656:
9650:
9520:{\displaystyle C}
9500:{\displaystyle X}
9350:{\displaystyle U}
9304:{\displaystyle X}
9284:{\displaystyle S}
9264:{\displaystyle X}
9216:{\displaystyle X}
9192:{\displaystyle X}
9146:{\displaystyle X}
9100:{\displaystyle X}
9085:coordinate charts
9048:{\displaystyle X}
9028:{\displaystyle X}
9015:induced on it by
9013:subspace topology
8823:
8764:{\displaystyle X}
8723:Consequently, if
8720:
8719:
8557:{\displaystyle T}
8408:{\displaystyle T}
8362:{\displaystyle S}
8304:{\displaystyle X}
8180:now implies that
8173:{\displaystyle X}
8153:{\displaystyle T}
7918:{\displaystyle T}
7851:{\displaystyle T}
7831:{\displaystyle S}
7741:{\displaystyle X}
7721:{\displaystyle T}
7565:
7559:
7515:{\displaystyle T}
7495:{\displaystyle S}
7199:{\displaystyle T}
7177:{\displaystyle T}
7135:
7134:
7032:{\displaystyle T}
6914:{\displaystyle C}
6801:{\displaystyle X}
6781:{\displaystyle C}
6679:{\displaystyle S}
6659:{\displaystyle T}
6646:closed subset of
6549:subspace topology
6540:{\displaystyle T}
6488:the intersection
6382:
6376:
6344:{\displaystyle X}
6324:{\displaystyle S}
6304:{\displaystyle T}
6284:{\displaystyle T}
6264:{\displaystyle S}
6192:{\displaystyle X}
6177:subspace topology
6168:{\displaystyle T}
6148:{\displaystyle X}
6124:{\displaystyle T}
6066:{\displaystyle I}
6053:is possible when
6004:
5970:
5785:{\displaystyle X}
5769:is the empty set;
5712:{\displaystyle X}
5688:{\displaystyle S}
5615:
5530:
5498:{\displaystyle X}
5405:
5320:
5288:{\displaystyle X}
5163:
4889:{\displaystyle X}
4725:
4558:topological space
4482:{\displaystyle X}
4433:
4407:{\displaystyle X}
4365:{\displaystyle S}
4340:
4321:{\displaystyle S}
4301:{\displaystyle S}
4276:
4257:{\displaystyle S}
4211:{\displaystyle S}
4097:relative topology
4088:{\displaystyle X}
3996:{\displaystyle A}
3976:{\displaystyle X}
3731:discrete topology
3503:{\displaystyle S}
3314:{\displaystyle X}
3168:{\displaystyle X}
3113:{\displaystyle X}
2966:{\displaystyle X}
2760:topological space
2608:{\displaystyle A}
2588:{\displaystyle S}
2568:{\displaystyle A}
2548:{\displaystyle A}
2407:{\displaystyle S}
2384:{\displaystyle S}
2293:{\displaystyle x}
2273:{\displaystyle S}
2161:{\displaystyle S}
2009:{\displaystyle S}
1989:{\displaystyle S}
1969:{\displaystyle S}
1949:{\displaystyle S}
1936:. (Each point of
1925:{\displaystyle S}
1860:points of closure
1745:
1675:{\displaystyle X}
1528:topological space
1519:{\displaystyle S}
1479:{\displaystyle S}
1459:{\displaystyle x}
1439:{\displaystyle S}
1426:is an element of
1419:{\displaystyle x}
1399:{\displaystyle S}
1379:{\displaystyle x}
1337:{\displaystyle S}
1314:{\displaystyle x}
1294:{\displaystyle S}
1274:{\displaystyle x}
1254:{\displaystyle S}
1234:{\displaystyle S}
1214:{\displaystyle x}
1174:{\displaystyle S}
1147:{\displaystyle S}
1127:{\displaystyle S}
1107:{\displaystyle S}
1087:{\displaystyle S}
1067:{\displaystyle x}
1047:{\displaystyle x}
1027:{\displaystyle S}
1007:{\displaystyle x}
987:{\displaystyle x}
965:{\displaystyle x}
944:{\displaystyle S}
924:{\displaystyle x}
904:{\displaystyle S}
884:{\displaystyle x}
794:{\displaystyle S}
774:{\displaystyle x}
754:{\displaystyle S}
720:{\displaystyle x}
677:{\displaystyle S}
579:
544:{\displaystyle S}
524:{\displaystyle x}
385:{\displaystyle S}
365:{\displaystyle x}
323:{\displaystyle X}
277:{\displaystyle S}
254:{\displaystyle x}
234:{\displaystyle S}
214:{\displaystyle x}
190:{\displaystyle S}
170:{\displaystyle x}
153:as a subset of a
146:{\displaystyle S}
110:to the notion of
58:. The closure of
40:topological space
12248:
12236:General topology
12213:
12212:
12186:
12185:
12176:
12166:
12156:
12155:
12144:
12143:
11938:
11851:
11844:
11837:
11828:
11827:
11823:
11798:
11795:Internet Archive
11762:
11755:(30 July 2002).
11748:
11739:
11738:, Academic Press
11730:
11721:
11710:
11699:
11681:
11670:
11659:
11632:
11626:
11620:
11614:
11608:
11594:
11588:
11582:
11576:
11570:
11564:
11558:
11552:
11546:
11529:
11528:
11526:
11525:
11520:
11504:
11498:
11491:
11490:
11473:
11467:
11448:
11447:
11436:
11430:
11419:
11417:
11416:
11411:
11382:
11381:
11365:
11363:
11362:
11357:
11334:it follows that
11333:
11331:
11330:
11325:
11295:
11293:
11292:
11287:
11253:
11238:
11211:
11188:universal arrows
11178:
11176:
11175:
11170:
11151:
11149:
11148:
11143:
11113:
11111:
11110:
11105:
11090:
11088:
11087:
11082:
11052:
11050:
11049:
11044:
11032:
11030:
11029:
11024:
11004:
11002:
11001:
10996:
10969:
10967:
10966:
10961:
10946:
10944:
10943:
10938:
10926:
10924:
10923:
10918:
10897:
10895:
10894:
10889:
10859:
10857:
10856:
10851:
10833:
10831:
10830:
10825:
10798:
10796:
10795:
10790:
10774:
10772:
10771:
10766:
10754:
10752:
10751:
10746:
10734:
10732:
10731:
10726:
10711:
10709:
10708:
10703:
10691:
10689:
10688:
10683:
10662:
10660:
10659:
10654:
10636:
10634:
10633:
10628:
10602:
10600:
10599:
10594:
10573:
10571:
10570:
10565:
10533:
10532:
10516:
10514:
10513:
10508:
10484:
10482:
10481:
10476:
10455:
10453:
10452:
10447:
10445:
10441:
10434:
10433:
10398:
10397:
10381:
10379:
10378:
10373:
10349:
10347:
10346:
10341:
10326:
10324:
10323:
10318:
10297:
10295:
10294:
10289:
10277:
10275:
10274:
10269:
10254:is a (strongly)
10253:
10251:
10250:
10245:
10208:
10206:
10205:
10200:
10176:
10174:
10173:
10168:
10147:
10145:
10144:
10139:
10118:
10116:
10115:
10110:
10098:
10096:
10095:
10090:
10072:
10070:
10069:
10064:
10051:
10049:
10048:
10043:
10019:
10017:
10016:
10011:
9999:
9997:
9996:
9991:
9980:
9978:
9977:
9972:
9951:
9949:
9948:
9943:
9927:
9925:
9924:
9919:
9908:
9907:
9885:
9883:
9882:
9877:
9855:
9853:
9852:
9847:
9835:
9833:
9832:
9827:
9812:
9810:
9809:
9804:
9783:
9781:
9780:
9775:
9755:
9753:
9752:
9747:
9726:
9724:
9723:
9718:
9700:
9698:
9697:
9692:
9666:
9665:
9654:
9648:
9647:
9643:
9636:
9635:
9612:
9610:
9609:
9604:
9583:
9581:
9580:
9575:
9549:
9547:
9546:
9541:
9526:
9524:
9523:
9518:
9506:
9504:
9503:
9498:
9486:
9484:
9483:
9478:
9467:
9466:
9439:
9437:
9436:
9431:
9389:
9387:
9386:
9381:
9376:
9375:
9356:
9354:
9353:
9348:
9336:
9334:
9333:
9328:
9310:
9308:
9307:
9302:
9290:
9288:
9287:
9282:
9270:
9268:
9267:
9262:
9246:
9244:
9243:
9238:
9236:
9235:
9222:
9220:
9219:
9214:
9198:
9196:
9195:
9190:
9178:
9176:
9175:
9170:
9152:
9150:
9149:
9144:
9132:
9130:
9129:
9124:
9106:
9104:
9103:
9098:
9082:
9080:
9079:
9074:
9072:
9071:
9054:
9052:
9051:
9046:
9034:
9032:
9031:
9026:
9010:
9008:
9007:
9002:
9000:
8999:
8980:
8978:
8977:
8972:
8970:
8969:
8950:
8948:
8947:
8942:
8934:
8933:
8897:
8896:
8880:
8878:
8877:
8872:
8852:
8851:
8841:
8840:
8839:
8813:
8812:
8796:
8794:
8793:
8788:
8770:
8768:
8767:
8762:
8746:
8744:
8743:
8738:
8736:
8735:
8715:
8713:
8712:
8707:
8696:
8694:
8693:
8688:
8677:
8676:
8646:
8645:
8623:
8621:
8620:
8615:
8586:
8585:
8563:
8561:
8560:
8555:
8543:
8541:
8540:
8535:
8500:
8499:
8483:
8481:
8480:
8475:
8449:
8448:
8414:
8412:
8411:
8406:
8394:
8392:
8391:
8386:
8368:
8366:
8365:
8360:
8348:
8346:
8345:
8340:
8310:
8308:
8307:
8302:
8290:
8288:
8287:
8282:
8228:
8226:
8225:
8220:
8206:is also open in
8205:
8203:
8202:
8197:
8179:
8177:
8176:
8171:
8159:
8157:
8156:
8151:
8139:
8137:
8136:
8131:
8116:
8114:
8113:
8108:
8090:
8088:
8087:
8082:
8052:
8050:
8049:
8044:
8024:
8023:
8001:
7999:
7998:
7993:
7970:
7969:
7947:
7945:
7944:
7939:
7924:
7922:
7921:
7916:
7905:and assume that
7904:
7902:
7901:
7896:
7862:
7861:
7857:
7855:
7854:
7849:
7837:
7835:
7834:
7829:
7817:
7815:
7814:
7809:
7801:
7800:
7764:
7763:
7747:
7745:
7744:
7739:
7727:
7725:
7724:
7719:
7707:
7705:
7704:
7699:
7669:
7667:
7666:
7661:
7625:
7623:
7622:
7617:
7615:
7597:
7595:
7594:
7589:
7581:
7580:
7563:
7557:
7538:
7537:
7521:
7519:
7518:
7513:
7501:
7499:
7498:
7493:
7481:
7479:
7478:
7473:
7447:
7445:
7444:
7439:
7406:
7404:
7403:
7398:
7366:
7364:
7363:
7358:
7353:
7335:
7333:
7332:
7327:
7316:
7315:
7299:
7297:
7296:
7291:
7283:
7282:
7258:
7257:
7241:
7239:
7238:
7233:
7222:
7221:
7205:
7203:
7202:
7197:
7183:
7181:
7180:
7175:
7163:
7161:
7160:
7155:
7138:It follows that
7130:
7128:
7127:
7122:
7111:
7109:
7108:
7103:
7092:
7091:
7061:
7060:
7038:
7036:
7035:
7030:
7018:
7016:
7015:
7010:
6993:
6992:
6976:
6974:
6973:
6968:
6960:
6959:
6943:
6941:
6940:
6935:
6920:
6918:
6917:
6912:
6900:
6898:
6897:
6892:
6878:
6877:
6855:
6853:
6852:
6847:
6824:
6823:
6807:
6805:
6804:
6799:
6787:
6785:
6784:
6779:
6767:
6765:
6764:
6759:
6741:
6739:
6738:
6733:
6718:
6716:
6715:
6710:
6702:
6701:
6685:
6683:
6682:
6677:
6665:
6663:
6662:
6657:
6641:
6639:
6638:
6633:
6625:
6624:
6608:
6606:
6605:
6600:
6592:
6591:
6567:
6566:
6546:
6544:
6543:
6538:
6526:
6524:
6523:
6518:
6510:
6509:
6487:
6485:
6484:
6479:
6464:
6462:
6461:
6456:
6448:
6447:
6421:
6420:
6417:
6415:
6414:
6409:
6398:
6397:
6380:
6374:
6367:
6366:
6350:
6348:
6347:
6342:
6330:
6328:
6327:
6322:
6310:
6308:
6307:
6302:
6290:
6288:
6287:
6282:
6270:
6268:
6267:
6262:
6250:
6248:
6247:
6242:
6234:
6233:
6215:
6214:
6198:
6196:
6195:
6190:
6174:
6172:
6171:
6166:
6154:
6152:
6151:
6146:
6130:
6128:
6127:
6122:
6110:
6108:
6107:
6102:
6072:
6070:
6069:
6064:
6052:
6050:
6049:
6044:
6042:
6041:
6029:
6028:
6018:
6000:
5996:
5995:
5994:
5984:
5961:
5960:
5944:
5942:
5941:
5936:
5922:
5921:
5897:
5896:
5863:
5862:
5814:
5812:
5811:
5806:
5791:
5789:
5788:
5783:
5760:
5758:
5757:
5752:
5735:
5734:
5718:
5716:
5715:
5710:
5694:
5692:
5691:
5686:
5665:
5663:
5662:
5657:
5652:
5648:
5647:
5643:
5642:
5641:
5631:
5630:
5606:
5605:
5588:
5587:
5575:
5571:
5570:
5569:
5557:
5556:
5546:
5545:
5521:
5520:
5504:
5502:
5501:
5496:
5484:
5482:
5481:
5476:
5474:
5473:
5455:
5453:
5452:
5447:
5442:
5438:
5437:
5433:
5432:
5431:
5421:
5420:
5396:
5395:
5378:
5377:
5365:
5361:
5360:
5359:
5347:
5346:
5336:
5335:
5311:
5310:
5294:
5292:
5291:
5286:
5274:
5272:
5271:
5266:
5264:
5263:
5244:
5242:
5241:
5236:
5218:
5216:
5215:
5210:
5202:
5201:
5189:
5188:
5172:
5169:
5152:
5150:
5149:
5144:
5122:
5120:
5119:
5114:
5091:
5090:
5066:
5065:
5043:
5041:
5040:
5035:
5012:
5011:
4987:
4986:
4967:
4965:
4964:
4959:
4954:
4953:
4929:
4927:
4926:
4921:
4919:
4918:
4895:
4893:
4892:
4887:
4875:
4873:
4872:
4867:
4850:
4838:
4836:
4835:
4830:
4808:
4806:
4805:
4800:
4785:
4783:
4782:
4777:
4775:
4763:
4761:
4760:
4755:
4753:
4752:
4736:
4734:
4733:
4728:
4726:
4718:
4709:
4707:
4706:
4701:
4690:
4689:
4673:
4671:
4670:
4665:
4647:
4645:
4644:
4639:
4607:
4606:
4590:
4588:
4587:
4582:
4551:
4549:
4548:
4543:
4532:
4531:
4519:
4517:
4516:
4511:
4488:
4486:
4485:
4480:
4466:closure operator
4455:Closure operator
4449:Closure operator
4444:
4442:
4441:
4436:
4434:
4429:
4413:
4411:
4410:
4405:
4393:
4391:
4390:
4385:
4383:
4371:
4369:
4368:
4363:
4351:
4349:
4348:
4343:
4341:
4336:
4327:
4325:
4324:
4319:
4307:
4305:
4304:
4299:
4287:
4285:
4284:
4279:
4277:
4272:
4263:
4261:
4260:
4255:
4244:because neither
4243:
4241:
4240:
4235:
4233:
4217:
4215:
4214:
4209:
4197:
4195:
4194:
4189:
4163:
4162:
4150:
4123:
4121:
4120:
4115:
4110:
4094:
4092:
4091:
4086:
4066:
4064:
4063:
4058:
4041:
4040:
4025:
4023:
4022:
4017:
4002:
4000:
3999:
3994:
3982:
3980:
3979:
3974:
3962:
3960:
3959:
3954:
3939:indiscrete space
3922:
3920:
3919:
3914:
3909:
3877:
3876:
3860:
3858:
3857:
3852:
3850:
3837:trivial topology
3834:
3832:
3831:
3826:
3824:
3804:
3802:
3801:
3796:
3749:
3748:
3728:
3726:
3725:
3720:
3718:
3698:
3696:
3695:
3690:
3643:
3642:
3622:
3620:
3619:
3614:
3612:
3578:
3576:
3575:
3570:
3553:
3552:
3536:
3534:
3533:
3528:
3509:
3507:
3506:
3501:
3487:
3485:
3484:
3479:
3465:
3457:
3449:
3432:
3428:
3418:
3410:
3402:
3380:
3379:
3363:
3361:
3360:
3355:
3350:
3349:
3344:
3335:
3320:
3318:
3317:
3312:
3298:
3296:
3295:
3290:
3285:
3269:
3267:
3266:
3261:
3259:
3247:
3245:
3244:
3239:
3234:
3221:rational numbers
3218:
3216:
3215:
3210:
3208:
3196:
3194:
3193:
3188:
3186:
3174:
3172:
3171:
3166:
3151:
3149:
3148:
3145:{\displaystyle }
3143:
3119:
3117:
3116:
3111:
3099:
3097:
3096:
3091:
3067:
3065:
3064:
3059:
3015:
3014:
2994:
2992:
2991:
2986:
2984:
2972:
2970:
2969:
2964:
2944:
2942:
2941:
2936:
2934:
2922:
2920:
2919:
2914:
2912:
2897:
2895:
2894:
2889:
2863:
2861:
2860:
2855:
2837:
2835:
2834:
2829:
2817:
2815:
2814:
2809:
2797:
2795:
2794:
2789:
2737:closure operator
2718:
2716:
2715:
2710:
2687:
2685:
2684:
2679:
2643:
2641:
2640:
2635:
2614:
2612:
2611:
2606:
2594:
2592:
2591:
2586:
2574:
2572:
2571:
2566:
2554:
2552:
2551:
2546:
2532:
2530:
2529:
2524:
2503:
2501:
2500:
2495:
2477:
2475:
2474:
2469:
2448:
2446:
2445:
2440:
2413:
2411:
2410:
2405:
2390:
2388:
2387:
2382:
2366:
2364:
2363:
2358:
2334:
2332:
2331:
2326:
2299:
2297:
2296:
2291:
2279:
2277:
2276:
2271:
2255:
2253:
2252:
2247:
2229:
2227:
2226:
2221:
2202:
2200:
2199:
2194:
2167:
2165:
2164:
2159:
2148:is the union of
2147:
2145:
2144:
2139:
2120:
2118:
2117:
2112:
2097:
2095:
2094:
2089:
2070:
2068:
2067:
2062:
2043:
2041:
2040:
2035:
2015:
2013:
2012:
2007:
1995:
1993:
1992:
1987:
1975:
1973:
1972:
1967:
1955:
1953:
1952:
1947:
1931:
1929:
1928:
1923:
1911:
1909:
1908:
1903:
1884:
1882:
1881:
1876:
1857:
1855:
1854:
1849:
1828:
1826:
1825:
1820:
1808:
1806:
1805:
1800:
1788:
1786:
1785:
1780:
1778:
1777:
1772:
1759:
1757:
1756:
1751:
1746:
1738:
1730:
1728:
1727:
1722:
1701:
1699:
1698:
1693:
1681:
1679:
1678:
1673:
1661:
1659:
1658:
1653:
1641:
1639:
1638:
1633:
1625:
1624:
1608:
1606:
1605:
1600:
1592:
1591:
1563:
1561:
1560:
1555:
1525:
1523:
1522:
1517:
1490:Closure of a set
1485:
1483:
1482:
1477:
1465:
1463:
1462:
1457:
1445:
1443:
1442:
1437:
1425:
1423:
1422:
1417:
1405:
1403:
1402:
1397:
1385:
1383:
1382:
1377:
1366:
1364:
1363:
1358:
1343:
1341:
1340:
1335:
1324:For a given set
1320:
1318:
1317:
1312:
1300:
1298:
1297:
1292:
1280:
1278:
1277:
1272:
1260:
1258:
1257:
1252:
1240:
1238:
1237:
1232:
1220:
1218:
1217:
1212:
1180:
1178:
1177:
1172:
1153:
1151:
1150:
1145:
1133:
1131:
1130:
1125:
1113:
1111:
1110:
1105:
1093:
1091:
1090:
1085:
1073:
1071:
1070:
1065:
1053:
1051:
1050:
1045:
1033:
1031:
1030:
1025:
1013:
1011:
1010:
1005:
993:
991:
990:
985:
971:
969:
968:
963:
950:
948:
947:
942:
930:
928:
927:
922:
910:
908:
907:
902:
890:
888:
887:
882:
852:
850:
849:
844:
826:
824:
823:
818:
800:
798:
797:
792:
780:
778:
777:
772:
760:
758:
757:
752:
731:point of closure
726:
724:
723:
718:
706:
704:
703:
698:
683:
681:
680:
675:
648:
646:
645:
640:
628:
626:
625:
620:
593:
551:if the distance
550:
548:
547:
542:
530:
528:
527:
522:
510:
508:
507:
502:
484:
482:
481:
476:
443:
441:
440:
435:
417:
415:
414:
409:
391:
389:
388:
383:
371:
369:
368:
363:
352:
350:
349:
344:
329:
327:
326:
321:
309:
307:
306:
301:
283:
281:
280:
275:
260:
258:
257:
252:
240:
238:
237:
232:
220:
218:
217:
212:
196:
194:
193:
188:
176:
174:
173:
168:
152:
150:
149:
144:
123:Point of closure
105:
100:point of closure
97:
93:
89:
85:
69:
61:
57:
49:
42:consists of all
37:
12256:
12255:
12251:
12250:
12249:
12247:
12246:
12245:
12226:
12225:
12224:
12219:
12150:
12132:
12128:Urysohn's lemma
12089:
12053:
11939:
11930:
11902:low-dimensional
11860:
11855:
11808:
11805:
11775:
11719:
11697:
11679:
11657:
11641:
11636:
11635:
11627:
11623:
11615:
11611:
11595:
11591:
11583:
11579:
11571:
11567:
11561:Kuratowski 1966
11559:
11555:
11547:
11543:
11538:
11533:
11532:
11486:
11482:
11443:
11439:
11425:
11422:
11421:
11420:which implies
11377:
11373:
11371:
11368:
11367:
11339:
11336:
11335:
11301:
11298:
11297:
11260:
11257:
11256:
11254:
11250:
11245:
11236:
11209:
11206:Closure algebra
11196:
11161:
11158:
11157:
11122:
11119:
11118:
11116:terminal object
11096:
11093:
11092:
11058:
11055:
11054:
11038:
11035:
11034:
11012:
11009:
11008:
10975:
10972:
10971:
10952:
10949:
10948:
10932:
10929:
10928:
10903:
10900:
10899:
10868:
10865:
10864:
10839:
10836:
10835:
10804:
10801:
10800:
10784:
10781:
10780:
10760:
10757:
10756:
10740:
10737:
10736:
10717:
10714:
10713:
10712:is a subset of
10697:
10694:
10693:
10671:
10668:
10667:
10648:
10645:
10644:
10622:
10619:
10618:
10608:
10579:
10576:
10575:
10528:
10524:
10522:
10519:
10518:
10490:
10487:
10486:
10461:
10458:
10457:
10429:
10425:
10424:
10420:
10393:
10389:
10387:
10384:
10383:
10355:
10352:
10351:
10332:
10329:
10328:
10303:
10300:
10299:
10283:
10280:
10279:
10263:
10260:
10259:
10227:
10224:
10223:
10220:
10214:
10182:
10179:
10178:
10153:
10150:
10149:
10124:
10121:
10120:
10104:
10101:
10100:
10078:
10075:
10074:
10058:
10055:
10054:
10025:
10022:
10021:
10005:
10002:
10001:
9985:
9982:
9981:
9957:
9954:
9953:
9937:
9934:
9933:
9903:
9899:
9891:
9888:
9887:
9865:
9862:
9861:
9841:
9838:
9837:
9818:
9815:
9814:
9789:
9786:
9785:
9760:
9757:
9756:
9732:
9729:
9728:
9706:
9703:
9702:
9661:
9657:
9631:
9627:
9626:
9622:
9617:
9614:
9613:
9589:
9586:
9585:
9557:
9554:
9553:
9532:
9529:
9528:
9512:
9509:
9508:
9492:
9489:
9488:
9459:
9455:
9453:
9450:
9449:
9413:
9410:
9409:
9406:
9400:
9395:
9371:
9370:
9362:
9359:
9358:
9342:
9339:
9338:
9316:
9313:
9312:
9311:if and only if
9296:
9293:
9292:
9276:
9273:
9272:
9256:
9253:
9252:
9231:
9230:
9228:
9225:
9224:
9208:
9205:
9204:
9184:
9181:
9180:
9158:
9155:
9154:
9138:
9135:
9134:
9112:
9109:
9108:
9092:
9089:
9088:
9083:are domains of
9067:
9066:
9064:
9061:
9060:
9040:
9037:
9036:
9020:
9017:
9016:
8995:
8994:
8986:
8983:
8982:
8965:
8964:
8956:
8953:
8952:
8929:
8925:
8892:
8888:
8886:
8883:
8882:
8847:
8843:
8835:
8834:
8827:
8808:
8804:
8802:
8799:
8798:
8776:
8773:
8772:
8756:
8753:
8752:
8731:
8730:
8728:
8725:
8724:
8721:
8701:
8698:
8697:
8672:
8668:
8641:
8637:
8629:
8626:
8625:
8581:
8577:
8569:
8566:
8565:
8549:
8546:
8545:
8495:
8491:
8489:
8486:
8485:
8444:
8440:
8420:
8417:
8416:
8400:
8397:
8396:
8374:
8371:
8370:
8354:
8351:
8350:
8316:
8313:
8312:
8296:
8293:
8292:
8234:
8231:
8230:
8211:
8208:
8207:
8185:
8182:
8181:
8165:
8162:
8161:
8145:
8142:
8141:
8122:
8119:
8118:
8096:
8093:
8092:
8058:
8055:
8054:
8019:
8015:
8007:
8004:
8003:
7965:
7961:
7953:
7950:
7949:
7930:
7927:
7926:
7910:
7907:
7906:
7878:
7875:
7874:
7867:
7843:
7840:
7839:
7823:
7820:
7819:
7796:
7792:
7759:
7755:
7753:
7750:
7749:
7733:
7730:
7729:
7713:
7710:
7709:
7708:), although if
7675:
7672:
7671:
7631:
7628:
7627:
7611:
7603:
7600:
7599:
7576:
7572:
7533:
7529:
7527:
7524:
7523:
7507:
7504:
7503:
7487:
7484:
7483:
7455:
7452:
7451:
7412:
7409:
7408:
7371:
7368:
7367:
7349:
7341:
7338:
7337:
7311:
7307:
7305:
7302:
7301:
7278:
7274:
7253:
7249:
7247:
7244:
7243:
7217:
7213:
7211:
7208:
7207:
7206:is a subset of
7191:
7188:
7187:
7169:
7166:
7165:
7143:
7140:
7139:
7136:
7116:
7113:
7112:
7087:
7083:
7056:
7052:
7044:
7041:
7040:
7024:
7021:
7020:
6988:
6984:
6982:
6979:
6978:
6955:
6951:
6949:
6946:
6945:
6926:
6923:
6922:
6906:
6903:
6902:
6873:
6869:
6861:
6858:
6857:
6819:
6815:
6813:
6810:
6809:
6793:
6790:
6789:
6773:
6770:
6769:
6747:
6744:
6743:
6724:
6721:
6720:
6697:
6693:
6691:
6688:
6687:
6671:
6668:
6667:
6651:
6648:
6647:
6620:
6616:
6614:
6611:
6610:
6587:
6583:
6562:
6558:
6556:
6553:
6552:
6532:
6529:
6528:
6505:
6501:
6493:
6490:
6489:
6470:
6467:
6466:
6443:
6439:
6437:
6434:
6433:
6426:
6393:
6389:
6362:
6358:
6356:
6353:
6352:
6336:
6333:
6332:
6316:
6313:
6312:
6296:
6293:
6292:
6276:
6273:
6272:
6256:
6253:
6252:
6229:
6225:
6210:
6206:
6204:
6201:
6200:
6184:
6181:
6180:
6160:
6157:
6156:
6140:
6137:
6136:
6116:
6113:
6112:
6084:
6081:
6080:
6058:
6055:
6054:
6037:
6033:
6024:
6020:
6008:
5990:
5986:
5974:
5969:
5965:
5956:
5952:
5950:
5947:
5946:
5917:
5913:
5892:
5888:
5858:
5854:
5852:
5849:
5848:
5797:
5794:
5793:
5777:
5774:
5773:
5772:The closure of
5761:In particular:
5730:
5726:
5724:
5721:
5720:
5719:if and only if
5704:
5701:
5700:
5680:
5677:
5676:
5673:
5668:
5637:
5633:
5626:
5619:
5614:
5610:
5601:
5597:
5596:
5592:
5583:
5579:
5565:
5561:
5552:
5548:
5541:
5534:
5529:
5525:
5516:
5512:
5510:
5507:
5506:
5490:
5487:
5486:
5469:
5465:
5463:
5460:
5459:
5427:
5423:
5416:
5409:
5404:
5400:
5391:
5387:
5386:
5382:
5373:
5369:
5355:
5351:
5342:
5338:
5331:
5324:
5319:
5315:
5306:
5302:
5300:
5297:
5296:
5280:
5277:
5276:
5259:
5255:
5253:
5250:
5249:
5227:
5224:
5223:
5197:
5193:
5184:
5180:
5178:
5175:
5174:
5170:
5167:
5135:
5132:
5131:
5086:
5082:
5061:
5057:
5055:
5052:
5051:
5007:
5003:
4982:
4978:
4976:
4973:
4972:
4949:
4945:
4943:
4940:
4939:
4914:
4910:
4908:
4905:
4904:
4881:
4878:
4877:
4846:
4844:
4841:
4840:
4818:
4815:
4814:
4791:
4788:
4787:
4771:
4769:
4766:
4765:
4748:
4744:
4742:
4739:
4738:
4717:
4715:
4712:
4711:
4685:
4681:
4679:
4676:
4675:
4653:
4650:
4649:
4602:
4598:
4596:
4593:
4592:
4564:
4561:
4560:
4527:
4526:
4524:
4521:
4520:
4502:
4499:
4498:
4474:
4471:
4470:
4461:
4451:
4428:
4426:
4423:
4422:
4399:
4396:
4395:
4379:
4377:
4374:
4373:
4357:
4354:
4353:
4335:
4333:
4330:
4329:
4313:
4310:
4309:
4293:
4290:
4289:
4271:
4269:
4266:
4265:
4249:
4246:
4245:
4229:
4227:
4224:
4223:
4203:
4200:
4199:
4158:
4154:
4146:
4129:
4126:
4125:
4106:
4104:
4101:
4100:
4080:
4077:
4076:
4036:
4032:
4030:
4027:
4026:
4008:
4005:
4004:
3988:
3985:
3984:
3968:
3965:
3964:
3945:
3942:
3941:
3905:
3872:
3868:
3866:
3863:
3862:
3846:
3844:
3841:
3840:
3820:
3812:
3809:
3808:
3744:
3740:
3738:
3735:
3734:
3714:
3706:
3703:
3702:
3638:
3634:
3632:
3629:
3628:
3608:
3600:
3597:
3596:
3584:
3548:
3544:
3542:
3539:
3538:
3519:
3516:
3515:
3495:
3492:
3491:
3461:
3453:
3445:
3414:
3406:
3398:
3388:
3384:
3375:
3371:
3369:
3366:
3365:
3345:
3340:
3339:
3331:
3329:
3326:
3325:
3306:
3303:
3302:
3281:
3279:
3276:
3275:
3255:
3253:
3250:
3249:
3230:
3228:
3225:
3224:
3204:
3202:
3199:
3198:
3182:
3180:
3177:
3176:
3160:
3157:
3156:
3125:
3122:
3121:
3105:
3102:
3101:
3100:as a subset of
3073:
3070:
3069:
3010:
3006:
3004:
3001:
3000:
2980:
2978:
2975:
2974:
2958:
2955:
2954:
2930:
2928:
2925:
2924:
2908:
2906:
2903:
2902:
2868:
2865:
2864:
2846:
2843:
2842:
2823:
2820:
2819:
2803:
2800:
2799:
2771:
2768:
2767:
2745:
2701:
2698:
2697:
2667:
2664:
2663:
2620:
2617:
2616:
2600:
2597:
2596:
2595:if and only if
2580:
2577:
2576:
2560:
2557:
2556:
2540:
2537:
2536:
2509:
2506:
2505:
2504:is a subset of
2483:
2480:
2479:
2457:
2454:
2453:
2422:
2419:
2418:
2399:
2396:
2395:
2376:
2373:
2372:
2346:
2343:
2342:
2305:
2302:
2301:
2285:
2282:
2281:
2265:
2262:
2261:
2235:
2232:
2231:
2209:
2206:
2205:
2176:
2173:
2172:
2153:
2150:
2149:
2127:
2124:
2123:
2103:
2100:
2099:
2077:
2074:
2073:
2053:
2050:
2049:
2023:
2020:
2019:
2001:
1998:
1997:
1981:
1978:
1977:
1961:
1958:
1957:
1941:
1938:
1937:
1917:
1914:
1913:
1891:
1888:
1887:
1867:
1864:
1863:
1837:
1834:
1833:
1814:
1811:
1810:
1794:
1791:
1790:
1773:
1771:
1770:
1765:
1762:
1761:
1737:
1735:
1732:
1731:
1707:
1704:
1703:
1687:
1684:
1683:
1667:
1664:
1663:
1647:
1644:
1643:
1620:
1616:
1614:
1611:
1610:
1609:or possibly by
1575:
1571:
1569:
1566:
1565:
1534:
1531:
1530:
1511:
1508:
1507:
1498:
1492:
1471:
1468:
1467:
1451:
1448:
1447:
1431:
1428:
1427:
1411:
1408:
1407:
1406:if and only if
1391:
1388:
1387:
1371:
1368:
1367:
1349:
1346:
1345:
1329:
1326:
1325:
1306:
1303:
1302:
1286:
1283:
1282:
1266:
1263:
1262:
1246:
1243:
1242:
1226:
1223:
1222:
1206:
1203:
1202:
1166:
1163:
1162:
1139:
1136:
1135:
1119:
1116:
1115:
1099:
1096:
1095:
1079:
1076:
1075:
1059:
1056:
1055:
1039:
1036:
1035:
1019:
1016:
1015:
999:
996:
995:
979:
976:
975:
957:
954:
953:
936:
933:
932:
916:
913:
912:
896:
893:
892:
876:
873:
872:
865:
859:
832:
829:
828:
806:
803:
802:
786:
783:
782:
766:
763:
762:
746:
743:
742:
712:
709:
708:
689:
686:
685:
669:
666:
665:
634:
631:
630:
583:
556:
553:
552:
536:
533:
532:
516:
513:
512:
490:
487:
486:
449:
446:
445:
423:
420:
419:
397:
394:
393:
377:
374:
373:
357:
354:
353:
335:
332:
331:
315:
312:
311:
292:
289:
288:
269:
266:
265:
246:
243:
242:
226:
223:
222:
206:
203:
202:
182:
179:
178:
162:
159:
158:
155:Euclidean space
138:
135:
134:
131:
125:
120:
103:
95:
91:
90:or "very near"
87:
83:
67:
59:
55:
47:
38:of points in a
35:
24:
17:
12:
11:
5:
12254:
12244:
12243:
12238:
12221:
12220:
12218:
12217:
12207:
12206:
12205:
12200:
12195:
12180:
12170:
12160:
12148:
12137:
12134:
12133:
12131:
12130:
12125:
12120:
12115:
12110:
12105:
12099:
12097:
12091:
12090:
12088:
12087:
12082:
12077:
12075:Winding number
12072:
12067:
12061:
12059:
12055:
12054:
12052:
12051:
12046:
12041:
12036:
12031:
12026:
12021:
12016:
12015:
12014:
12009:
12007:homotopy group
11999:
11998:
11997:
11992:
11987:
11982:
11977:
11967:
11962:
11957:
11947:
11945:
11941:
11940:
11933:
11931:
11929:
11928:
11923:
11918:
11917:
11916:
11906:
11905:
11904:
11894:
11889:
11884:
11879:
11874:
11868:
11866:
11862:
11861:
11854:
11853:
11846:
11839:
11831:
11825:
11824:
11804:
11803:External links
11801:
11800:
11799:
11773:
11749:
11740:
11731:
11722:
11717:
11700:
11695:
11682:
11677:
11660:
11655:
11640:
11637:
11634:
11633:
11629:Zălinescu 2002
11621:
11609:
11597:Gemignani 1990
11589:
11577:
11565:
11553:
11540:
11539:
11537:
11534:
11531:
11530:
11518:
11515:
11512:
11509:
11503:
11497:
11494:
11489:
11485:
11481:
11478:
11472:
11466:
11463:
11460:
11457:
11454:
11451:
11446:
11442:
11435:
11429:
11409:
11406:
11403:
11400:
11397:
11394:
11391:
11388:
11385:
11380:
11376:
11355:
11352:
11349:
11346:
11343:
11323:
11320:
11317:
11314:
11311:
11308:
11305:
11285:
11282:
11279:
11276:
11273:
11270:
11267:
11264:
11247:
11246:
11244:
11241:
11240:
11239:
11230:
11224:
11218:
11212:
11203:
11200:Adherent point
11195:
11192:
11168:
11165:
11141:
11138:
11135:
11132:
11129:
11126:
11103:
11100:
11080:
11077:
11074:
11071:
11068:
11065:
11062:
11042:
11022:
11019:
11016:
10994:
10991:
10988:
10985:
10982:
10979:
10959:
10956:
10936:
10916:
10913:
10910:
10907:
10887:
10884:
10881:
10878:
10875:
10872:
10862:comma category
10849:
10846:
10843:
10823:
10820:
10817:
10814:
10811:
10808:
10788:
10764:
10744:
10724:
10721:
10701:
10681:
10678:
10675:
10665:inclusion maps
10652:
10626:
10607:
10604:
10592:
10589:
10586:
10583:
10563:
10560:
10557:
10554:
10551:
10548:
10545:
10542:
10539:
10536:
10531:
10527:
10506:
10503:
10500:
10497:
10494:
10485:Equivalently,
10474:
10471:
10468:
10465:
10444:
10440:
10437:
10432:
10428:
10423:
10419:
10416:
10413:
10410:
10407:
10404:
10401:
10396:
10392:
10371:
10368:
10365:
10362:
10359:
10339:
10336:
10316:
10313:
10310:
10307:
10287:
10267:
10243:
10240:
10237:
10234:
10231:
10216:Main article:
10213:
10210:
10198:
10195:
10192:
10189:
10186:
10166:
10163:
10160:
10157:
10137:
10134:
10131:
10128:
10108:
10088:
10085:
10082:
10062:
10041:
10038:
10035:
10032:
10029:
10009:
9989:
9970:
9967:
9964:
9961:
9941:
9917:
9914:
9911:
9906:
9902:
9898:
9895:
9875:
9872:
9869:
9859:
9845:
9825:
9822:
9802:
9799:
9796:
9793:
9773:
9770:
9767:
9764:
9745:
9742:
9739:
9736:
9716:
9713:
9710:
9690:
9687:
9684:
9681:
9678:
9675:
9672:
9669:
9664:
9660:
9653:
9646:
9642:
9639:
9634:
9630:
9625:
9621:
9602:
9599:
9596:
9593:
9573:
9570:
9567:
9564:
9561:
9539:
9536:
9516:
9496:
9476:
9473:
9470:
9465:
9462:
9458:
9429:
9426:
9423:
9420:
9417:
9402:Main article:
9399:
9396:
9394:
9391:
9379:
9374:
9369:
9366:
9346:
9326:
9323:
9320:
9300:
9280:
9260:
9234:
9212:
9201:locally closed
9188:
9168:
9165:
9162:
9142:
9122:
9119:
9116:
9107:of any subset
9096:
9070:
9044:
9024:
8998:
8993:
8990:
8968:
8963:
8960:
8940:
8937:
8932:
8928:
8924:
8921:
8918:
8915:
8912:
8909:
8906:
8903:
8900:
8895:
8891:
8870:
8867:
8864:
8861:
8858:
8855:
8850:
8846:
8838:
8833:
8830:
8826:
8822:
8819:
8816:
8811:
8807:
8786:
8783:
8780:
8760:
8734:
8718:
8717:
8705:
8686:
8683:
8680:
8675:
8671:
8667:
8664:
8661:
8658:
8655:
8652:
8649:
8644:
8640:
8636:
8633:
8613:
8610:
8607:
8604:
8601:
8598:
8595:
8592:
8589:
8584:
8580:
8576:
8573:
8553:
8533:
8530:
8527:
8524:
8521:
8518:
8515:
8512:
8509:
8506:
8503:
8498:
8494:
8473:
8470:
8467:
8464:
8461:
8458:
8455:
8452:
8447:
8443:
8439:
8436:
8433:
8430:
8427:
8424:
8404:
8384:
8381:
8378:
8358:
8338:
8335:
8332:
8329:
8326:
8323:
8320:
8300:
8280:
8277:
8274:
8271:
8268:
8265:
8262:
8259:
8256:
8253:
8250:
8247:
8244:
8241:
8238:
8218:
8215:
8195:
8192:
8189:
8169:
8160:being open in
8149:
8129:
8126:
8106:
8103:
8100:
8080:
8077:
8074:
8071:
8068:
8065:
8062:
8042:
8039:
8036:
8033:
8030:
8027:
8022:
8018:
8014:
8011:
7991:
7988:
7985:
7982:
7979:
7976:
7973:
7968:
7964:
7960:
7957:
7937:
7934:
7914:
7894:
7891:
7888:
7885:
7882:
7869:
7868:
7865:
7860:
7847:
7827:
7807:
7804:
7799:
7795:
7791:
7788:
7785:
7782:
7779:
7776:
7773:
7770:
7767:
7762:
7758:
7737:
7717:
7697:
7694:
7691:
7688:
7685:
7682:
7679:
7659:
7656:
7653:
7650:
7647:
7644:
7641:
7638:
7635:
7614:
7610:
7607:
7587:
7584:
7579:
7575:
7571:
7568:
7562:
7556:
7553:
7550:
7547:
7544:
7541:
7536:
7532:
7511:
7491:
7471:
7468:
7465:
7462:
7459:
7437:
7434:
7431:
7428:
7425:
7422:
7419:
7416:
7396:
7393:
7390:
7387:
7384:
7381:
7378:
7375:
7356:
7352:
7348:
7345:
7325:
7322:
7319:
7314:
7310:
7289:
7286:
7281:
7277:
7273:
7270:
7267:
7264:
7261:
7256:
7252:
7231:
7228:
7225:
7220:
7216:
7195:
7185:if and only if
7173:
7153:
7150:
7147:
7133:
7132:
7120:
7101:
7098:
7095:
7090:
7086:
7082:
7079:
7076:
7073:
7070:
7067:
7064:
7059:
7055:
7051:
7048:
7028:
7008:
7005:
7002:
6999:
6996:
6991:
6987:
6966:
6963:
6958:
6954:
6933:
6930:
6910:
6890:
6887:
6884:
6881:
6876:
6872:
6868:
6865:
6845:
6842:
6839:
6836:
6833:
6830:
6827:
6822:
6818:
6797:
6777:
6757:
6754:
6751:
6731:
6728:
6708:
6705:
6700:
6696:
6675:
6655:
6645:
6631:
6628:
6623:
6619:
6598:
6595:
6590:
6586:
6582:
6579:
6576:
6573:
6570:
6565:
6561:
6536:
6516:
6513:
6508:
6504:
6500:
6497:
6477:
6474:
6454:
6451:
6446:
6442:
6428:
6427:
6424:
6419:
6407:
6404:
6401:
6396:
6392:
6388:
6385:
6379:
6373:
6370:
6365:
6361:
6340:
6320:
6300:
6280:
6260:
6240:
6237:
6232:
6228:
6224:
6221:
6218:
6213:
6209:
6188:
6164:
6155:(meaning that
6144:
6120:
6100:
6097:
6094:
6091:
6088:
6077:
6076:
6075:
6074:
6062:
6040:
6036:
6032:
6027:
6023:
6017:
6014:
6011:
6007:
6003:
5999:
5993:
5989:
5983:
5980:
5977:
5973:
5968:
5964:
5959:
5955:
5934:
5931:
5928:
5925:
5920:
5916:
5912:
5909:
5906:
5903:
5900:
5895:
5891:
5887:
5884:
5881:
5878:
5875:
5872:
5869:
5866:
5861:
5857:
5838:
5826:
5815:
5804:
5801:
5781:
5770:
5750:
5747:
5744:
5741:
5738:
5733:
5729:
5708:
5684:
5672:
5669:
5667:
5666:
5655:
5651:
5646:
5640:
5636:
5629:
5625:
5622:
5618:
5613:
5609:
5604:
5600:
5595:
5591:
5586:
5582:
5578:
5574:
5568:
5564:
5560:
5555:
5551:
5544:
5540:
5537:
5533:
5528:
5524:
5519:
5515:
5494:
5472:
5468:
5456:
5445:
5441:
5436:
5430:
5426:
5419:
5415:
5412:
5408:
5403:
5399:
5394:
5390:
5385:
5381:
5376:
5372:
5368:
5364:
5358:
5354:
5350:
5345:
5341:
5334:
5330:
5327:
5323:
5318:
5314:
5309:
5305:
5284:
5262:
5258:
5234:
5231:
5208:
5205:
5200:
5196:
5192:
5187:
5183:
5161:
5142:
5139:
5124:
5123:
5112:
5109:
5106:
5103:
5100:
5097:
5094:
5089:
5085:
5081:
5078:
5075:
5072:
5069:
5064:
5060:
5045:
5044:
5033:
5030:
5027:
5024:
5021:
5018:
5015:
5010:
5006:
5002:
4999:
4996:
4993:
4990:
4985:
4981:
4957:
4952:
4948:
4917:
4913:
4885:
4865:
4862:
4859:
4856:
4853:
4849:
4828:
4825:
4822:
4798:
4795:
4774:
4751:
4747:
4724:
4721:
4699:
4696:
4693:
4688:
4684:
4663:
4660:
4657:
4637:
4634:
4631:
4628:
4625:
4622:
4619:
4616:
4613:
4610:
4605:
4601:
4580:
4577:
4574:
4571:
4568:
4541:
4538:
4535:
4530:
4509:
4506:
4478:
4468:
4450:
4447:
4432:
4421:
4417:
4403:
4382:
4361:
4339:
4317:
4297:
4275:
4253:
4232:
4207:
4187:
4184:
4181:
4178:
4175:
4172:
4169:
4166:
4161:
4157:
4153:
4149:
4145:
4142:
4139:
4136:
4133:
4113:
4109:
4084:
4073:
4072:
4056:
4053:
4050:
4047:
4044:
4039:
4035:
4015:
4012:
3992:
3972:
3952:
3949:
3935:
3932:discrete space
3924:
3923:
3912:
3908:
3904:
3901:
3898:
3895:
3892:
3889:
3886:
3883:
3880:
3875:
3871:
3849:
3823:
3819:
3816:
3805:
3794:
3791:
3788:
3785:
3782:
3779:
3776:
3773:
3770:
3767:
3764:
3761:
3758:
3755:
3752:
3747:
3743:
3717:
3713:
3710:
3699:
3688:
3685:
3682:
3679:
3676:
3673:
3670:
3667:
3664:
3661:
3658:
3655:
3652:
3649:
3646:
3641:
3637:
3611:
3607:
3604:
3589:
3588:
3582:
3568:
3565:
3562:
3559:
3556:
3551:
3547:
3526:
3523:
3499:
3488:
3477:
3474:
3471:
3468:
3464:
3460:
3456:
3452:
3448:
3444:
3441:
3438:
3435:
3431:
3427:
3424:
3421:
3417:
3413:
3409:
3405:
3401:
3397:
3394:
3391:
3387:
3383:
3378:
3374:
3353:
3348:
3343:
3338:
3334:
3310:
3299:
3288:
3284:
3258:
3237:
3233:
3207:
3185:
3164:
3153:
3141:
3138:
3135:
3132:
3129:
3109:
3089:
3086:
3083:
3080:
3077:
3057:
3054:
3051:
3048:
3045:
3042:
3039:
3036:
3033:
3030:
3027:
3024:
3021:
3018:
3013:
3009:
2983:
2962:
2933:
2911:
2899:
2898:
2887:
2884:
2881:
2878:
2875:
2872:
2853:
2850:
2839:
2827:
2807:
2787:
2784:
2781:
2778:
2775:
2766:In any space,
2744:
2741:
2708:
2705:
2677:
2674:
2671:
2650:
2645:
2644:
2633:
2630:
2627:
2624:
2604:
2584:
2564:
2544:
2533:
2522:
2519:
2516:
2513:
2493:
2490:
2487:
2467:
2464:
2461:
2450:
2438:
2435:
2432:
2429:
2426:
2416:if and only if
2403:
2392:
2380:
2356:
2353:
2350:
2336:
2335:
2324:
2321:
2318:
2315:
2312:
2309:
2289:
2269:
2245:
2242:
2239:
2219:
2216:
2213:
2203:
2192:
2189:
2186:
2183:
2180:
2157:
2137:
2134:
2131:
2121:
2110:
2107:
2087:
2084:
2081:
2071:
2060:
2057:
2033:
2030:
2027:
2017:
2005:
1985:
1965:
1945:
1932:together with
1921:
1901:
1898:
1895:
1885:
1874:
1871:
1847:
1844:
1841:
1818:
1798:
1776:
1769:
1749:
1744:
1741:
1720:
1717:
1714:
1711:
1691:
1671:
1651:
1631:
1628:
1623:
1619:
1598:
1595:
1590:
1587:
1584:
1581:
1578:
1574:
1553:
1550:
1547:
1544:
1541:
1538:
1515:
1505:
1491:
1488:
1475:
1455:
1435:
1415:
1395:
1375:
1356:
1353:
1333:
1310:
1290:
1270:
1250:
1230:
1210:
1199:isolated point
1181:
1170:
1154:is called the
1143:
1123:
1103:
1083:
1063:
1043:
1023:
1003:
994:obviously has
983:
973:
961:
940:
920:
900:
880:
861:Main article:
858:
855:
842:
839:
836:
816:
813:
810:
790:
770:
750:
740:
738:adherent point
734:
716:
696:
693:
673:
638:
618:
615:
612:
609:
606:
603:
600:
597:
592:
589:
586:
582:
578:
575:
572:
569:
566:
563:
560:
540:
520:
500:
497:
494:
474:
471:
468:
465:
462:
459:
456:
453:
433:
430:
427:
407:
404:
401:
381:
361:
342:
339:
319:
299:
296:
273:
250:
230:
210:
186:
166:
142:
129:Adherent point
127:Main article:
124:
121:
119:
116:
15:
9:
6:
4:
3:
2:
12253:
12242:
12239:
12237:
12234:
12233:
12231:
12216:
12208:
12204:
12201:
12199:
12196:
12194:
12191:
12190:
12189:
12181:
12179:
12175:
12171:
12169:
12165:
12161:
12159:
12154:
12149:
12147:
12139:
12138:
12135:
12129:
12126:
12124:
12121:
12119:
12116:
12114:
12111:
12109:
12106:
12104:
12101:
12100:
12098:
12096:
12092:
12086:
12085:Orientability
12083:
12081:
12078:
12076:
12073:
12071:
12068:
12066:
12063:
12062:
12060:
12056:
12050:
12047:
12045:
12042:
12040:
12037:
12035:
12032:
12030:
12027:
12025:
12022:
12020:
12017:
12013:
12010:
12008:
12005:
12004:
12003:
12000:
11996:
11993:
11991:
11988:
11986:
11983:
11981:
11978:
11976:
11973:
11972:
11971:
11968:
11966:
11963:
11961:
11958:
11956:
11952:
11949:
11948:
11946:
11942:
11937:
11927:
11924:
11922:
11921:Set-theoretic
11919:
11915:
11912:
11911:
11910:
11907:
11903:
11900:
11899:
11898:
11895:
11893:
11890:
11888:
11885:
11883:
11882:Combinatorial
11880:
11878:
11875:
11873:
11870:
11869:
11867:
11863:
11859:
11852:
11847:
11845:
11840:
11838:
11833:
11832:
11829:
11821:
11817:
11816:
11811:
11807:
11806:
11796:
11792:
11788:
11784:
11780:
11776:
11770:
11766:
11761:
11760:
11754:
11750:
11746:
11741:
11737:
11732:
11728:
11723:
11720:
11718:0-486-65676-4
11714:
11709:
11708:
11701:
11698:
11696:0-486-66522-4
11692:
11688:
11683:
11680:
11678:0-03-012813-7
11674:
11669:
11668:
11661:
11658:
11656:0-697-05972-3
11652:
11648:
11643:
11642:
11631:, p. 33.
11630:
11625:
11618:
11613:
11606:
11602:
11598:
11593:
11586:
11581:
11574:
11569:
11562:
11557:
11550:
11549:Schubert 1968
11545:
11541:
11516:
11510:
11501:
11495:
11492:
11487:
11483:
11479:
11476:
11470:
11461:
11458:
11455:
11449:
11444:
11440:
11433:
11427:
11407:
11398:
11395:
11389:
11386:
11383:
11378:
11374:
11350:
11347:
11344:
11341:
11315:
11312:
11306:
11303:
11280:
11277:
11271:
11265:
11262:
11252:
11248:
11234:
11231:
11228:
11225:
11222:
11219:
11216:
11213:
11207:
11204:
11201:
11198:
11197:
11191:
11189:
11185:
11179:
11166:
11163:
11155:
11139:
11133:
11127:
11124:
11117:
11101:
11098:
11075:
11069:
11063:
11040:
11020:
11014:
11005:
10992:
10989:
10986:
10983:
10977:
10957:
10954:
10934:
10914:
10911:
10908:
10905:
10885:
10879:
10873:
10863:
10847:
10844:
10841:
10821:
10818:
10812:
10809:
10806:
10786:
10778:
10762:
10742:
10722:
10719:
10699:
10679:
10673:
10666:
10650:
10643:
10640:
10639:partial order
10624:
10616:
10611:
10603:
10590:
10587:
10584:
10581:
10558:
10552:
10549:
10543:
10537:
10534:
10529:
10525:
10504:
10498:
10495:
10492:
10472:
10469:
10466:
10463:
10442:
10438:
10435:
10430:
10426:
10421:
10417:
10414:
10408:
10402:
10399:
10394:
10390:
10369:
10363:
10360:
10357:
10337:
10334:
10311:
10305:
10285:
10265:
10257:
10241:
10235:
10232:
10229:
10219:
10209:
10196:
10190:
10184:
10161:
10155:
10135:
10132:
10129:
10126:
10106:
10086:
10083:
10080:
10060:
10039:
10033:
10027:
10007:
9987:
9968:
9965:
9962:
9959:
9939:
9931:
9930:plain English
9915:
9912:
9909:
9904:
9900:
9896:
9893:
9873:
9870:
9867:
9857:
9843:
9823:
9820:
9797:
9791:
9768:
9762:
9743:
9740:
9737:
9734:
9714:
9711:
9708:
9688:
9679:
9673:
9667:
9662:
9658:
9651:
9644:
9640:
9637:
9632:
9628:
9623:
9619:
9600:
9597:
9594:
9591:
9571:
9565:
9562:
9559:
9550:
9537:
9534:
9514:
9494:
9487:is closed in
9471:
9463:
9460:
9456:
9447:
9443:
9427:
9421:
9418:
9415:
9405:
9390:
9377:
9367:
9364:
9344:
9337:is closed in
9324:
9321:
9318:
9298:
9291:is closed in
9278:
9258:
9250:
9210:
9202:
9186:
9179:is closed in
9166:
9163:
9160:
9140:
9120:
9117:
9114:
9094:
9086:
9058:
9042:
9022:
9014:
8991:
8988:
8981:(where every
8961:
8958:
8938:
8935:
8930:
8926:
8922:
8919:
8916:
8910:
8907:
8904:
8898:
8893:
8889:
8865:
8862:
8859:
8853:
8848:
8844:
8831:
8828:
8824:
8820:
8817:
8814:
8809:
8805:
8784:
8781:
8778:
8758:
8750:
8716:
8703:
8684:
8681:
8678:
8673:
8669:
8665:
8659:
8656:
8653:
8647:
8642:
8638:
8634:
8631:
8611:
8608:
8605:
8602:
8599:
8596:
8593:
8590:
8587:
8582:
8578:
8574:
8571:
8551:
8531:
8528:
8525:
8519:
8513:
8507:
8504:
8501:
8496:
8492:
8471:
8468:
8462:
8459:
8456:
8450:
8445:
8441:
8437:
8434:
8431:
8428:
8425:
8422:
8402:
8382:
8379:
8376:
8356:
8336:
8333:
8327:
8321:
8298:
8278:
8275:
8269:
8263:
8257:
8251:
8245:
8236:
8229:Consequently
8216:
8213:
8193:
8187:
8167:
8147:
8127:
8124:
8104:
8098:
8078:
8075:
8072:
8069:
8066:
8063:
8060:
8037:
8034:
8031:
8025:
8020:
8016:
8012:
8009:
7989:
7983:
7980:
7977:
7971:
7966:
7962:
7958:
7955:
7935:
7932:
7912:
7892:
7889:
7886:
7883:
7880:
7871:
7870:
7864:
7863:
7859:
7845:
7825:
7805:
7802:
7797:
7793:
7789:
7786:
7783:
7777:
7774:
7771:
7765:
7760:
7756:
7735:
7715:
7689:
7686:
7680:
7677:
7657:
7651:
7648:
7642:
7636:
7633:
7608:
7605:
7585:
7582:
7577:
7573:
7569:
7566:
7560:
7551:
7548:
7545:
7539:
7534:
7530:
7509:
7489:
7469:
7466:
7463:
7460:
7457:
7448:
7435:
7426:
7423:
7417:
7414:
7394:
7388:
7385:
7382:
7376:
7373:
7354:
7346:
7343:
7323:
7320:
7317:
7312:
7308:
7287:
7284:
7279:
7275:
7271:
7268:
7265:
7262:
7259:
7254:
7250:
7229:
7226:
7223:
7218:
7214:
7193:
7186:
7171:
7151:
7148:
7145:
7131:
7118:
7099:
7096:
7093:
7088:
7084:
7080:
7077:
7074:
7071:
7068:
7065:
7062:
7057:
7053:
7049:
7046:
7026:
7006:
7003:
7000:
6997:
6994:
6989:
6985:
6977:implies that
6964:
6961:
6956:
6952:
6931:
6928:
6921:is closed in
6908:
6888:
6885:
6882:
6879:
6874:
6870:
6866:
6863:
6843:
6840:
6837:
6834:
6831:
6828:
6825:
6820:
6816:
6795:
6788:is closed in
6775:
6755:
6752:
6749:
6729:
6726:
6706:
6703:
6698:
6694:
6673:
6653:
6643:
6629:
6626:
6621:
6617:
6596:
6593:
6588:
6584:
6580:
6577:
6574:
6571:
6568:
6563:
6559:
6550:
6534:
6514:
6511:
6506:
6502:
6498:
6495:
6475:
6472:
6452:
6449:
6444:
6440:
6430:
6429:
6423:
6422:
6418:
6405:
6402:
6399:
6394:
6390:
6386:
6383:
6377:
6371:
6368:
6363:
6359:
6338:
6318:
6298:
6278:
6258:
6238:
6235:
6230:
6226:
6222:
6219:
6216:
6211:
6207:
6186:
6178:
6162:
6142:
6134:
6118:
6098:
6095:
6092:
6089:
6086:
6060:
6038:
6034:
6030:
6025:
6021:
6015:
6012:
6009:
6005:
6001:
5997:
5991:
5987:
5981:
5978:
5975:
5971:
5966:
5962:
5957:
5953:
5932:
5926:
5923:
5918:
5914:
5907:
5901:
5898:
5893:
5889:
5882:
5876:
5873:
5870:
5864:
5859:
5855:
5846:
5845:
5843:
5839:
5835:
5831:
5827:
5824:
5820:
5816:
5802:
5799:
5779:
5771:
5768:
5764:
5763:
5762:
5748:
5745:
5742:
5739:
5736:
5731:
5727:
5706:
5698:
5682:
5653:
5649:
5644:
5638:
5634:
5623:
5620:
5616:
5611:
5607:
5602:
5598:
5593:
5589:
5584:
5580:
5576:
5572:
5566:
5562:
5558:
5553:
5549:
5538:
5535:
5531:
5526:
5522:
5517:
5513:
5492:
5470:
5466:
5457:
5443:
5439:
5434:
5428:
5424:
5413:
5410:
5406:
5401:
5397:
5392:
5388:
5383:
5379:
5374:
5370:
5366:
5362:
5356:
5352:
5348:
5343:
5339:
5328:
5325:
5321:
5316:
5312:
5307:
5303:
5282:
5275:is closed in
5260:
5256:
5247:
5246:
5245:
5232:
5229:
5222:
5206:
5203:
5198:
5194:
5190:
5185:
5181:
5160:
5158:
5153:
5140:
5137:
5129:
5110:
5104:
5098:
5092:
5087:
5083:
5076:
5073:
5070:
5067:
5062:
5058:
5050:
5049:
5048:
5031:
5025:
5019:
5013:
5008:
5004:
4997:
4994:
4991:
4988:
4983:
4979:
4971:
4970:
4969:
4955:
4950:
4946:
4937:
4933:
4915:
4911:
4901:
4899:
4883:
4863:
4860:
4854:
4839:that satisfy
4826:
4823:
4820:
4812:
4796:
4793:
4749:
4745:
4719:
4697:
4694:
4691:
4686:
4682:
4661:
4658:
4655:
4632:
4617:
4608:
4603:
4599:
4575:
4572:
4569:
4559:
4555:
4536:
4507:
4504:
4496:
4492:
4476:
4467:
4464:
4460:
4456:
4446:
4430:
4419:
4418:greater than
4415:
4401:
4359:
4337:
4315:
4295:
4273:
4251:
4221:
4205:
4185:
4179:
4176:
4173:
4170:
4167:
4164:
4159:
4155:
4151:
4143:
4140:
4134:
4131:
4111:
4098:
4082:
4070:
4054:
4051:
4048:
4045:
4042:
4037:
4033:
4013:
4010:
3990:
3970:
3950:
3947:
3940:
3936:
3933:
3929:
3928:
3927:
3910:
3902:
3893:
3890:
3887:
3878:
3873:
3869:
3861:itself, then
3838:
3817:
3814:
3806:
3792:
3786:
3783:
3780:
3774:
3765:
3762:
3759:
3750:
3745:
3741:
3732:
3711:
3708:
3700:
3686:
3680:
3677:
3674:
3668:
3659:
3656:
3653:
3644:
3639:
3635:
3626:
3605:
3602:
3594:
3593:
3592:
3586:
3566:
3563:
3560:
3557:
3554:
3549:
3545:
3524:
3521:
3513:
3497:
3489:
3475:
3469:
3466:
3458:
3450:
3442:
3439:
3433:
3429:
3422:
3419:
3411:
3403:
3395:
3392:
3385:
3381:
3376:
3372:
3351:
3346:
3336:
3324:
3323:complex plane
3308:
3300:
3286:
3273:
3235:
3222:
3162:
3154:
3136:
3133:
3130:
3107:
3084:
3081:
3078:
3052:
3049:
3046:
3040:
3031:
3028:
3025:
3016:
3011:
3007:
2998:
2960:
2952:
2951:
2950:
2948:
2885:
2882:
2879:
2876:
2873:
2870:
2851:
2848:
2841:In any space
2840:
2825:
2805:
2782:
2779:
2776:
2773:
2765:
2764:
2763:
2761:
2756:
2754:
2750:
2740:
2738:
2732:
2730:
2726:
2722:
2706:
2703:
2696:of points in
2695:
2691:
2675:
2672:
2669:
2661:
2657:
2652:
2648:
2631:
2628:
2625:
2622:
2602:
2582:
2562:
2542:
2534:
2520:
2517:
2514:
2511:
2491:
2488:
2485:
2465:
2462:
2459:
2451:
2436:
2433:
2430:
2427:
2424:
2417:
2401:
2393:
2378:
2370:
2354:
2351:
2348:
2341:
2340:
2339:
2322:
2316:
2313:
2310:
2287:
2267:
2259:
2243:
2240:
2237:
2217:
2214:
2211:
2204:
2190:
2184:
2171:
2155:
2135:
2132:
2129:
2122:
2108:
2105:
2085:
2082:
2079:
2072:
2058:
2055:
2047:
2031:
2028:
2025:
2018:
2003:
1983:
1963:
1943:
1935:
1919:
1899:
1896:
1893:
1886:
1872:
1869:
1861:
1845:
1842:
1839:
1832:
1831:
1830:
1816:
1796:
1774:
1767:
1747:
1739:
1718:
1715:
1712:
1709:
1689:
1669:
1649:
1629:
1626:
1621:
1617:
1596:
1593:
1585:
1582:
1579:
1572:
1551:
1545:
1542:
1539:
1529:
1513:
1504:
1501:
1497:
1487:
1473:
1453:
1433:
1413:
1393:
1373:
1354:
1351:
1331:
1322:
1308:
1288:
1268:
1248:
1228:
1208:
1200:
1196:
1191:
1190:of the set.
1189:
1185:
1184:cluster point
1168:
1160:
1159:
1155:
1141:
1121:
1101:
1081:
1061:
1054:in order for
1041:
1021:
1001:
981:
959:
951:
938:
918:
898:
878:
870:
864:
854:
840:
837:
834:
814:
811:
808:
788:
768:
748:
739:
736:
733:
732:
728:
714:
694:
691:
671:
663:
662:neighbourhood
659:
654:
652:
616:
613:
607:
604:
601:
595:
590:
587:
584:
576:
570:
567:
564:
558:
538:
518:
498:
495:
492:
472:
469:
463:
460:
457:
451:
431:
428:
425:
405:
402:
399:
392:if for every
379:
359:
340:
337:
317:
297:
294:
287:
271:
262:
248:
228:
208:
200:
184:
164:
156:
140:
130:
115:
113:
109:
101:
81:
77:
73:
65:
53:
45:
41:
33:
29:
22:
12215:Publications
12080:Chern number
12070:Betti number
11953: /
11944:Key concepts
11892:Differential
11813:
11793:– via
11758:
11744:
11735:
11726:
11706:
11686:
11666:
11646:
11639:Bibliography
11624:
11612:
11603:, p. 40 and
11592:
11580:
11568:
11556:
11544:
11251:
11180:
11006:
10612:
10609:
10221:
10177:is close to
9551:
9407:
8722:
8564:proves that
7872:
7449:
7137:
6431:
6331:computed in
6271:computed in
6078:
6073:is infinite.
5819:intersection
5674:
5168:(C. Ursescu)
5162:
5154:
5125:
5046:
4902:
4465:
4462:
4416:real numbers
4074:
3925:
3590:
3248:We say that
2997:real numbers
2900:
2757:
2746:
2733:
2660:metric space
2653:
2646:
2371:superset of
2337:
2260:(valued) in
1506:of a subset
1502:
1499:
1323:
1194:
1192:
1187:
1183:
1156:
866:
737:
729:
655:
286:metric space
263:
201:centered at
132:
76:intersection
52:limit points
34:of a subset
31:
25:
12178:Wikiversity
12095:Key results
11617:Pervin 1965
11601:Pervin 1965
10777:subcategory
10222:A function
10212:Closed maps
9408:A function
8117:is open in
7925:is open in
7522:then only
7039:shows that
6686:). Because
6666:containing
5485:is open in
5128:complements
4811:closed sets
4556:. Given a
4420:or equal to
2747:Consider a
2658:(such as a
2048:containing
2046:closed sets
1912:is the set
1789:(Moreover,
1564:denoted by
1486:(or both).
1158:derived set
952:other than
857:Limit point
118:Definitions
82:containing
80:closed sets
12230:Categories
12024:CW complex
11965:Continuity
11955:Closed set
11914:cohomology
11605:Baker 1991
11585:Croom 1989
11536:References
10256:closed map
9442:continuous
9398:Continuity
9357:for every
9249:open cover
8951:for every
8749:open cover
6768:such that
5792:itself is
4453:See also:
2649:definition
2414:is closed
1494:See also:
1344:and point
12203:geometric
12198:algebraic
12049:Cobordism
11985:Hausdorff
11980:connected
11897:Geometric
11887:Continuum
11877:Algebraic
11820:EMS Press
11791:285163112
11711:, Dover,
11599:, p. 55,
11493:
11480:∩
11471:≠
11459:∩
11450:
11428:∅
11402:∞
11384:
11354:∅
11345:∩
11319:∞
11275:∞
11272:−
11128:
11073:∖
11067:↓
11018:∖
10987:
10981:→
10909:
10877:↓
10845:⊆
10816:→
10692:whenever
10677:→
10617:of a set
10585:⊆
10550:⊆
10535:
10502:→
10467:⊆
10436:
10415:⊆
10400:
10367:→
10239:→
10130:⊆
10084:∈
9963:⊆
9910:
9897:∈
9871:⊆
9860:a subset
9738:⊆
9712:∈
9668:
9652:⊆
9638:
9595:⊆
9569:→
9507:whenever
9461:−
9425:→
9368:∈
9322:∩
9164:⊆
9118:⊆
8992:∈
8962:∈
8936:
8923:∩
8908:∩
8899:
8863:∩
8854:
8832:∈
8825:⋃
8815:
8782:⊆
8704:◼
8679:
8666:⊆
8657:∩
8648:
8635:⊆
8600:∩
8594:⊆
8588:
8575:∩
8526:∪
8517:∖
8508:⊆
8502:
8460:∩
8451:
8438:⊆
8432:∩
8426:∈
8380:∈
8349:contains
8334:∪
8325:∖
8276:∪
8267:∖
8249:∖
8240:∖
8191:∖
8102:∖
8076:⊆
8070:⊆
8064:∩
8053:(because
8035:∩
8026:
8013:∩
7981:∩
7972:
7890:⊆
7803:
7790:∩
7775:∩
7766:
7693:∞
7646:∞
7643:−
7583:
7570:∩
7561:⊆
7549:∩
7540:
7467:⊆
7430:∞
7318:
7285:
7272:∩
7260:
7224:
7149:⊆
7119:◼
7094:
7075:∩
7069:⊆
7063:
7050:∩
7001:⊆
6995:
6962:
6886:⊆
6880:
6867:⊆
6838:∩
6826:
6753:⊆
6704:
6627:
6609:(because
6594:
6581:∩
6575:⊆
6569:
6512:
6499:∩
6450:
6400:
6387:∩
6369:
6236:
6223:⊆
6217:
6096:⊆
6090:⊆
6031:
6013:∈
6006:⋃
6002:≠
5979:∈
5972:⋃
5963:
5924:
5908:∪
5899:
5874:∪
5865:
5767:empty set
5737:
5675:A subset
5624:∈
5617:⋂
5608:
5590:
5559:
5539:∈
5532:⋂
5523:
5414:∈
5407:⋃
5398:
5380:
5349:
5329:∈
5322:⋃
5313:
5207:…
5102:∖
5093:
5080:∖
5068:
5047:and also
5023:∖
5014:
5001:∖
4989:
4898:open sets
4824:⊆
4750:−
4723:¯
4692:
4659:⊆
4627:℘
4624:→
4612:℘
4576:τ
4495:power set
4469:on a set
4144:∈
4043:
3879:
3751:
3645:
3555:
3467:≥
3443:∈
3396:∈
3382:
3017:
2880:
2826:∅
2806:∅
2786:∅
2783:
2774:∅
2694:sequences
2673:
2626:
2615:contains
2575:contains
2515:
2489:
2463:⊆
2434:
2352:
2317:τ
2241:∈
2215:
2179:∂
2133:
2083:
2029:
1897:
1843:
1775:−
1743:¯
1713:
1690:τ
1650:τ
1627:
1594:
1586:τ
1546:τ
891:of a set
838:∈
588:∈
429:∈
261:itself).
199:open ball
197:if every
12168:Wikibook
12146:Category
12034:Manifold
12002:Homotopy
11960:Interior
11951:Open set
11909:Homology
11858:Topology
11745:Topology
11727:Topology
11707:Topology
11587:, p. 104
11194:See also
11154:interior
10642:category
10615:powerset
9858:close to
9446:preimage
9057:manifold
8881:because
6856:Because
6644:smallest
6432:Because
6133:subspace
5842:superset
5834:finitely
5458:If each
5248:If each
4936:interior
4328:because
2743:Examples
2394:The set
2170:boundary
2168:and its
1321:itself.
801:(again,
112:interior
72:boundary
70:and its
28:topology
12193:general
11995:uniform
11975:compact
11926:Digital
11822:, 2001
11783:1921556
11619:, p. 41
11563:, p. 75
11551:, p. 20
9247:is any
8771:and if
8747:is any
6642:is the
6111:and if
5164:Theorem
4934:to the
4493:of the
4491:mapping
4124:and if
3937:In any
3930:In any
3627:, then
3321:is the
2999:, then
2901:Giving
2838:itself.
2739:below.
1503:closure
664:". Let
651:infimum
649:is the
78:of all
32:closure
12188:Topics
11990:metric
11865:Fields
11789:
11781:
11771:
11715:
11693:
11675:
11653:
11575:, p. 4
11505:
11499:
11474:
11468:
11437:
11431:
9655:
9649:
8395:is in
8311:where
8140:where
7564:
7558:
6381:
6375:
5823:subset
5697:closed
5166:
3512:finite
2749:sphere
2725:filter
2723:" or "
2690:limits
2369:closed
1193:Thus,
972:itself
629:where
44:points
30:, the
11970:Space
11255:From
11243:Notes
11114:with
10775:is a
10298:then
10148:then
9271:then
9055:is a
8415:then
7866:Proof
6425:Proof
6179:that
6131:is a
5837:case.
5830:union
5828:In a
5505:then
5295:then
4489:is a
4198:then
4069:dense
3585:axiom
3537:then
3510:is a
3364:then
3272:dense
2654:In a
2478:then
2367:is a
1526:of a
1301:than
727:is a
707:Then
284:of a
98:is a
64:union
11787:OCLC
11769:ISBN
11713:ISBN
11691:ISBN
11673:ISBN
11651:ISBN
11366:and
11296:and
11152:the
10613:The
7948:Let
7873:Let
7858:).
7838:and
7670:and
7482:but
7407:and
6901:and
6808:and
5173:Let
4932:dual
4457:and
4177:>
4165:>
3835:the
3729:the
3420:>
2945:the
2923:and
2753:ball
1682:and
1642:(if
1500:The
827:for
470:<
403:>
133:For
108:dual
11156:of
11125:int
10947:to
10779:of
10755:on
9886:if
9856:is
9813:in
9251:of
9203:in
8751:of
7450:If
6351::
6135:of
6079:If
5832:of
5699:in
5695:is
5581:int
5514:int
5389:int
5340:int
5130:in
5059:int
5005:int
4947:int
4930:is
4737:or
4674:to
4497:of
4222:in
4218:is
4003:of
3595:If
3490:If
3301:If
3274:in
3270:is
3219:of
3155:If
3120:is
2995:of
2953:If
2818:is
2758:In
2731:).
2721:net
2662:),
2535:If
2452:If
2300:in
2258:net
1862:of
1760:or
1446:or
1186:or
1161:of
741:of
735:or
653:.
637:inf
581:inf
102:of
66:of
54:of
46:in
26:In
12232::
11818:,
11812:,
11785:.
11779:MR
11777:.
11767:.
11484:cl
11441:cl
11375:cl
11307::=
11266::=
11190:.
10984:cl
10906:cl
10526:cl
10427:cl
10391:cl
9901:cl
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9629:cl
8927:cl
8890:cl
8845:cl
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8670:cl
8639:cl
8579:cl
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7963:cl
7959::=
7794:cl
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7531:cl
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4683:cl
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4463:A
4445:.
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2026:cl
2016:.)
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1817:Cl
1797:cl
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1573:cl
577::=
157:,
114:.
11850:e
11843:t
11836:v
11797:.
11517:.
11514:}
11511:0
11508:{
11502:=
11496:S
11488:X
11477:T
11465:)
11462:T
11456:S
11453:(
11445:T
11434:=
11408:,
11405:)
11399:,
11396:0
11393:[
11390:=
11387:S
11379:X
11351:=
11348:T
11342:S
11322:)
11316:,
11313:0
11310:(
11304:S
11284:]
11281:0
11278:,
11269:(
11263:T
11167:.
11164:A
11140:,
11137:)
11134:A
11131:(
11102:,
11099:A
11079:)
11076:A
11070:X
11064:I
11061:(
11041:A
11021:A
11015:X
10993:.
10990:A
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10958:,
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10935:A
10915:.
10912:A
10886:.
10883:)
10880:I
10874:A
10871:(
10848:X
10842:A
10822:.
10819:P
10813:T
10810::
10807:I
10787:P
10763:X
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10723:.
10720:B
10700:A
10680:B
10674:A
10651:P
10625:X
10591:.
10588:X
10582:C
10562:)
10559:C
10556:(
10553:f
10547:)
10544:C
10541:(
10538:f
10530:Y
10505:Y
10499:X
10496::
10493:f
10473:.
10470:X
10464:A
10443:)
10439:A
10431:X
10422:(
10418:f
10412:)
10409:A
10406:(
10403:f
10395:Y
10370:Y
10364:X
10361::
10358:f
10338:.
10335:Y
10315:)
10312:C
10309:(
10306:f
10286:X
10266:C
10242:Y
10236:X
10233::
10230:f
10197:.
10194:)
10191:A
10188:(
10185:f
10165:)
10162:x
10159:(
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10127:A
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10037:)
10034:A
10031:(
10028:f
10008:A
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9969:,
9966:X
9960:A
9940:f
9916:,
9913:A
9905:X
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9868:A
9844:x
9824:.
9821:Y
9801:)
9798:A
9795:(
9792:f
9772:)
9769:x
9766:(
9763:f
9744:,
9741:X
9735:A
9715:X
9709:x
9689:.
9686:)
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9680:A
9677:(
9674:f
9671:(
9663:Y
9645:)
9641:A
9633:X
9624:(
9620:f
9601:,
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9592:A
9572:Y
9566:X
9563::
9560:f
9538:.
9535:Y
9515:C
9495:X
9475:)
9472:C
9469:(
9464:1
9457:f
9428:Y
9422:X
9419::
9416:f
9378:.
9373:U
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9325:U
9319:S
9299:X
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9259:X
9233:U
9211:X
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9095:X
9069:U
9043:X
9023:X
8997:U
8989:U
8967:U
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8939:S
8931:X
8920:U
8917:=
8914:)
8911:U
8905:S
8902:(
8894:U
8869:)
8866:S
8860:U
8857:(
8849:U
8837:U
8829:U
8821:=
8818:S
8810:X
8785:X
8779:S
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8733:U
8685:.
8682:S
8674:X
8663:)
8660:S
8654:T
8651:(
8643:X
8632:C
8612:.
8609:C
8606:=
8603:C
8597:T
8591:S
8583:X
8572:T
8552:T
8532:.
8529:C
8523:)
8520:T
8514:X
8511:(
8505:S
8497:X
8472:C
8469:=
8466:)
8463:S
8457:T
8454:(
8446:T
8435:S
8429:T
8423:s
8403:T
8383:S
8377:s
8357:S
8337:C
8331:)
8328:T
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8319:(
8299:X
8279:C
8273:)
8270:T
8264:X
8261:(
8258:=
8255:)
8252:C
8246:T
8243:(
8237:X
8217:.
8214:X
8194:C
8188:T
8168:X
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8073:T
8067:S
8061:T
8041:)
8038:S
8032:T
8029:(
8021:X
8010:T
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7987:)
7984:S
7978:T
7975:(
7967:T
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7936:.
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7784:=
7781:)
7778:T
7772:S
7769:(
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7736:X
7716:T
7696:)
7690:,
7687:0
7684:(
7681:=
7678:S
7658:,
7655:]
7652:0
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7640:(
7637:=
7634:T
7613:R
7609:=
7606:X
7586:S
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7567:T
7555:)
7552:T
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7543:(
7535:T
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7433:)
7427:,
7424:0
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7418:=
7415:T
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7392:)
7389:1
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7380:(
7377:=
7374:S
7355:,
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7324:;
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7230:.
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7097:S
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7081:=
7078:C
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7027:T
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6965:S
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6730:,
6727:T
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5880:)
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