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It does not make a difference if we restrict the condition to open neighbourhoods only. It is often convenient to use the "open neighbourhood" form of the definition to show that a point is a limit point and to use the "general neighbourhood" form of the definition to derive facts from a known limit
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488:). Importantly, although "limit point of a set" is synonymous with "cluster/accumulation point of a set", this is not true for sequences (nor nets or filters). That is, the term "limit point of a sequence" is
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If no element occurs infinitely many times in the sequence, for example if all the elements are distinct, any accumulation point of the sequence is an
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is an accumulation point of any of the corresponding sequences (because any neighborhood of the point will contain infinitely many elements of
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We use the fact that a point is in the closure of a set if and only if every neighborhood of the point meets the set. Now,
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of a non-constant sequence is an accumulation point of the sequence. And by definition, every limit point is an
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3902:{\displaystyle \operatorname {cl} (S)=L(S)\cup I(S)\quad {\text{and}}\quad L(S)\cap I(S)=\emptyset .}
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cannot be an accumulation point of any of the associated sequences without infinite repeats (because
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is discrete, then every point is isolated and cannot be a limit point of any set. Conversely, if
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Cluster points in nets encompass the idea of both condensation points and ω-accumulation points.
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However, a set can not have a non-trivial intersection with its complement. Conversely, assume
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contains all but finitely many elements of the sequence). That is why we do not use the term
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with standard topology (for a less trivial example of a limit point, see the first caption).
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here), viz. -1 and +1. Thus, thinking of sets, these points are limit points of the set
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can be expressed as a union of open neighbourhoods of the points in the complement of
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has a neighborhood that contains only finitely many (possibly even none) points of
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Relation between accumulation point of a sequence and accumulation point of a set
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866:(i.e. does not converge), but has two accumulation points (which are considered
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This definition of a cluster or accumulation point of a sequence generalizes to
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itself. A limit point can be characterized as an adherent point that is not an
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30:"Limit point" redirects here. For uses where the word "point" is optional, see
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and that neighborhood can only contain finitely many terms of such sequences).
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A corollary of this result gives us a characterisation of closed sets: A set
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in many ways, even with repeats, and thus associate with it many sequences
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contains all its limit points. We shall show that the complement of
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The set of all cluster points of a sequence is sometimes called the
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In fact, Fréchet–Urysohn spaces are characterized by this property.
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is not a limit point, and hence there exists an open neighbourhood
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of a sequence as a synonym for accumulation point of the sequence.
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and hence also infinitely many terms in any associated sequence).
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For example, if the sequence is the constant sequence with value
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to which the sequence converges (that is, every neighborhood of
2887:{\displaystyle x_{\bullet }=\left(x_{n}\right)_{n=1}^{\infty }}
1984:{\displaystyle x_{\bullet }=\left(x_{n}\right)_{n=1}^{\infty }}
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is closed if and only if it contains all of its limit points.
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then we have the following characterization of the closure of
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synonymous with "cluster/accumulation point of a sequence".
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need not be an accumulation point of the corresponding set
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is a limit point (though not a boundary point) of interval
6502:. Berlin New York: Springer Science & Business Media.
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may have a neighbourhood not containing points other than
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Pages displaying short descriptions of redirect targets
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is a boundary point (but not a limit point) of the set
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Limit points of a set should also not be confused with
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The limit points of a set should not be confused with
6415:"Difference between boundary point & limit point"
6360: – Point to which functions converge in analysis
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that is not open. Hence, every open neighbourhood of
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This concept profitably generalizes the notion of a
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3344:{\displaystyle \operatorname {Im} x_{\bullet }.}
3284:{\displaystyle \operatorname {Im} x_{\bullet }.}
3127:{\displaystyle \operatorname {Im} x_{\bullet }.}
3248:{\displaystyle \operatorname {Im} x_{\bullet }}
1853:Net (mathematics) § Cluster point of a net
920:
3078:is an accumulation point of the sequence. But
2948:{\displaystyle x_{\bullet }:\mathbb {N} \to X}
853:{\displaystyle x_{n}=(-1)^{n}{\frac {n}{n+1}}}
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550:. Unlike for limit points, an adherent point
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1500:Special types of accumulation point of a set
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1303:spaces are characterized by this property.
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883:
774:and is the underpinning of concepts such as
654:
648:
246:There is also a closely related concept for
5879:is not discrete, then there is a singleton
3354:Conversely, given a countable infinite set
1811:is a specific type of limit point called a
1667:is a specific type of limit point called a
476:) by definition refers to a point that the
302:{\displaystyle (x_{n})_{n\in \mathbb {N} }}
6994:
6967:
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3315:-accumulation point of the associated set
395:there are infinitely many natural numbers
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2384:Note that there is already the notion of
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2051:
1845:Accumulation points of sequences and nets
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675:
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223:does not itself have to be an element of
36:Limit (disambiguation) § Mathematics
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6389:
6342: – Set of all limit points of a set
6324: – a stronger analog of limit point
5575:Since this argument holds for arbitrary
924:
784:
119:that can be "approximated" by points of
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5033:is equal to its closure if and only if
14:
7012:
3715:{\displaystyle \operatorname {cl} (S)}
2172:{\displaystyle n_{0}\in \mathbb {N} ,}
1233:if and only if every neighbourhood of
6614:
6372: – The limit of some subsequence
4256:to denote the set of limit points of
4114:if and only if every neighborhood of
4051:if and only if every neighborhood of
3403:we can enumerate all the elements of
6133:is a singleton, then every point of
4371:This fact is sometimes taken as the
929:A sequence enumerating all positive
27:Cluster point in a topological space
5762:that contains no points other than
5552:lies entirely in the complement of
5257:comprises an open neighbourhood of
1524:contains infinitely many points of
1253:contains infinitely many points of
793:, the sequence of rational numbers
506:) for which every neighbourhood of
352:such that, for every neighbourhood
24:
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3028:can be defined in the usual way.
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2347:is a limit of some subsequence of
1976:
25:
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6443:"Examples of Accumulation Points"
6220:is nonempty, its closure will be
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4217:{\displaystyle S\setminus \{x\}.}
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4162:{\displaystyle S\setminus \{x\},}
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3995:{\displaystyle S\setminus \{x\}.}
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2483:{\displaystyle f:(P,\leq )\to X,}
2058:{\displaystyle n\in \mathbb {N} }
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6213:{\displaystyle S\setminus \{x\}}
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2546:is a topological space. A point
1922:accumulation point of a sequence
1413:{\displaystyle S\setminus \{x\}}
462:The similarly named notion of a
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1086:contains at least one point of
203:itself. A limit point of a set
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3847:
3838:
3832:
3823:
3817:
3791:
3785:
3762:
3756:
3709:
3703:
2939:
2739:
2733:
2509:
2497:
2471:
2468:
2456:
823:
813:
732:
720:
282:
268:
13:
1:
6476:
5689:is a limit point of any set.
3674:
2370:{\displaystyle x_{\bullet }.}
1039:accumulation point of the set
915:
6640:
6445:. 2021-01-13. Archived from
6376:
6152:{\displaystyle X\setminus S}
5812:if and only if no subset of
5067:{\displaystyle S=S\cup L(S)}
4856:then every neighbourhood of
4718:then every neighbourhood of
4515:then every neighbourhood of
3963:if and only if it is in the
3956:{\displaystyle S\subseteq X}
3443:{\displaystyle x_{\bullet }}
3373:{\displaystyle A\subseteq X}
2914:is by definition just a map
2320:{\displaystyle x_{\bullet }}
921:Accumulation points of a set
905:{\displaystyle S=\{x_{n}\}.}
760:{\displaystyle \mathbb {R} }
682:{\displaystyle \mathbb {R} }
7:
6599:Encyclopedia of Mathematics
6532:. Boston: Allyn and Bacon.
6309:
5716:is an isolated point, then
4936:is again in the closure of
3032:If there exists an element
2716:{\displaystyle p\geq p_{0}}
2683:{\displaystyle p_{0}\in P,}
2587:accumulation point of a net
2241:{\displaystyle x_{n}\in V.}
2205:{\displaystyle n\geq n_{0}}
2094:{\displaystyle x_{n}\in V.}
1815:complete accumulation point
1450:The set of limit points of
444:{\displaystyle x_{n}\in V.}
10:
7046:
6887:Banach fixed-point theorem
5323:any open neighbourhood of
4959:This completes the proof.
2754:{\displaystyle f(p)\in V,}
2439:generalizes the idea of a
2037:there are infinitely many
1850:
1768:equals the cardinality of
1600:If every neighbourhood of
1504:If every neighbourhood of
1384:if and only if there is a
789:With respect to the usual
29:
6920:
6877:
6841:
6727:
6716:
6648:
6340:Derived set (mathematics)
6283:is the unique element of
5013:is closed if and only if
4558:and this point cannot be
4319:is equal to the union of
2515:{\displaystyle (P,\leq )}
1857:Cluster point of a filter
465:limit point of a sequence
6500:Éléments de mathématique
5953:{\displaystyle y\neq x,}
5661:Hence the complement of
5509:that does not intersect
4386:("Left subset") Suppose
3222:is an isolated point of
139:in the sense that every
6429:"What is a limit point"
5237:then the complement to
3610:-accumulation point of
3603:{\displaystyle \omega }
3515:-accumulation point of
3508:{\displaystyle \omega }
3308:{\displaystyle \omega }
2276:(or, more generally, a
1861:In a topological space
1761:{\displaystyle U\cap S}
1698:If every neighbourhood
470:limit point of a filter
6942:Mathematics portal
6842:Metrics and properties
6828:Second-countable space
6594:"Limit point of a set"
6564:Upper Saddle River, NJ
6300:
6277:
6257:
6243:It is only empty when
6237:
6214:
6176:
6153:
6127:
6107:
6084:
6064:
6044:
6020:
5997:
5974:
5954:
5925:
5899:
5873:
5853:
5826:
5802:
5779:
5756:
5742:is a neighbourhood of
5736:
5710:
5675:
5655:
5632:
5612:
5589:
5569:
5546:
5526:
5503:
5483:
5463:
5443:
5420:
5400:
5380:
5360:
5337:
5317:
5294:
5274:
5251:
5231:
5208:
5188:
5165:
5145:
5120:
5097:
5068:
5027:
5007:
4977:
4953:
4930:
4910:
4890:
4870:
4850:
4818:
4798:
4775:
4755:
4732:
4712:
4689:
4667:
4635:
4615:
4595:
4575:
4552:
4529:
4509:
4486:
4466:
4443:
4423:
4400:
4365:
4333:
4313:
4293:
4273:
4250:
4218:
4183:
4163:
4128:
4108:
4085:
4065:
4045:
4022:
3996:
3957:
3931:
3930:{\displaystyle x\in X}
3903:
3798:
3769:
3736:
3716:
3664:
3644:
3624:
3604:
3580:
3579:{\displaystyle x\in X}
3549:
3529:
3509:
3486:
3444:
3417:
3397:
3374:
3345:
3309:
3285:
3249:
3216:
3196:
3151:
3128:
3092:
3072:
3052:
3051:{\displaystyle x\in X}
3022:
2949:
2908:
2888:
2802:
2775:
2755:
2717:
2684:
2648:
2628:
2605:
2566:
2565:{\displaystyle x\in X}
2540:
2516:
2484:
2443:. A net is a function
2422:
2402:
2371:
2341:
2321:
2300:is a cluster point of
2294:
2262:
2242:
2206:
2173:
2135:
2115:
2095:
2059:
2031:
2008:
1985:
1904:
1903:{\displaystyle x\in X}
1878:
1836:
1805:
1785:
1762:
1732:
1712:
1690:
1661:
1641:
1614:
1592:
1561:
1541:
1518:
1491:
1464:
1441:
1414:
1378:
1358:
1357:{\displaystyle x\in X}
1336:first-countable spaces
1320:
1297:
1270:
1247:
1227:
1207:
1206:{\displaystyle x\in X}
1175:
1147:
1120:
1100:
1080:
1056:
1021:
1001:
981:
955:
938:
912:
906:
854:
761:
739:
707:
683:
661:
635:
604:
584:
564:
544:
520:
445:
409:
389:
366:
346:
326:
303:
240:
217:
197:
177:
157:
133:
113:
93:
70:
6301:
6278:
6258:
6238:
6215:
6177:
6154:
6128:
6108:
6085:
6065:
6045:
6021:
5998:
5975:
5955:
5926:
5924:{\displaystyle \{x\}}
5900:
5898:{\displaystyle \{x\}}
5874:
5854:
5827:
5803:
5780:
5757:
5737:
5735:{\displaystyle \{x\}}
5711:
5676:
5656:
5633:
5613:
5595:in the complement of
5590:
5570:
5547:
5527:
5504:
5484:
5464:
5444:
5421:
5401:
5381:
5361:
5338:
5318:
5295:
5275:
5252:
5232:
5209:
5189:
5166:
5146:
5121:
5098:
5069:
5028:
5008:
4978:
4954:
4931:
4911:
4891:
4871:
4851:
4849:{\displaystyle L(S),}
4819:
4799:
4781:is in the closure of
4776:
4756:
4733:
4713:
4690:
4668:
4666:{\displaystyle L(S).}
4636:
4616:
4596:
4576:
4553:
4530:
4510:
4487:
4467:
4444:
4424:
4406:is in the closure of
4401:
4366:
4364:{\displaystyle L(S).}
4334:
4314:
4294:
4274:
4251:
4219:
4189:is in the closure of
4184:
4164:
4129:
4109:
4086:
4066:
4046:
4023:
3997:
3958:
3932:
3904:
3799:
3770:
3737:
3717:
3665:
3645:
3625:
3605:
3581:
3550:
3530:
3510:
3487:
3445:
3418:
3398:
3375:
3346:
3310:
3286:
3250:
3217:
3197:
3152:
3129:
3093:
3073:
3053:
3023:
2950:
2909:
2889:
2816:are also defined for
2803:
2776:
2756:
2718:
2685:
2649:
2629:
2606:
2567:
2541:
2517:
2485:
2423:
2403:
2372:
2342:
2322:
2295:
2278:Fréchet–Urysohn space
2274:first-countable space
2263:
2243:
2207:
2174:
2136:
2116:
2096:
2060:
2032:
2009:
1986:
1905:
1879:
1837:
1806:
1786:
1763:
1733:
1713:
1691:
1662:
1642:
1615:
1593:
1562:
1542:
1519:
1492:
1465:
1442:
1415:
1379:
1359:
1328:Fréchet–Urysohn space
1321:
1298:
1296:{\displaystyle T_{1}}
1271:
1248:
1228:
1208:
1176:
1174:{\displaystyle T_{1}}
1148:
1121:
1101:
1081:
1057:
1022:
1002:
982:
956:
928:
907:
855:
788:
762:
740:
708:
684:
662:
660:{\displaystyle \{0\}}
636:
605:
585:
565:
545:
521:
478:sequence converges to
446:
410:
390:
367:
347:
327:
304:
241:
218:
198:
178:
158:
134:
114:
94:
71:
6897:Invariance of domain
6849:Euler characteristic
6823:Bundle (mathematics)
6287:
6267:
6247:
6224:
6192:
6163:
6159:is a limit point of
6137:
6117:
6094:
6090:are limit points of
6074:
6054:
6034:
6010:
5984:
5980:is a limit point of
5964:
5935:
5909:
5883:
5863:
5843:
5816:
5792:
5766:
5746:
5720:
5700:
5665:
5642:
5622:
5599:
5579:
5556:
5536:
5513:
5493:
5473:
5453:
5430:
5410:
5406:is an open set. Let
5390:
5370:
5347:
5327:
5304:
5300:is a limit point of
5284:
5261:
5241:
5218:
5198:
5175:
5155:
5151:be a closed set and
5135:
5107:
5096:{\displaystyle L(S)}
5078:
5037:
5017:
4997:
4967:
4940:
4920:
4900:
4880:
4876:contains a point of
4860:
4828:
4808:
4785:
4765:
4742:
4722:
4699:
4679:
4675:("Right subset") If
4645:
4625:
4605:
4601:is a limit point of
4585:
4562:
4539:
4535:contains a point of
4519:
4496:
4476:
4453:
4433:
4410:
4390:
4343:
4323:
4303:
4283:
4260:
4249:{\displaystyle L(S)}
4231:
4193:
4173:
4138:
4134:contains a point of
4118:
4095:
4075:
4071:contains a point of
4055:
4032:
4028:is a limit point of
4012:
3971:
3941:
3937:is a limit point of
3915:
3808:
3797:{\displaystyle I(S)}
3779:
3775:and isolated points
3768:{\displaystyle L(S)}
3750:
3746:of its limit points
3726:
3694:
3654:
3634:
3614:
3594:
3564:
3539:
3519:
3499:
3454:
3427:
3407:
3384:
3358:
3319:
3299:
3259:
3226:
3206:
3161:
3138:
3102:
3082:
3062:
3036:
2962:
2918:
2898:
2832:
2789:
2765:
2727:
2694:
2658:
2638:
2618:
2595:
2550:
2530:
2494:
2447:
2412:
2392:
2351:
2331:
2304:
2284:
2252:
2216:
2183:
2145:
2125:
2105:
2069:
2041:
2018:
1998:
1929:
1888:
1865:
1823:
1795:
1772:
1746:
1722:
1702:
1677:
1651:
1628:
1604:
1579:
1571:ω-accumulation point
1551:
1528:
1508:
1478:
1454:
1428:
1392:
1368:
1364:is a limit point of
1342:
1310:
1280:
1257:
1237:
1217:
1213:is a limit point of
1191:
1158:
1137:
1110:
1090:
1070:
1046:
1011:
991:
968:
945:
874:
797:
749:
717:
697:
671:
645:
625:
594:
574:
554:
534:
510:
474:limit point of a net
419:
399:
376:
356:
336:
316:
265:
227:
207:
187:
167:
163:contains a point of
147:
123:
103:
83:
60:
18:Limit point of a set
6907:Tychonoff's theorem
6902:Poincaré conjecture
6656:General (point-set)
6562:(Second ed.).
6392:, pp. 209–210.
6370:Subsequential limit
6364:Limit of a sequence
6358:Limit of a function
6346:Filters in topology
5832:has a limit point.
2883:
2785:which converges to
2386:limit of a sequence
1980:
937:is a cluster point.
780:topological closure
706:{\displaystyle 0.5}
482:filter converges to
480:(respectively, the
32:Limit (mathematics)
6892:De Rham cohomology
6813:Polyhedral complex
6803:Simplicial complex
6568:Prentice Hall, Inc
6495:Topologie Générale
6469:, pp. 97–102.
6322:Condensation point
6299:{\displaystyle S.}
6296:
6273:
6253:
6236:{\displaystyle X.}
6233:
6210:
6186:
6175:{\displaystyle S.}
6172:
6149:
6123:
6106:{\displaystyle S.}
6103:
6080:
6060:
6040:
6016:
5996:{\displaystyle X.}
5993:
5970:
5950:
5921:
5895:
5869:
5849:
5837:
5822:
5798:
5778:{\displaystyle x.}
5775:
5752:
5732:
5706:
5694:
5671:
5654:{\displaystyle S.}
5651:
5628:
5618:the complement of
5611:{\displaystyle S,}
5608:
5585:
5568:{\displaystyle S.}
5565:
5542:
5525:{\displaystyle S,}
5522:
5499:
5479:
5459:
5442:{\displaystyle S.}
5439:
5416:
5396:
5376:
5359:{\displaystyle S.}
5356:
5333:
5316:{\displaystyle S,}
5313:
5290:
5273:{\displaystyle x.}
5270:
5247:
5230:{\displaystyle S,}
5227:
5204:
5187:{\displaystyle S.}
5184:
5161:
5141:
5119:{\displaystyle S.}
5116:
5093:
5064:
5023:
5003:
4988:
4973:
4952:{\displaystyle S.}
4949:
4926:
4906:
4886:
4866:
4846:
4814:
4797:{\displaystyle S.}
4794:
4771:
4754:{\displaystyle S,}
4751:
4728:
4711:{\displaystyle S,}
4708:
4685:
4663:
4631:
4611:
4591:
4574:{\displaystyle x.}
4571:
4551:{\displaystyle S,}
4548:
4525:
4508:{\displaystyle S,}
4505:
4482:
4465:{\displaystyle S,}
4462:
4439:
4422:{\displaystyle S.}
4419:
4396:
4384:
4361:
4329:
4309:
4289:
4272:{\displaystyle S,}
4269:
4246:
4214:
4179:
4159:
4124:
4107:{\displaystyle x,}
4104:
4081:
4061:
4044:{\displaystyle S,}
4041:
4018:
4006:
3992:
3953:
3927:
3899:
3794:
3765:
3732:
3712:
3660:
3640:
3620:
3600:
3576:
3545:
3525:
3505:
3482:
3450:that will satisfy
3440:
3413:
3396:{\displaystyle X,}
3393:
3370:
3341:
3305:
3281:
3245:
3212:
3192:
3150:{\displaystyle x,}
3147:
3124:
3088:
3068:
3048:
3018:
2945:
2904:
2884:
2848:
2801:{\displaystyle x.}
2798:
2771:
2751:
2713:
2680:
2644:
2624:
2601:
2562:
2536:
2512:
2480:
2418:
2398:
2367:
2337:
2317:
2290:
2258:
2238:
2202:
2169:
2131:
2111:
2091:
2055:
2030:{\displaystyle x,}
2027:
2004:
1981:
1945:
1900:
1877:{\displaystyle X,}
1874:
1835:{\displaystyle S.}
1832:
1801:
1784:{\displaystyle S,}
1781:
1758:
1728:
1708:
1689:{\displaystyle S.}
1686:
1670:condensation point
1657:
1640:{\displaystyle S,}
1637:
1610:
1591:{\displaystyle S.}
1588:
1557:
1540:{\displaystyle S,}
1537:
1514:
1490:{\displaystyle S.}
1487:
1460:
1440:{\displaystyle x.}
1437:
1410:
1374:
1354:
1316:
1293:
1269:{\displaystyle S.}
1266:
1243:
1223:
1203:
1171:
1143:
1116:
1096:
1076:
1052:
1017:
997:
980:{\displaystyle X.}
977:
951:
939:
913:
902:
850:
791:Euclidean topology
757:
735:
703:
679:
657:
631:
600:
580:
560:
540:
516:
441:
405:
388:{\displaystyle x,}
385:
362:
342:
322:
299:
256:accumulation point
239:{\displaystyle S.}
236:
213:
193:
173:
153:
129:
109:
89:
66:
47:accumulation point
41:In mathematics, a
7007:
7006:
6796:fundamental group
6577:978-0-13-181629-9
6556:Munkres, James R.
6539:978-0-697-06889-7
6509:978-3-540-64241-1
6484:Bourbaki, Nicolas
6404:, pp. 68–83.
6331:Convergent filter
6276:{\displaystyle x}
6256:{\displaystyle S}
6184:
6126:{\displaystyle S}
6083:{\displaystyle X}
6063:{\displaystyle X}
6043:{\displaystyle S}
6019:{\displaystyle X}
5973:{\displaystyle x}
5931:contains a point
5872:{\displaystyle X}
5852:{\displaystyle X}
5835:
5825:{\displaystyle X}
5801:{\displaystyle X}
5755:{\displaystyle x}
5709:{\displaystyle x}
5692:
5674:{\displaystyle S}
5631:{\displaystyle S}
5588:{\displaystyle x}
5545:{\displaystyle U}
5502:{\displaystyle x}
5482:{\displaystyle U}
5462:{\displaystyle x}
5419:{\displaystyle x}
5399:{\displaystyle S}
5379:{\displaystyle S}
5336:{\displaystyle x}
5293:{\displaystyle x}
5250:{\displaystyle S}
5207:{\displaystyle x}
5171:a limit point of
5164:{\displaystyle x}
5144:{\displaystyle S}
5026:{\displaystyle S}
5006:{\displaystyle S}
4986:
4976:{\displaystyle S}
4929:{\displaystyle x}
4909:{\displaystyle x}
4889:{\displaystyle S}
4869:{\displaystyle x}
4817:{\displaystyle x}
4774:{\displaystyle x}
4731:{\displaystyle x}
4688:{\displaystyle x}
4634:{\displaystyle x}
4614:{\displaystyle S}
4594:{\displaystyle x}
4528:{\displaystyle x}
4485:{\displaystyle x}
4442:{\displaystyle x}
4399:{\displaystyle x}
4382:
4332:{\displaystyle S}
4312:{\displaystyle S}
4299:: The closure of
4292:{\displaystyle S}
4182:{\displaystyle x}
4127:{\displaystyle x}
4084:{\displaystyle S}
4064:{\displaystyle x}
4021:{\displaystyle x}
4004:
3860:
3735:{\displaystyle S}
3663:{\displaystyle A}
3643:{\displaystyle x}
3623:{\displaystyle A}
3548:{\displaystyle A}
3528:{\displaystyle A}
3416:{\displaystyle A}
3215:{\displaystyle x}
3091:{\displaystyle x}
3071:{\displaystyle x}
2907:{\displaystyle X}
2774:{\displaystyle f}
2761:equivalently, if
2647:{\displaystyle x}
2627:{\displaystyle V}
2604:{\displaystyle f}
2539:{\displaystyle X}
2435:The concept of a
2421:{\displaystyle x}
2401:{\displaystyle x}
2340:{\displaystyle x}
2293:{\displaystyle x}
2261:{\displaystyle X}
2134:{\displaystyle x}
2114:{\displaystyle V}
2007:{\displaystyle V}
1804:{\displaystyle x}
1738:is such that the
1731:{\displaystyle x}
1711:{\displaystyle U}
1660:{\displaystyle x}
1613:{\displaystyle x}
1560:{\displaystyle x}
1517:{\displaystyle x}
1463:{\displaystyle S}
1377:{\displaystyle S}
1319:{\displaystyle X}
1246:{\displaystyle x}
1226:{\displaystyle S}
1146:{\displaystyle X}
1119:{\displaystyle x}
1099:{\displaystyle S}
1079:{\displaystyle x}
1055:{\displaystyle S}
1020:{\displaystyle X}
1000:{\displaystyle x}
963:topological space
961:be a subset of a
954:{\displaystyle S}
848:
691:standard topology
634:{\displaystyle 0}
603:{\displaystyle x}
583:{\displaystyle S}
563:{\displaystyle x}
543:{\displaystyle S}
519:{\displaystyle x}
468:(respectively, a
408:{\displaystyle n}
365:{\displaystyle V}
345:{\displaystyle x}
325:{\displaystyle X}
311:topological space
216:{\displaystyle S}
196:{\displaystyle x}
176:{\displaystyle S}
156:{\displaystyle x}
132:{\displaystyle S}
112:{\displaystyle x}
92:{\displaystyle X}
78:topological space
69:{\displaystyle S}
16:(Redirected from
7037:
7030:General topology
6997:
6996:
6970:
6969:
6960:
6950:
6940:
6939:
6928:
6927:
6722:
6635:
6628:
6621:
6612:
6611:
6607:
6589:
6551:
6521:
6470:
6464:
6458:
6457:
6455:
6454:
6439:
6433:
6432:
6425:
6419:
6418:
6411:
6405:
6399:
6393:
6387:
6336:
6327:
6305:
6303:
6302:
6297:
6282:
6280:
6279:
6274:
6262:
6260:
6259:
6254:
6242:
6240:
6239:
6234:
6219:
6217:
6216:
6211:
6181:
6179:
6178:
6173:
6158:
6156:
6155:
6150:
6132:
6130:
6129:
6124:
6112:
6110:
6109:
6104:
6089:
6087:
6086:
6081:
6069:
6067:
6066:
6061:
6049:
6047:
6046:
6041:
6028:trivial topology
6025:
6023:
6022:
6017:
6002:
6000:
5999:
5994:
5979:
5977:
5976:
5971:
5959:
5957:
5956:
5951:
5930:
5928:
5927:
5922:
5904:
5902:
5901:
5896:
5878:
5876:
5875:
5870:
5858:
5856:
5855:
5850:
5831:
5829:
5828:
5823:
5807:
5805:
5804:
5799:
5784:
5782:
5781:
5776:
5761:
5759:
5758:
5753:
5741:
5739:
5738:
5733:
5715:
5713:
5712:
5707:
5680:
5678:
5677:
5672:
5660:
5658:
5657:
5652:
5637:
5635:
5634:
5629:
5617:
5615:
5614:
5609:
5594:
5592:
5591:
5586:
5574:
5572:
5571:
5566:
5551:
5549:
5548:
5543:
5531:
5529:
5528:
5523:
5508:
5506:
5505:
5500:
5488:
5486:
5485:
5480:
5468:
5466:
5465:
5460:
5448:
5446:
5445:
5440:
5425:
5423:
5422:
5417:
5405:
5403:
5402:
5397:
5385:
5383:
5382:
5377:
5365:
5363:
5362:
5357:
5342:
5340:
5339:
5334:
5322:
5320:
5319:
5314:
5299:
5297:
5296:
5291:
5279:
5277:
5276:
5271:
5256:
5254:
5253:
5248:
5236:
5234:
5233:
5228:
5213:
5211:
5210:
5205:
5193:
5191:
5190:
5185:
5170:
5168:
5167:
5162:
5150:
5148:
5147:
5142:
5125:
5123:
5122:
5117:
5103:is contained in
5102:
5100:
5099:
5094:
5073:
5071:
5070:
5065:
5032:
5030:
5029:
5024:
5012:
5010:
5009:
5004:
4982:
4980:
4979:
4974:
4958:
4956:
4955:
4950:
4935:
4933:
4932:
4927:
4915:
4913:
4912:
4907:
4895:
4893:
4892:
4887:
4875:
4873:
4872:
4867:
4855:
4853:
4852:
4847:
4823:
4821:
4820:
4815:
4803:
4801:
4800:
4795:
4780:
4778:
4777:
4772:
4760:
4758:
4757:
4752:
4737:
4735:
4734:
4729:
4717:
4715:
4714:
4709:
4694:
4692:
4691:
4686:
4672:
4670:
4669:
4664:
4640:
4638:
4637:
4632:
4620:
4618:
4617:
4612:
4600:
4598:
4597:
4592:
4581:In other words,
4580:
4578:
4577:
4572:
4557:
4555:
4554:
4549:
4534:
4532:
4531:
4526:
4514:
4512:
4511:
4506:
4491:
4489:
4488:
4483:
4472:we are done. If
4471:
4469:
4468:
4463:
4448:
4446:
4445:
4440:
4428:
4426:
4425:
4420:
4405:
4403:
4402:
4397:
4370:
4368:
4367:
4362:
4338:
4336:
4335:
4330:
4318:
4316:
4315:
4310:
4298:
4296:
4295:
4290:
4278:
4276:
4275:
4270:
4255:
4253:
4252:
4247:
4223:
4221:
4220:
4215:
4188:
4186:
4185:
4180:
4168:
4166:
4165:
4160:
4133:
4131:
4130:
4125:
4113:
4111:
4110:
4105:
4090:
4088:
4087:
4082:
4070:
4068:
4067:
4062:
4050:
4048:
4047:
4042:
4027:
4025:
4024:
4019:
4001:
3999:
3998:
3993:
3962:
3960:
3959:
3954:
3936:
3934:
3933:
3928:
3908:
3906:
3905:
3900:
3861:
3858:
3803:
3801:
3800:
3795:
3774:
3772:
3771:
3766:
3741:
3739:
3738:
3733:
3721:
3719:
3718:
3713:
3669:
3667:
3666:
3661:
3649:
3647:
3646:
3641:
3629:
3627:
3626:
3621:
3609:
3607:
3606:
3601:
3585:
3583:
3582:
3577:
3554:
3552:
3551:
3546:
3534:
3532:
3531:
3526:
3514:
3512:
3511:
3506:
3491:
3489:
3488:
3483:
3478:
3477:
3449:
3447:
3446:
3441:
3439:
3438:
3422:
3420:
3419:
3414:
3402:
3400:
3399:
3394:
3379:
3377:
3376:
3371:
3350:
3348:
3347:
3342:
3337:
3336:
3314:
3312:
3311:
3306:
3290:
3288:
3287:
3282:
3277:
3276:
3254:
3252:
3251:
3246:
3244:
3243:
3221:
3219:
3218:
3213:
3201:
3199:
3198:
3193:
3179:
3178:
3156:
3154:
3153:
3148:
3133:
3131:
3130:
3125:
3120:
3119:
3097:
3095:
3094:
3089:
3077:
3075:
3074:
3069:
3057:
3055:
3054:
3049:
3027:
3025:
3024:
3019:
3017:
3013:
3012:
2998:
2997:
2980:
2979:
2954:
2952:
2951:
2946:
2938:
2930:
2929:
2913:
2911:
2910:
2905:
2893:
2891:
2890:
2885:
2882:
2877:
2866:
2862:
2861:
2844:
2843:
2807:
2805:
2804:
2799:
2780:
2778:
2777:
2772:
2760:
2758:
2757:
2752:
2722:
2720:
2719:
2714:
2712:
2711:
2689:
2687:
2686:
2681:
2670:
2669:
2653:
2651:
2650:
2645:
2633:
2631:
2630:
2625:
2610:
2608:
2607:
2602:
2589:
2588:
2579:
2578:
2572:is said to be a
2571:
2569:
2568:
2563:
2545:
2543:
2542:
2537:
2521:
2519:
2518:
2513:
2489:
2487:
2486:
2481:
2427:
2425:
2424:
2419:
2407:
2405:
2404:
2399:
2388:to mean a point
2376:
2374:
2373:
2368:
2363:
2362:
2346:
2344:
2343:
2338:
2326:
2324:
2323:
2318:
2316:
2315:
2299:
2297:
2296:
2291:
2267:
2265:
2264:
2259:
2247:
2245:
2244:
2239:
2228:
2227:
2211:
2209:
2208:
2203:
2201:
2200:
2178:
2176:
2175:
2170:
2165:
2157:
2156:
2140:
2138:
2137:
2132:
2120:
2118:
2117:
2112:
2100:
2098:
2097:
2092:
2081:
2080:
2064:
2062:
2061:
2056:
2054:
2036:
2034:
2033:
2028:
2013:
2011:
2010:
2005:
1990:
1988:
1987:
1982:
1979:
1974:
1963:
1959:
1958:
1941:
1940:
1924:
1923:
1916:
1915:
1910:is said to be a
1909:
1907:
1906:
1901:
1883:
1881:
1880:
1875:
1841:
1839:
1838:
1833:
1817:
1816:
1810:
1808:
1807:
1802:
1790:
1788:
1787:
1782:
1767:
1765:
1764:
1759:
1737:
1735:
1734:
1729:
1717:
1715:
1714:
1709:
1695:
1693:
1692:
1687:
1666:
1664:
1663:
1658:
1646:
1644:
1643:
1638:
1622:uncountably many
1619:
1617:
1616:
1611:
1597:
1595:
1594:
1589:
1573:
1572:
1566:
1564:
1563:
1558:
1546:
1544:
1543:
1538:
1523:
1521:
1520:
1515:
1496:
1494:
1493:
1488:
1469:
1467:
1466:
1461:
1446:
1444:
1443:
1438:
1419:
1417:
1416:
1411:
1383:
1381:
1380:
1375:
1363:
1361:
1360:
1355:
1325:
1323:
1322:
1317:
1302:
1300:
1299:
1294:
1292:
1291:
1275:
1273:
1272:
1267:
1252:
1250:
1249:
1244:
1232:
1230:
1229:
1224:
1212:
1210:
1209:
1204:
1180:
1178:
1177:
1172:
1170:
1169:
1152:
1150:
1149:
1144:
1125:
1123:
1122:
1117:
1105:
1103:
1102:
1097:
1085:
1083:
1082:
1077:
1061:
1059:
1058:
1053:
1041:
1040:
1026:
1024:
1023:
1018:
1006:
1004:
1003:
998:
986:
984:
983:
978:
960:
958:
957:
952:
933:. Each positive
931:rational numbers
911:
909:
908:
903:
895:
894:
859:
857:
856:
851:
849:
847:
833:
831:
830:
809:
808:
766:
764:
763:
758:
756:
744:
742:
741:
738:{\displaystyle }
736:
712:
710:
709:
704:
688:
686:
685:
680:
678:
666:
664:
663:
658:
640:
638:
637:
632:
609:
607:
606:
601:
589:
587:
586:
581:
569:
567:
566:
561:
549:
547:
546:
541:
525:
523:
522:
517:
486:net converges to
450:
448:
447:
442:
431:
430:
414:
412:
411:
406:
394:
392:
391:
386:
371:
369:
368:
363:
351:
349:
348:
343:
331:
329:
328:
323:
308:
306:
305:
300:
298:
297:
296:
280:
279:
245:
243:
242:
237:
222:
220:
219:
214:
202:
200:
199:
194:
182:
180:
179:
174:
162:
160:
159:
154:
138:
136:
135:
130:
118:
116:
115:
110:
98:
96:
95:
90:
75:
73:
72:
67:
21:
7045:
7044:
7040:
7039:
7038:
7036:
7035:
7034:
7010:
7009:
7008:
7003:
6934:
6916:
6912:Urysohn's lemma
6873:
6837:
6723:
6714:
6686:low-dimensional
6644:
6639:
6592:
6578:
6540:
6526:Dugundji, James
6510:
6479:
6474:
6473:
6465:
6461:
6452:
6450:
6441:
6440:
6436:
6427:
6426:
6422:
6413:
6412:
6408:
6400:
6396:
6388:
6384:
6379:
6334:
6325:
6312:
6307:
6288:
6285:
6284:
6268:
6265:
6264:
6248:
6245:
6244:
6225:
6222:
6221:
6193:
6190:
6189:
6164:
6161:
6160:
6138:
6135:
6134:
6118:
6115:
6114:
6095:
6092:
6091:
6075:
6072:
6071:
6055:
6052:
6051:
6050:is a subset of
6035:
6032:
6031:
6011:
6008:
6007:
6004:
5985:
5982:
5981:
5965:
5962:
5961:
5936:
5933:
5932:
5910:
5907:
5906:
5884:
5881:
5880:
5864:
5861:
5860:
5844:
5841:
5840:
5817:
5814:
5813:
5793:
5790:
5789:
5786:
5767:
5764:
5763:
5747:
5744:
5743:
5721:
5718:
5717:
5701:
5698:
5697:
5683:
5666:
5663:
5662:
5643:
5640:
5639:
5623:
5620:
5619:
5600:
5597:
5596:
5580:
5577:
5576:
5557:
5554:
5553:
5537:
5534:
5533:
5514:
5511:
5510:
5494:
5491:
5490:
5474:
5471:
5470:
5454:
5451:
5450:
5449:By assumption,
5431:
5428:
5427:
5411:
5408:
5407:
5391:
5388:
5387:
5371:
5368:
5367:
5348:
5345:
5344:
5328:
5325:
5324:
5305:
5302:
5301:
5285:
5282:
5281:
5262:
5259:
5258:
5242:
5239:
5238:
5219:
5216:
5215:
5199:
5196:
5195:
5176:
5173:
5172:
5156:
5153:
5152:
5136:
5133:
5132:
5108:
5105:
5104:
5079:
5076:
5075:
5074:if and only if
5038:
5035:
5034:
5018:
5015:
5014:
4998:
4995:
4994:
4968:
4965:
4964:
4961:
4941:
4938:
4937:
4921:
4918:
4917:
4901:
4898:
4897:
4881:
4878:
4877:
4861:
4858:
4857:
4829:
4826:
4825:
4809:
4806:
4805:
4786:
4783:
4782:
4766:
4763:
4762:
4743:
4740:
4739:
4723:
4720:
4719:
4700:
4697:
4696:
4680:
4677:
4676:
4646:
4643:
4642:
4626:
4623:
4622:
4606:
4603:
4602:
4586:
4583:
4582:
4563:
4560:
4559:
4540:
4537:
4536:
4520:
4517:
4516:
4497:
4494:
4493:
4477:
4474:
4473:
4454:
4451:
4450:
4434:
4431:
4430:
4411:
4408:
4407:
4391:
4388:
4387:
4344:
4341:
4340:
4324:
4321:
4320:
4304:
4301:
4300:
4284:
4281:
4280:
4261:
4258:
4257:
4232:
4229:
4228:
4225:
4194:
4191:
4190:
4174:
4171:
4170:
4169:if and only if
4139:
4136:
4135:
4119:
4116:
4115:
4096:
4093:
4092:
4076:
4073:
4072:
4056:
4053:
4052:
4033:
4030:
4029:
4013:
4010:
4009:
3972:
3969:
3968:
3942:
3939:
3938:
3916:
3913:
3912:
3857:
3809:
3806:
3805:
3780:
3777:
3776:
3751:
3748:
3747:
3727:
3724:
3723:
3695:
3692:
3691:
3677:
3655:
3652:
3651:
3635:
3632:
3631:
3615:
3612:
3611:
3595:
3592:
3591:
3565:
3562:
3561:
3540:
3537:
3536:
3520:
3517:
3516:
3500:
3497:
3496:
3473:
3469:
3455:
3452:
3451:
3434:
3430:
3428:
3425:
3424:
3408:
3405:
3404:
3385:
3382:
3381:
3359:
3356:
3355:
3332:
3328:
3320:
3317:
3316:
3300:
3297:
3296:
3272:
3268:
3260:
3257:
3256:
3239:
3235:
3227:
3224:
3223:
3207:
3204:
3203:
3174:
3170:
3162:
3159:
3158:
3139:
3136:
3135:
3115:
3111:
3103:
3100:
3099:
3083:
3080:
3079:
3063:
3060:
3059:
3037:
3034:
3033:
3008:
2993:
2989:
2988:
2984:
2975:
2971:
2963:
2960:
2959:
2934:
2925:
2921:
2919:
2916:
2915:
2899:
2896:
2895:
2878:
2867:
2857:
2853:
2849:
2839:
2835:
2833:
2830:
2829:
2828:Every sequence
2826:
2790:
2787:
2786:
2766:
2763:
2762:
2728:
2725:
2724:
2707:
2703:
2695:
2692:
2691:
2665:
2661:
2659:
2656:
2655:
2639:
2636:
2635:
2619:
2616:
2615:
2596:
2593:
2592:
2586:
2585:
2576:
2575:
2551:
2548:
2547:
2531:
2528:
2527:
2495:
2492:
2491:
2448:
2445:
2444:
2413:
2410:
2409:
2393:
2390:
2389:
2358:
2354:
2352:
2349:
2348:
2332:
2329:
2328:
2327:if and only if
2311:
2307:
2305:
2302:
2301:
2285:
2282:
2281:
2253:
2250:
2249:
2223:
2219:
2217:
2214:
2213:
2196:
2192:
2184:
2181:
2180:
2161:
2152:
2148:
2146:
2143:
2142:
2126:
2123:
2122:
2106:
2103:
2102:
2076:
2072:
2070:
2067:
2066:
2050:
2042:
2039:
2038:
2019:
2016:
2015:
1999:
1996:
1995:
1975:
1964:
1954:
1950:
1946:
1936:
1932:
1930:
1927:
1926:
1921:
1920:
1913:
1912:
1889:
1886:
1885:
1866:
1863:
1862:
1859:
1847:
1824:
1821:
1820:
1814:
1813:
1796:
1793:
1792:
1773:
1770:
1769:
1747:
1744:
1743:
1723:
1720:
1719:
1703:
1700:
1699:
1678:
1675:
1674:
1652:
1649:
1648:
1629:
1626:
1625:
1605:
1602:
1601:
1580:
1577:
1576:
1570:
1569:
1552:
1549:
1548:
1529:
1526:
1525:
1509:
1506:
1505:
1502:
1479:
1476:
1475:
1455:
1452:
1451:
1429:
1426:
1425:
1393:
1390:
1389:
1369:
1366:
1365:
1343:
1340:
1339:
1311:
1308:
1307:
1287:
1283:
1281:
1278:
1277:
1258:
1255:
1254:
1238:
1235:
1234:
1218:
1215:
1214:
1192:
1189:
1188:
1165:
1161:
1159:
1156:
1155:
1138:
1135:
1134:
1111:
1108:
1107:
1106:different from
1091:
1088:
1087:
1071:
1068:
1067:
1047:
1044:
1043:
1038:
1037:
1012:
1009:
1008:
992:
989:
988:
969:
966:
965:
946:
943:
942:
923:
918:
890:
886:
875:
872:
871:
837:
832:
826:
822:
804:
800:
798:
795:
794:
752:
750:
747:
746:
718:
715:
714:
698:
695:
694:
674:
672:
669:
668:
646:
643:
642:
626:
623:
622:
621:. For example,
619:boundary points
595:
592:
591:
575:
572:
571:
555:
552:
551:
535:
532:
531:
511:
508:
507:
497:adherent points
426:
422:
420:
417:
416:
400:
397:
396:
377:
374:
373:
357:
354:
353:
337:
334:
333:
317:
314:
313:
292:
285:
281:
275:
271:
266:
263:
262:
228:
225:
224:
208:
205:
204:
188:
185:
184:
168:
165:
164:
148:
145:
144:
124:
121:
120:
104:
101:
100:
84:
81:
80:
61:
58:
57:
39:
28:
23:
22:
15:
12:
11:
5:
7043:
7033:
7032:
7027:
7022:
7005:
7004:
7002:
7001:
6991:
6990:
6989:
6984:
6979:
6964:
6954:
6944:
6932:
6921:
6918:
6917:
6915:
6914:
6909:
6904:
6899:
6894:
6889:
6883:
6881:
6875:
6874:
6872:
6871:
6866:
6861:
6859:Winding number
6856:
6851:
6845:
6843:
6839:
6838:
6836:
6835:
6830:
6825:
6820:
6815:
6810:
6805:
6800:
6799:
6798:
6793:
6791:homotopy group
6783:
6782:
6781:
6776:
6771:
6766:
6761:
6751:
6746:
6741:
6731:
6729:
6725:
6724:
6717:
6715:
6713:
6712:
6707:
6702:
6701:
6700:
6690:
6689:
6688:
6678:
6673:
6668:
6663:
6658:
6652:
6650:
6646:
6645:
6638:
6637:
6630:
6623:
6615:
6609:
6608:
6590:
6576:
6552:
6538:
6522:
6508:
6478:
6475:
6472:
6471:
6459:
6434:
6420:
6406:
6394:
6381:
6380:
6378:
6375:
6374:
6373:
6367:
6361:
6355:
6352:Isolated point
6349:
6343:
6337:
6328:
6319:
6316:Adherent point
6311:
6308:
6295:
6292:
6272:
6252:
6232:
6229:
6209:
6206:
6203:
6200:
6197:
6183:
6171:
6168:
6148:
6145:
6142:
6122:
6102:
6099:
6079:
6059:
6039:
6015:
5992:
5989:
5969:
5949:
5946:
5943:
5940:
5920:
5917:
5914:
5894:
5891:
5888:
5868:
5848:
5834:
5821:
5797:
5774:
5771:
5751:
5731:
5728:
5725:
5705:
5691:
5687:isolated point
5670:
5650:
5647:
5627:
5607:
5604:
5584:
5564:
5561:
5541:
5521:
5518:
5498:
5478:
5458:
5438:
5435:
5415:
5395:
5375:
5355:
5352:
5332:
5312:
5309:
5289:
5269:
5266:
5246:
5226:
5223:
5203:
5183:
5180:
5160:
5140:
5115:
5112:
5092:
5089:
5086:
5083:
5063:
5060:
5057:
5054:
5051:
5048:
5045:
5042:
5022:
5002:
4985:
4972:
4948:
4945:
4925:
4905:
4885:
4865:
4845:
4842:
4839:
4836:
4833:
4813:
4793:
4790:
4770:
4750:
4747:
4738:clearly meets
4727:
4707:
4704:
4684:
4662:
4659:
4656:
4653:
4650:
4630:
4610:
4590:
4570:
4567:
4547:
4544:
4524:
4504:
4501:
4481:
4461:
4458:
4438:
4418:
4415:
4395:
4381:
4374:
4360:
4357:
4354:
4351:
4348:
4328:
4308:
4288:
4268:
4265:
4245:
4242:
4239:
4236:
4213:
4210:
4207:
4204:
4201:
4198:
4178:
4158:
4155:
4152:
4149:
4146:
4143:
4123:
4103:
4100:
4080:
4060:
4040:
4037:
4017:
4003:
3991:
3988:
3985:
3982:
3979:
3976:
3952:
3949:
3946:
3926:
3923:
3920:
3898:
3895:
3892:
3889:
3886:
3883:
3880:
3877:
3874:
3871:
3868:
3865:
3855:
3852:
3849:
3846:
3843:
3840:
3837:
3834:
3831:
3828:
3825:
3822:
3819:
3816:
3813:
3793:
3790:
3787:
3784:
3764:
3761:
3758:
3755:
3744:disjoint union
3731:
3711:
3708:
3705:
3702:
3699:
3685:adherent point
3676:
3673:
3672:
3671:
3659:
3639:
3619:
3599:
3589:
3575:
3572:
3569:
3557:
3556:
3544:
3524:
3504:
3481:
3476:
3472:
3468:
3465:
3462:
3459:
3437:
3433:
3412:
3392:
3389:
3369:
3366:
3363:
3352:
3351:
3340:
3335:
3331:
3327:
3324:
3304:
3292:
3291:
3280:
3275:
3271:
3267:
3264:
3242:
3238:
3234:
3231:
3211:
3191:
3188:
3185:
3182:
3177:
3173:
3169:
3166:
3146:
3143:
3123:
3118:
3114:
3110:
3107:
3087:
3067:
3047:
3044:
3041:
3016:
3011:
3007:
3004:
3001:
2996:
2992:
2987:
2983:
2978:
2974:
2970:
2967:
2944:
2941:
2937:
2933:
2928:
2924:
2903:
2881:
2876:
2873:
2870:
2865:
2860:
2856:
2852:
2847:
2842:
2838:
2825:
2822:
2797:
2794:
2770:
2750:
2747:
2744:
2741:
2738:
2735:
2732:
2710:
2706:
2702:
2699:
2690:there is some
2679:
2676:
2673:
2668:
2664:
2643:
2623:
2611:if, for every
2600:
2561:
2558:
2555:
2535:
2511:
2508:
2505:
2502:
2499:
2479:
2476:
2473:
2470:
2467:
2464:
2461:
2458:
2455:
2452:
2431:
2417:
2397:
2366:
2361:
2357:
2336:
2314:
2310:
2289:
2257:
2237:
2234:
2231:
2226:
2222:
2199:
2195:
2191:
2188:
2179:there is some
2168:
2164:
2160:
2155:
2151:
2130:
2110:
2090:
2087:
2084:
2079:
2075:
2053:
2049:
2046:
2026:
2023:
2003:
1991:if, for every
1978:
1973:
1970:
1967:
1962:
1957:
1953:
1949:
1944:
1939:
1935:
1899:
1896:
1893:
1873:
1870:
1846:
1843:
1831:
1828:
1800:
1780:
1777:
1757:
1754:
1751:
1727:
1707:
1685:
1682:
1656:
1636:
1633:
1609:
1587:
1584:
1556:
1536:
1533:
1513:
1501:
1498:
1486:
1483:
1470:is called the
1459:
1436:
1433:
1409:
1406:
1403:
1400:
1397:
1373:
1353:
1350:
1347:
1315:
1290:
1286:
1265:
1262:
1242:
1222:
1202:
1199:
1196:
1168:
1164:
1142:
1115:
1095:
1075:
1051:
1016:
996:
976:
973:
950:
922:
919:
917:
914:
901:
898:
893:
889:
885:
882:
879:
869:
865:
846:
843:
840:
836:
829:
825:
821:
818:
815:
812:
807:
803:
755:
734:
731:
728:
725:
722:
702:
677:
656:
653:
650:
630:
612:isolated point
599:
579:
559:
539:
515:
505:
491:
467:
440:
437:
434:
429:
425:
404:
384:
381:
361:
341:
321:
295:
291:
288:
284:
278:
274:
270:
235:
232:
212:
192:
172:
152:
128:
108:
88:
65:
26:
9:
6:
4:
3:
2:
7042:
7031:
7028:
7026:
7023:
7021:
7018:
7017:
7015:
7000:
6992:
6988:
6985:
6983:
6980:
6978:
6975:
6974:
6973:
6965:
6963:
6959:
6955:
6953:
6949:
6945:
6943:
6938:
6933:
6931:
6923:
6922:
6919:
6913:
6910:
6908:
6905:
6903:
6900:
6898:
6895:
6893:
6890:
6888:
6885:
6884:
6882:
6880:
6876:
6870:
6869:Orientability
6867:
6865:
6862:
6860:
6857:
6855:
6852:
6850:
6847:
6846:
6844:
6840:
6834:
6831:
6829:
6826:
6824:
6821:
6819:
6816:
6814:
6811:
6809:
6806:
6804:
6801:
6797:
6794:
6792:
6789:
6788:
6787:
6784:
6780:
6777:
6775:
6772:
6770:
6767:
6765:
6762:
6760:
6757:
6756:
6755:
6752:
6750:
6747:
6745:
6742:
6740:
6736:
6733:
6732:
6730:
6726:
6721:
6711:
6708:
6706:
6705:Set-theoretic
6703:
6699:
6696:
6695:
6694:
6691:
6687:
6684:
6683:
6682:
6679:
6677:
6674:
6672:
6669:
6667:
6666:Combinatorial
6664:
6662:
6659:
6657:
6654:
6653:
6651:
6647:
6643:
6636:
6631:
6629:
6624:
6622:
6617:
6616:
6613:
6605:
6601:
6600:
6595:
6591:
6587:
6583:
6579:
6573:
6569:
6565:
6561:
6557:
6553:
6549:
6545:
6541:
6535:
6531:
6527:
6523:
6519:
6515:
6511:
6505:
6501:
6497:
6496:
6491:
6490:
6485:
6481:
6480:
6468:
6463:
6449:on 2021-04-21
6448:
6444:
6438:
6431:. 2021-01-13.
6430:
6424:
6417:. 2021-01-13.
6416:
6410:
6403:
6402:Bourbaki 1989
6398:
6391:
6390:Dugundji 1966
6386:
6382:
6371:
6368:
6365:
6362:
6359:
6356:
6353:
6350:
6347:
6344:
6341:
6338:
6332:
6329:
6323:
6320:
6317:
6314:
6313:
6306:
6293:
6290:
6270:
6250:
6230:
6227:
6204:
6195:
6182:
6169:
6166:
6146:
6140:
6120:
6100:
6097:
6077:
6057:
6037:
6029:
6013:
6003:
5990:
5987:
5967:
5947:
5944:
5941:
5938:
5915:
5889:
5866:
5846:
5833:
5819:
5811:
5795:
5785:
5772:
5769:
5749:
5726:
5703:
5690:
5688:
5682:
5668:
5648:
5645:
5625:
5605:
5602:
5582:
5562:
5559:
5539:
5519:
5516:
5496:
5476:
5456:
5436:
5433:
5413:
5393:
5373:
5353:
5350:
5330:
5310:
5307:
5287:
5267:
5264:
5244:
5224:
5221:
5201:
5181:
5178:
5158:
5138:
5130:
5126:
5113:
5110:
5087:
5081:
5058:
5052:
5049:
5046:
5043:
5040:
5020:
5000:
4992:
4984:
4970:
4960:
4946:
4943:
4923:
4903:
4883:
4863:
4843:
4837:
4831:
4811:
4791:
4788:
4768:
4748:
4745:
4725:
4705:
4702:
4682:
4673:
4660:
4654:
4648:
4628:
4608:
4588:
4568:
4565:
4545:
4542:
4522:
4502:
4499:
4479:
4459:
4456:
4436:
4416:
4413:
4393:
4380:
4378:
4372:
4358:
4352:
4346:
4326:
4306:
4286:
4266:
4263:
4240:
4234:
4224:
4211:
4205:
4196:
4176:
4156:
4150:
4141:
4121:
4101:
4098:
4078:
4058:
4038:
4035:
4015:
4002:
3989:
3983:
3974:
3966:
3950:
3947:
3944:
3924:
3921:
3918:
3909:
3896:
3890:
3884:
3878:
3875:
3869:
3863:
3850:
3844:
3841:
3835:
3829:
3826:
3820:
3814:
3811:
3788:
3782:
3759:
3753:
3745:
3729:
3706:
3700:
3697:
3688:
3686:
3682:
3657:
3637:
3617:
3597:
3587:
3573:
3570:
3567:
3559:
3558:
3542:
3522:
3502:
3494:
3493:
3492:
3479:
3474:
3470:
3466:
3463:
3460:
3457:
3435:
3431:
3410:
3390:
3387:
3367:
3364:
3361:
3338:
3333:
3329:
3325:
3322:
3302:
3294:
3293:
3278:
3273:
3269:
3265:
3262:
3240:
3236:
3232:
3229:
3209:
3186:
3180:
3175:
3171:
3167:
3164:
3144:
3141:
3121:
3116:
3112:
3108:
3105:
3085:
3065:
3045:
3042:
3039:
3031:
3030:
3029:
3014:
3005:
3002:
2999:
2994:
2990:
2985:
2981:
2976:
2972:
2968:
2965:
2958:
2942:
2931:
2926:
2922:
2901:
2874:
2871:
2868:
2863:
2858:
2854:
2850:
2845:
2840:
2836:
2821:
2819:
2815:
2811:
2795:
2792:
2784:
2768:
2748:
2745:
2742:
2736:
2730:
2708:
2704:
2700:
2697:
2677:
2674:
2671:
2666:
2662:
2641:
2621:
2614:
2613:neighbourhood
2598:
2591:
2590:
2581:
2580:
2577:cluster point
2559:
2556:
2553:
2533:
2525:
2506:
2503:
2500:
2477:
2474:
2465:
2462:
2459:
2453:
2450:
2442:
2438:
2433:
2429:
2415:
2395:
2387:
2382:
2380:
2364:
2359:
2355:
2334:
2312:
2308:
2287:
2279:
2275:
2271:
2255:
2235:
2232:
2229:
2224:
2220:
2197:
2193:
2189:
2186:
2166:
2158:
2153:
2149:
2128:
2108:
2088:
2085:
2082:
2077:
2073:
2047:
2044:
2024:
2021:
2001:
1994:
1993:neighbourhood
1971:
1968:
1965:
1960:
1955:
1951:
1947:
1942:
1937:
1933:
1925:
1917:
1914:cluster point
1897:
1894:
1891:
1871:
1868:
1858:
1854:
1849:
1842:
1829:
1826:
1818:
1798:
1778:
1775:
1755:
1752:
1749:
1741:
1725:
1705:
1696:
1683:
1680:
1672:
1671:
1654:
1634:
1631:
1623:
1607:
1598:
1585:
1582:
1574:
1554:
1534:
1531:
1511:
1497:
1484:
1481:
1473:
1457:
1448:
1434:
1431:
1423:
1404:
1395:
1388:of points in
1387:
1371:
1351:
1348:
1345:
1337:
1333:
1332:metric spaces
1329:
1313:
1304:
1288:
1284:
1263:
1260:
1240:
1220:
1200:
1197:
1194:
1186:
1182:
1166:
1162:
1140:
1131:
1127:
1113:
1093:
1073:
1065:
1064:neighbourhood
1049:
1042:
1034:
1033:cluster point
1030:
1014:
994:
974:
971:
964:
948:
936:
932:
927:
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887:
880:
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500:
499:(also called
498:
493:
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286:
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252:cluster point
249:
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230:
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150:
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141:neighbourhood
126:
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79:
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56:
52:
51:cluster point
48:
44:
37:
33:
19:
6999:Publications
6864:Chern number
6854:Betti number
6737: /
6728:Key concepts
6676:Differential
6597:
6559:
6529:
6493:
6488:
6467:Munkres 2000
6462:
6451:. Retrieved
6447:the original
6437:
6423:
6409:
6397:
6385:
6263:is empty or
6187:
6005:
5838:
5787:
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5128:
5127:
4990:
4989:
4962:
4896:(other than
4674:
4385:
4226:
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3690:The closure
3689:
3678:
3353:
2955:so that its
2827:
2814:limit points
2584:
2574:
2524:directed set
2434:
2383:
2270:metric space
1919:
1911:
1860:
1848:
1812:
1697:
1668:
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1568:
1503:
1449:
1305:
1185:metric space
1132:
1128:
1036:
1032:
1028:
940:
868:limit points
769:
616:
527:
494:
461:
255:
251:
50:
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42:
40:
6962:Wikiversity
6879:Key results
6188:As long as
6006:If a space
4091:other than
3804:; that is,
2430:limit point
1740:cardinality
1472:derived set
1338:are), then
1330:(which all
1183:(such as a
1029:limit point
935:real number
693:. However,
332:is a point
183:other than
99:is a point
43:limit point
7020:Limit sets
7014:Categories
6808:CW complex
6749:Continuity
6739:Closed set
6698:cohomology
6477:References
6453:2021-01-14
5214:is not in
4492:is not in
4373:definition
4227:If we use
3675:Properties
2810:Clustering
2723:such that
2654:and every
2212:such that
2141:and every
2065:such that
1851:See also:
1624:points of
916:Definition
776:closed set
501:points of
415:such that
6987:geometric
6982:algebraic
6833:Cobordism
6769:Hausdorff
6764:connected
6681:Geometric
6671:Continuum
6661:Algebraic
6604:EMS Press
6548:395340485
6486:(1989) .
6377:Citations
6199:∖
6144:∖
5942:≠
5681:is open.
5050:∪
4200:∖
4145:∖
3978:∖
3948:⊆
3922:∈
3894:∅
3876:∩
3842:∪
3815:
3722:of a set
3701:
3598:ω
3571:∈
3503:ω
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2507:≤
2472:→
2466:≤
2379:limit set
2360:∙
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2190:≥
2159:∈
2083:∈
2048:∈
1977:∞
1938:∙
1895:∈
1753:∩
1620:contains
1399:∖
1349:∈
1276:In fact,
1198:∈
1126:itself.
1062:if every
817:−
530:point of
526:contains
433:∈
290:∈
248:sequences
7025:Topology
6952:Wikibook
6930:Category
6818:Manifold
6786:Homotopy
6744:Interior
6735:Open set
6693:Homology
6642:Topology
6586:42683260
6560:Topology
6558:(2000).
6530:Topology
6528:(1966).
6518:18588129
6310:See also
6026:has the
5810:discrete
5788:A space
3911:A point
3586:that is
3560:A point
3157:we have
2441:sequence
2280:), then
1884:a point
1386:sequence
1187:), then
1130:point.
987:A point
260:sequence
6977:general
6779:uniform
6759:compact
6710:Digital
6606:, 2001
6498:].
5960:and so
5532:and so
5131:2: Let
4377:closure
3965:closure
2818:filters
860:has no
503:closure
457:filters
6972:Topics
6774:metric
6649:Fields
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6536:
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6506:
5280:Since
4916:), so
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4695:is in
4641:is in
4449:is in
3679:Every
2783:subnet
2781:has a
2490:where
1855:, and
1420:whose
484:, the
6754:Space
6492:[
6185:Proof
5836:Proof
5693:Proof
5129:Proof
4991:Proof
4987:Proof
4383:Proof
4005:Proof
3742:is a
3681:limit
2957:image
2522:is a
2272:or a
2268:is a
1791:then
1647:then
1547:then
1422:limit
1326:is a
1181:space
1153:is a
1027:is a
863:limit
772:limit
689:with
309:in a
258:of a
76:in a
53:of a
49:, or
6582:OCLC
6572:ISBN
6544:OCLC
6534:ISBN
6514:OCLC
6504:ISBN
6030:and
4621:and
4339:and
3495:Any
3202:and
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528:some
472:, a
455:and
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