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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
3896:
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477:). 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
771:. Indeed, a set is closed if and only if it contains all of its limit points, and the topological closure operation can be thought of as an operation that enriches a set by uniting it with its limit points.
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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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3891:{\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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3010:{\displaystyle \operatorname {Im} x_{\bullet }:=\left\{x_{n}:n\in \mathbb {N} \right\}}
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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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855:(i.e. does not converge), but has two accumulation points (which are considered
440:
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
19:"Limit point" redirects here. For uses where the word "point" is optional, see
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3659:
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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236:
6343: – Point of a subset S around which there are no other points of S
5375:
contains all its limit points. We shall show that the complement of
2366:
The set of all cluster points of a sequence is sometimes called the
1436:
In fact, Fréchet–Urysohn spaces are characterized by this property.
5458:
is not a limit point, and hence there exists an open neighbourhood
2421:
of a sequence as a synonym for accumulation point of the sequence.
913:
3544:
and hence also infinitely many terms in any associated sequence).
3123:
For example, if the sequence is the constant sequence with value
2397:
to which the sequence converges (that is, every neighborhood of
6603:
2876:{\displaystyle x_{\bullet }=\left(x_{n}\right)_{n=1}^{\infty }}
1973:{\displaystyle x_{\bullet }=\left(x_{n}\right)_{n=1}^{\infty }}
4972:
is closed if and only if it contains all of its limit points.
4268:
then we have the following characterization of the closure of
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synonymous with "cluster/accumulation point of a sequence".
6599:
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need not be an accumulation point of the corresponding set
702:
is a limit point (though not a boundary point) of interval
6491:. Berlin New York: Springer Science & Business Media.
579:
may have a neighbourhood not containing points other than
6324:
Pages displaying short descriptions of redirect targets
630:
is a boundary point (but not a limit point) of the set
606:
Limit points of a set should also not be confused with
484:
The limit points of a set should not be confused with
6404:"Difference between boundary point & limit point"
6349: – Point to which functions converge in analysis
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that is not open. Hence, every open neighbourhood of
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3184:{\displaystyle \operatorname {Im} x_{\bullet }=\{x\}}
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It is equivalent to say that for every neighbourhood
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This concept profitably generalizes the notion of a
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6355: – Value to which tends an infinite sequence
6059:with more than one element, then all elements of
3333:{\displaystyle \operatorname {Im} x_{\bullet }.}
3273:{\displaystyle \operatorname {Im} x_{\bullet }.}
3116:{\displaystyle \operatorname {Im} x_{\bullet }.}
3237:{\displaystyle \operatorname {Im} x_{\bullet }}
1842:Net (mathematics) § Cluster point of a net
3067:is an accumulation point of the sequence. But
2937:{\displaystyle x_{\bullet }:\mathbb {N} \to X}
842:{\displaystyle x_{n}=(-1)^{n}{\frac {n}{n+1}}}
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539:. Unlike for limit points, an adherent point
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1489:Special types of accumulation point of a set
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1292:spaces are characterized by this property.
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763:and is the underpinning of concepts such as
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235:There is also a closely related concept for
5868:is not discrete, then there is a singleton
3343:Conversely, given a countable infinite set
1800:is a specific type of limit point called a
1656:is a specific type of limit point called a
465:) by definition refers to a point that the
291:{\displaystyle (x_{n})_{n\in \mathbb {N} }}
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3304:-accumulation point of the associated set
384:there are infinitely many natural numbers
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2274:
2242:
2212:
2206:
2185:
2173:
2151:
2150:
2141:
2135:
2115:
2095:
2065:
2059:
2040:
2039:
2031:
2008:
1988:
1964:
1953:
1943:
1925:
1919:
1878:
1855:
1834:Accumulation points of sequences and nets
1813:
1785:
1762:
1736:
1712:
1692:
1667:
1641:
1618:
1594:
1569:
1541:
1518:
1498:
1468:
1444:
1418:
1382:
1358:
1332:
1300:
1276:
1270:
1247:
1227:
1207:
1181:
1154:
1148:
1127:
1100:
1080:
1060:
1036:
1001:
981:
958:
935:
879:
864:
821:
815:
793:
787:
742:
741:
739:
707:
687:
664:
663:
661:
635:
615:
584:
564:
544:
524:
500:
415:
409:
389:
366:
346:
326:
306:
282:
281:
274:
264:
255:
217:
212:does not itself have to be an element of
197:
177:
157:
137:
113:
93:
73:
50:
25:Limit (disambiguation) § Mathematics
6390:
6378:
6331: – Set of all limit points of a set
6313: – a stronger analog of limit point
5564:Since this argument holds for arbitrary
773:
108:that can be "approximated" by points of
6455:
6371:
6187:
6132:
5022:is equal to its closure if and only if
4188:
4133:
3966:
1387:
3704:{\displaystyle \operatorname {cl} (S)}
2161:{\displaystyle n_{0}\in \mathbb {N} ,}
1222:if and only if every neighbourhood of
6361: – The limit of some subsequence
4245:to denote the set of limit points of
4103:if and only if every neighborhood of
4040:if and only if every neighborhood of
3392:we can enumerate all the elements of
7:
6122:is a singleton, then every point of
4360:This fact is sometimes taken as the
918:A sequence enumerating all positive
16:Cluster point in a topological space
5751:that contains no points other than
5541:lies entirely in the complement of
5246:comprises an open neighbourhood of
1513:contains infinitely many points of
1242:contains infinitely many points of
782:, the sequence of rational numbers
495:) for which every neighbourhood of
341:such that, for every neighbourhood
3882:
3017:can be defined in the usual way.
2868:
2336:is a limit of some subsequence of
1965:
14:
6432:"Examples of Accumulation Points"
6209:is nonempty, its closure will be
4206:{\displaystyle S\setminus \{x\}.}
4151:{\displaystyle S\setminus \{x\},}
3984:{\displaystyle S\setminus \{x\}.}
3244:and not an accumulation point of
2472:{\displaystyle f:(P,\leq )\to X,}
2047:{\displaystyle n\in \mathbb {N} }
6982:
6955:
6945:
6935:
6924:
6914:
6913:
6707:
6202:{\displaystyle S\setminus \{x\}}
5415:be a point in the complement of
2535:is a topological space. A point
1911:accumulation point of a sequence
1402:{\displaystyle S\setminus \{x\}}
451:The similarly named notion of a
3851:
3845:
1075:contains at least one point of
192:itself. A limit point of a set
6478:General Topology: Chapters 1–4
5079:
5073:
5050:
5044:
4829:
4823:
4646:
4640:
4344:
4338:
4232:
4226:
3876:
3870:
3861:
3855:
3842:
3836:
3827:
3821:
3812:
3806:
3780:
3774:
3751:
3745:
3698:
3692:
2928:
2728:
2722:
2498:
2486:
2460:
2457:
2445:
812:
802:
721:
709:
271:
257:
1:
5678:is a limit point of any set.
2359:{\displaystyle x_{\bullet }.}
1028:accumulation point of the set
6434:. 2021-01-13. Archived from
6141:{\displaystyle X\setminus S}
5801:if and only if no subset of
5056:{\displaystyle S=S\cup L(S)}
4845:then every neighbourhood of
4707:then every neighbourhood of
4504:then every neighbourhood of
3952:if and only if it is in the
3945:{\displaystyle S\subseteq X}
3432:{\displaystyle x_{\bullet }}
3362:{\displaystyle A\subseteq X}
2903:is by definition just a map
2309:{\displaystyle x_{\bullet }}
910:Accumulation points of a set
894:{\displaystyle S=\{x_{n}\}.}
749:{\displaystyle \mathbb {R} }
671:{\displaystyle \mathbb {R} }
6588:Encyclopedia of Mathematics
6521:. Boston: Allyn and Bacon.
5705:is an isolated point, then
4925:is again in the closure of
3021:If there exists an element
2705:{\displaystyle p\geq p_{0}}
2672:{\displaystyle p_{0}\in P,}
2576:accumulation point of a net
2230:{\displaystyle x_{n}\in V.}
2194:{\displaystyle n\geq n_{0}}
2083:{\displaystyle x_{n}\in V.}
1804:complete accumulation point
1439:The set of limit points of
433:{\displaystyle x_{n}\in V.}
7035:
6876:Banach fixed-point theorem
5312:any open neighbourhood of
4948:This completes the proof.
2743:{\displaystyle f(p)\in V,}
2428:generalizes the idea of a
2026:there are infinitely many
1839:
1757:equals the cardinality of
1589:If every neighbourhood of
1493:If every neighbourhood of
1373:if and only if there is a
778:With respect to the usual
18:
6909:
6705:
6329:Derived set (mathematics)
6272:is the unique element of
5002:is closed if and only if
4547:and this point cannot be
4308:is equal to the union of
2504:{\displaystyle (P,\leq )}
1846:Cluster point of a filter
454:limit point of a sequence
6489:Éléments de mathématique
5942:{\displaystyle y\neq x,}
5650:Hence the complement of
5498:that does not intersect
4375:("Left subset") Suppose
3211:is an isolated point of
128:in the sense that every
6418:"What is a limit point"
5226:then the complement to
3599:-accumulation point of
3592:{\displaystyle \omega }
3504:-accumulation point of
3497:{\displaystyle \omega }
3297:{\displaystyle \omega }
2265:(or, more generally, a
1850:In a topological space
1750:{\displaystyle U\cap S}
1687:If every neighbourhood
459:limit point of a filter
6931:Mathematics portal
6831:Metrics and properties
6817:Second-countable space
6583:"Limit point of a set"
6553:Upper Saddle River, NJ
6289:
6266:
6246:
6232:It is only empty when
6226:
6203:
6165:
6142:
6116:
6096:
6073:
6053:
6033:
6009:
5986:
5963:
5943:
5914:
5888:
5862:
5842:
5815:
5791:
5768:
5745:
5731:is a neighbourhood of
5725:
5699:
5664:
5644:
5621:
5601:
5578:
5558:
5535:
5515:
5492:
5472:
5452:
5432:
5409:
5389:
5369:
5349:
5326:
5306:
5283:
5263:
5240:
5220:
5197:
5177:
5154:
5134:
5109:
5086:
5057:
5016:
4996:
4966:
4942:
4919:
4899:
4879:
4859:
4839:
4807:
4787:
4764:
4744:
4721:
4701:
4678:
4656:
4624:
4604:
4584:
4564:
4541:
4518:
4498:
4475:
4455:
4432:
4412:
4389:
4354:
4322:
4302:
4282:
4262:
4239:
4207:
4172:
4152:
4117:
4097:
4074:
4054:
4034:
4011:
3985:
3946:
3920:
3919:{\displaystyle x\in X}
3892:
3787:
3758:
3725:
3705:
3653:
3633:
3613:
3593:
3569:
3568:{\displaystyle x\in X}
3538:
3518:
3498:
3475:
3433:
3406:
3386:
3363:
3334:
3298:
3274:
3238:
3205:
3185:
3140:
3117:
3081:
3061:
3041:
3040:{\displaystyle x\in X}
3011:
2938:
2897:
2877:
2791:
2764:
2744:
2706:
2673:
2637:
2617:
2594:
2555:
2554:{\displaystyle x\in X}
2529:
2505:
2473:
2432:. A net is a function
2411:
2391:
2360:
2330:
2310:
2289:is a cluster point of
2283:
2251:
2231:
2195:
2162:
2124:
2104:
2084:
2048:
2020:
1997:
1974:
1893:
1892:{\displaystyle x\in X}
1867:
1825:
1794:
1774:
1751:
1721:
1701:
1679:
1650:
1630:
1603:
1581:
1550:
1530:
1507:
1480:
1453:
1430:
1403:
1367:
1347:
1346:{\displaystyle x\in X}
1325:first-countable spaces
1309:
1286:
1259:
1236:
1216:
1196:
1195:{\displaystyle x\in X}
1164:
1136:
1109:
1089:
1069:
1045:
1010:
990:
970:
944:
927:
901:
895:
843:
750:
728:
696:
672:
650:
624:
593:
573:
553:
533:
509:
434:
398:
378:
355:
335:
315:
292:
229:
206:
186:
166:
146:
122:
102:
82:
59:
6290:
6267:
6247:
6227:
6204:
6166:
6143:
6117:
6097:
6074:
6054:
6034:
6010:
5987:
5964:
5944:
5915:
5913:{\displaystyle \{x\}}
5889:
5887:{\displaystyle \{x\}}
5863:
5843:
5816:
5792:
5769:
5746:
5726:
5724:{\displaystyle \{x\}}
5700:
5665:
5645:
5622:
5602:
5584:in the complement of
5579:
5559:
5536:
5516:
5493:
5473:
5453:
5433:
5410:
5390:
5370:
5350:
5327:
5307:
5284:
5264:
5241:
5221:
5198:
5178:
5155:
5135:
5110:
5087:
5058:
5017:
4997:
4967:
4943:
4920:
4900:
4880:
4860:
4840:
4838:{\displaystyle L(S),}
4808:
4788:
4770:is in the closure of
4765:
4745:
4722:
4702:
4679:
4657:
4655:{\displaystyle L(S).}
4625:
4605:
4585:
4565:
4542:
4519:
4499:
4476:
4456:
4433:
4413:
4395:is in the closure of
4390:
4355:
4353:{\displaystyle L(S).}
4323:
4303:
4283:
4263:
4240:
4208:
4178:is in the closure of
4173:
4153:
4118:
4098:
4075:
4055:
4035:
4012:
3986:
3947:
3921:
3893:
3788:
3759:
3726:
3706:
3654:
3634:
3614:
3594:
3570:
3539:
3519:
3499:
3476:
3434:
3407:
3387:
3364:
3335:
3299:
3275:
3239:
3206:
3186:
3141:
3118:
3082:
3062:
3042:
3012:
2939:
2898:
2878:
2805:are also defined for
2792:
2765:
2745:
2707:
2674:
2638:
2618:
2595:
2556:
2530:
2506:
2474:
2412:
2392:
2361:
2331:
2311:
2284:
2267:Fréchet–Urysohn space
2263:first-countable space
2252:
2232:
2196:
2163:
2125:
2105:
2085:
2049:
2021:
1998:
1975:
1894:
1868:
1826:
1795:
1775:
1752:
1722:
1702:
1680:
1651:
1631:
1604:
1582:
1551:
1531:
1508:
1481:
1454:
1431:
1404:
1368:
1348:
1317:Fréchet–Urysohn space
1310:
1287:
1285:{\displaystyle T_{1}}
1260:
1237:
1217:
1197:
1165:
1163:{\displaystyle T_{1}}
1137:
1110:
1090:
1070:
1046:
1011:
991:
971:
945:
917:
896:
844:
777:
751:
729:
697:
673:
651:
649:{\displaystyle \{0\}}
625:
594:
574:
554:
534:
510:
467:sequence converges to
435:
399:
379:
356:
336:
316:
293:
230:
207:
187:
167:
147:
123:
103:
83:
60:
6886:Invariance of domain
6838:Euler characteristic
6812:Bundle (mathematics)
6276:
6256:
6236:
6213:
6181:
6152:
6148:is a limit point of
6126:
6106:
6083:
6079:are limit points of
6063:
6043:
6023:
5999:
5973:
5969:is a limit point of
5953:
5924:
5898:
5872:
5852:
5832:
5805:
5781:
5755:
5735:
5709:
5689:
5654:
5631:
5611:
5588:
5568:
5545:
5525:
5502:
5482:
5462:
5442:
5419:
5399:
5395:is an open set. Let
5379:
5359:
5336:
5316:
5293:
5289:is a limit point of
5273:
5250:
5230:
5207:
5187:
5164:
5144:
5140:be a closed set and
5124:
5096:
5085:{\displaystyle L(S)}
5067:
5026:
5006:
4986:
4956:
4929:
4909:
4889:
4869:
4865:contains a point of
4849:
4817:
4797:
4774:
4754:
4731:
4711:
4688:
4668:
4664:("Right subset") If
4634:
4614:
4594:
4590:is a limit point of
4574:
4551:
4528:
4524:contains a point of
4508:
4485:
4465:
4442:
4422:
4399:
4379:
4332:
4312:
4292:
4272:
4249:
4238:{\displaystyle L(S)}
4220:
4182:
4162:
4127:
4123:contains a point of
4107:
4084:
4064:
4060:contains a point of
4044:
4021:
4017:is a limit point of
4001:
3960:
3930:
3926:is a limit point of
3904:
3797:
3786:{\displaystyle I(S)}
3768:
3764:and isolated points
3757:{\displaystyle L(S)}
3739:
3735:of its limit points
3715:
3683:
3643:
3623:
3603:
3583:
3553:
3528:
3508:
3488:
3443:
3416:
3396:
3373:
3347:
3308:
3288:
3248:
3215:
3195:
3150:
3127:
3091:
3071:
3051:
3025:
2951:
2907:
2887:
2821:
2778:
2754:
2716:
2683:
2647:
2627:
2607:
2584:
2539:
2519:
2483:
2436:
2401:
2381:
2340:
2320:
2293:
2273:
2241:
2205:
2172:
2134:
2114:
2094:
2058:
2030:
2007:
1987:
1918:
1877:
1854:
1812:
1784:
1761:
1735:
1711:
1691:
1666:
1640:
1617:
1593:
1568:
1560:ω-accumulation point
1540:
1517:
1497:
1467:
1443:
1417:
1381:
1357:
1353:is a limit point of
1331:
1299:
1269:
1246:
1226:
1206:
1202:is a limit point of
1180:
1147:
1126:
1099:
1079:
1059:
1035:
1000:
980:
957:
934:
863:
786:
738:
706:
686:
660:
634:
614:
583:
563:
543:
523:
499:
463:limit point of a net
408:
388:
365:
345:
325:
305:
254:
216:
196:
176:
156:
152:contains a point of
136:
112:
92:
72:
49:
6896:Tychonoff's theorem
6891:Poincaré conjecture
6645:General (point-set)
6551:(Second ed.).
6381:, pp. 209–210.
6359:Subsequential limit
6353:Limit of a sequence
6347:Limit of a function
6335:Filters in topology
5821:has a limit point.
2872:
2774:which converges to
2375:limit of a sequence
1969:
926:is a cluster point.
769:topological closure
695:{\displaystyle 0.5}
471:filter converges to
469:(respectively, the
21:Limit (mathematics)
6881:De Rham cohomology
6802:Polyhedral complex
6792:Simplicial complex
6557:Prentice Hall, Inc
6484:Topologie Générale
6458:, pp. 97–102.
6311:Condensation point
6288:{\displaystyle S.}
6285:
6262:
6242:
6225:{\displaystyle X.}
6222:
6199:
6175:
6164:{\displaystyle S.}
6161:
6138:
6112:
6095:{\displaystyle S.}
6092:
6069:
6049:
6029:
6005:
5985:{\displaystyle X.}
5982:
5959:
5939:
5910:
5884:
5858:
5838:
5826:
5811:
5787:
5767:{\displaystyle x.}
5764:
5741:
5721:
5695:
5683:
5660:
5643:{\displaystyle S.}
5640:
5617:
5607:the complement of
5600:{\displaystyle S,}
5597:
5574:
5557:{\displaystyle S.}
5554:
5531:
5514:{\displaystyle S,}
5511:
5488:
5468:
5448:
5431:{\displaystyle S.}
5428:
5405:
5385:
5365:
5348:{\displaystyle S.}
5345:
5322:
5305:{\displaystyle S,}
5302:
5279:
5262:{\displaystyle x.}
5259:
5236:
5219:{\displaystyle S,}
5216:
5193:
5176:{\displaystyle S.}
5173:
5150:
5130:
5108:{\displaystyle S.}
5105:
5082:
5053:
5012:
4992:
4977:
4962:
4941:{\displaystyle S.}
4938:
4915:
4895:
4875:
4855:
4835:
4803:
4786:{\displaystyle S.}
4783:
4760:
4743:{\displaystyle S,}
4740:
4717:
4700:{\displaystyle S,}
4697:
4674:
4652:
4620:
4600:
4580:
4563:{\displaystyle x.}
4560:
4540:{\displaystyle S,}
4537:
4514:
4497:{\displaystyle S,}
4494:
4471:
4454:{\displaystyle S,}
4451:
4428:
4411:{\displaystyle S.}
4408:
4385:
4373:
4350:
4318:
4298:
4278:
4261:{\displaystyle S,}
4258:
4235:
4203:
4168:
4148:
4113:
4096:{\displaystyle x,}
4093:
4070:
4050:
4033:{\displaystyle S,}
4030:
4007:
3995:
3981:
3942:
3916:
3888:
3783:
3754:
3721:
3701:
3649:
3629:
3609:
3589:
3565:
3534:
3514:
3494:
3471:
3439:that will satisfy
3429:
3402:
3385:{\displaystyle X,}
3382:
3359:
3330:
3294:
3270:
3234:
3201:
3181:
3139:{\displaystyle x,}
3136:
3113:
3077:
3057:
3037:
3007:
2934:
2893:
2873:
2837:
2790:{\displaystyle x.}
2787:
2760:
2740:
2702:
2669:
2633:
2613:
2590:
2551:
2525:
2501:
2469:
2407:
2387:
2356:
2326:
2306:
2279:
2247:
2227:
2191:
2158:
2120:
2100:
2080:
2044:
2019:{\displaystyle x,}
2016:
1993:
1970:
1934:
1889:
1866:{\displaystyle X,}
1863:
1824:{\displaystyle S.}
1821:
1790:
1773:{\displaystyle S,}
1770:
1747:
1717:
1697:
1678:{\displaystyle S.}
1675:
1659:condensation point
1646:
1629:{\displaystyle S,}
1626:
1599:
1580:{\displaystyle S.}
1577:
1546:
1529:{\displaystyle S,}
1526:
1503:
1479:{\displaystyle S.}
1476:
1449:
1429:{\displaystyle x.}
1426:
1399:
1363:
1343:
1305:
1282:
1258:{\displaystyle S.}
1255:
1232:
1212:
1192:
1160:
1132:
1105:
1085:
1065:
1041:
1006:
986:
969:{\displaystyle X.}
966:
940:
928:
902:
891:
839:
780:Euclidean topology
746:
724:
692:
668:
646:
620:
589:
569:
549:
529:
505:
430:
394:
377:{\displaystyle x,}
374:
351:
331:
311:
288:
245:accumulation point
228:{\displaystyle S.}
225:
202:
182:
162:
142:
118:
98:
78:
55:
36:accumulation point
30:In mathematics, a
6996:
6995:
6785:fundamental group
6566:978-0-13-181629-9
6545:Munkres, James R.
6528:978-0-697-06889-7
6498:978-3-540-64241-1
6473:Bourbaki, Nicolas
6393:, pp. 68–83.
6320:Convergent filter
6265:{\displaystyle x}
6245:{\displaystyle S}
6173:
6115:{\displaystyle S}
6072:{\displaystyle X}
6052:{\displaystyle X}
6032:{\displaystyle S}
6008:{\displaystyle X}
5962:{\displaystyle x}
5920:contains a point
5861:{\displaystyle X}
5841:{\displaystyle X}
5824:
5814:{\displaystyle X}
5790:{\displaystyle X}
5744:{\displaystyle x}
5698:{\displaystyle x}
5681:
5663:{\displaystyle S}
5620:{\displaystyle S}
5577:{\displaystyle x}
5534:{\displaystyle U}
5491:{\displaystyle x}
5471:{\displaystyle U}
5451:{\displaystyle x}
5408:{\displaystyle x}
5388:{\displaystyle S}
5368:{\displaystyle S}
5325:{\displaystyle x}
5282:{\displaystyle x}
5239:{\displaystyle S}
5196:{\displaystyle x}
5160:a limit point of
5153:{\displaystyle x}
5133:{\displaystyle S}
5015:{\displaystyle S}
4995:{\displaystyle S}
4975:
4965:{\displaystyle S}
4918:{\displaystyle x}
4898:{\displaystyle x}
4878:{\displaystyle S}
4858:{\displaystyle x}
4806:{\displaystyle x}
4763:{\displaystyle x}
4720:{\displaystyle x}
4677:{\displaystyle x}
4623:{\displaystyle x}
4603:{\displaystyle S}
4583:{\displaystyle x}
4517:{\displaystyle x}
4474:{\displaystyle x}
4431:{\displaystyle x}
4388:{\displaystyle x}
4371:
4321:{\displaystyle S}
4301:{\displaystyle S}
4288:: The closure of
4281:{\displaystyle S}
4171:{\displaystyle x}
4116:{\displaystyle x}
4073:{\displaystyle S}
4053:{\displaystyle x}
4010:{\displaystyle x}
3993:
3849:
3724:{\displaystyle S}
3652:{\displaystyle A}
3632:{\displaystyle x}
3612:{\displaystyle A}
3537:{\displaystyle A}
3517:{\displaystyle A}
3405:{\displaystyle A}
3204:{\displaystyle x}
3080:{\displaystyle x}
3060:{\displaystyle x}
2896:{\displaystyle X}
2763:{\displaystyle f}
2750:equivalently, if
2636:{\displaystyle x}
2616:{\displaystyle V}
2593:{\displaystyle f}
2528:{\displaystyle X}
2424:The concept of a
2410:{\displaystyle x}
2390:{\displaystyle x}
2329:{\displaystyle x}
2282:{\displaystyle x}
2250:{\displaystyle X}
2123:{\displaystyle x}
2103:{\displaystyle V}
1996:{\displaystyle V}
1793:{\displaystyle x}
1727:is such that the
1720:{\displaystyle x}
1700:{\displaystyle U}
1649:{\displaystyle x}
1602:{\displaystyle x}
1549:{\displaystyle x}
1506:{\displaystyle x}
1452:{\displaystyle S}
1366:{\displaystyle S}
1308:{\displaystyle X}
1235:{\displaystyle x}
1215:{\displaystyle S}
1135:{\displaystyle X}
1108:{\displaystyle x}
1088:{\displaystyle S}
1068:{\displaystyle x}
1044:{\displaystyle S}
1009:{\displaystyle X}
989:{\displaystyle x}
952:topological space
950:be a subset of a
943:{\displaystyle S}
837:
680:standard topology
623:{\displaystyle 0}
592:{\displaystyle x}
572:{\displaystyle S}
552:{\displaystyle x}
532:{\displaystyle S}
508:{\displaystyle x}
457:(respectively, a
397:{\displaystyle n}
354:{\displaystyle V}
334:{\displaystyle x}
314:{\displaystyle X}
300:topological space
205:{\displaystyle S}
185:{\displaystyle x}
165:{\displaystyle S}
145:{\displaystyle x}
121:{\displaystyle S}
101:{\displaystyle x}
81:{\displaystyle X}
67:topological space
58:{\displaystyle S}
7026:
7019:General topology
6986:
6985:
6959:
6958:
6949:
6939:
6929:
6928:
6917:
6916:
6711:
6624:
6617:
6610:
6601:
6596:
6578:
6540:
6510:
6459:
6453:
6447:
6446:
6444:
6443:
6428:
6422:
6421:
6414:
6408:
6407:
6400:
6394:
6388:
6382:
6376:
6325:
6316:
6294:
6292:
6291:
6286:
6271:
6269:
6268:
6263:
6251:
6249:
6248:
6243:
6231:
6229:
6228:
6223:
6208:
6206:
6205:
6200:
6170:
6168:
6167:
6162:
6147:
6145:
6144:
6139:
6121:
6119:
6118:
6113:
6101:
6099:
6098:
6093:
6078:
6076:
6075:
6070:
6058:
6056:
6055:
6050:
6038:
6036:
6035:
6030:
6017:trivial topology
6014:
6012:
6011:
6006:
5991:
5989:
5988:
5983:
5968:
5966:
5965:
5960:
5948:
5946:
5945:
5940:
5919:
5917:
5916:
5911:
5893:
5891:
5890:
5885:
5867:
5865:
5864:
5859:
5847:
5845:
5844:
5839:
5820:
5818:
5817:
5812:
5796:
5794:
5793:
5788:
5773:
5771:
5770:
5765:
5750:
5748:
5747:
5742:
5730:
5728:
5727:
5722:
5704:
5702:
5701:
5696:
5669:
5667:
5666:
5661:
5649:
5647:
5646:
5641:
5626:
5624:
5623:
5618:
5606:
5604:
5603:
5598:
5583:
5581:
5580:
5575:
5563:
5561:
5560:
5555:
5540:
5538:
5537:
5532:
5520:
5518:
5517:
5512:
5497:
5495:
5494:
5489:
5477:
5475:
5474:
5469:
5457:
5455:
5454:
5449:
5437:
5435:
5434:
5429:
5414:
5412:
5411:
5406:
5394:
5392:
5391:
5386:
5374:
5372:
5371:
5366:
5354:
5352:
5351:
5346:
5331:
5329:
5328:
5323:
5311:
5309:
5308:
5303:
5288:
5286:
5285:
5280:
5268:
5266:
5265:
5260:
5245:
5243:
5242:
5237:
5225:
5223:
5222:
5217:
5202:
5200:
5199:
5194:
5182:
5180:
5179:
5174:
5159:
5157:
5156:
5151:
5139:
5137:
5136:
5131:
5114:
5112:
5111:
5106:
5092:is contained in
5091:
5089:
5088:
5083:
5062:
5060:
5059:
5054:
5021:
5019:
5018:
5013:
5001:
4999:
4998:
4993:
4971:
4969:
4968:
4963:
4947:
4945:
4944:
4939:
4924:
4922:
4921:
4916:
4904:
4902:
4901:
4896:
4884:
4882:
4881:
4876:
4864:
4862:
4861:
4856:
4844:
4842:
4841:
4836:
4812:
4810:
4809:
4804:
4792:
4790:
4789:
4784:
4769:
4767:
4766:
4761:
4749:
4747:
4746:
4741:
4726:
4724:
4723:
4718:
4706:
4704:
4703:
4698:
4683:
4681:
4680:
4675:
4661:
4659:
4658:
4653:
4629:
4627:
4626:
4621:
4609:
4607:
4606:
4601:
4589:
4587:
4586:
4581:
4570:In other words,
4569:
4567:
4566:
4561:
4546:
4544:
4543:
4538:
4523:
4521:
4520:
4515:
4503:
4501:
4500:
4495:
4480:
4478:
4477:
4472:
4461:we are done. If
4460:
4458:
4457:
4452:
4437:
4435:
4434:
4429:
4417:
4415:
4414:
4409:
4394:
4392:
4391:
4386:
4359:
4357:
4356:
4351:
4327:
4325:
4324:
4319:
4307:
4305:
4304:
4299:
4287:
4285:
4284:
4279:
4267:
4265:
4264:
4259:
4244:
4242:
4241:
4236:
4212:
4210:
4209:
4204:
4177:
4175:
4174:
4169:
4157:
4155:
4154:
4149:
4122:
4120:
4119:
4114:
4102:
4100:
4099:
4094:
4079:
4077:
4076:
4071:
4059:
4057:
4056:
4051:
4039:
4037:
4036:
4031:
4016:
4014:
4013:
4008:
3990:
3988:
3987:
3982:
3951:
3949:
3948:
3943:
3925:
3923:
3922:
3917:
3897:
3895:
3894:
3889:
3850:
3847:
3792:
3790:
3789:
3784:
3763:
3761:
3760:
3755:
3730:
3728:
3727:
3722:
3710:
3708:
3707:
3702:
3658:
3656:
3655:
3650:
3638:
3636:
3635:
3630:
3618:
3616:
3615:
3610:
3598:
3596:
3595:
3590:
3574:
3572:
3571:
3566:
3543:
3541:
3540:
3535:
3523:
3521:
3520:
3515:
3503:
3501:
3500:
3495:
3480:
3478:
3477:
3472:
3467:
3466:
3438:
3436:
3435:
3430:
3428:
3427:
3411:
3409:
3408:
3403:
3391:
3389:
3388:
3383:
3368:
3366:
3365:
3360:
3339:
3337:
3336:
3331:
3326:
3325:
3303:
3301:
3300:
3295:
3279:
3277:
3276:
3271:
3266:
3265:
3243:
3241:
3240:
3235:
3233:
3232:
3210:
3208:
3207:
3202:
3190:
3188:
3187:
3182:
3168:
3167:
3145:
3143:
3142:
3137:
3122:
3120:
3119:
3114:
3109:
3108:
3086:
3084:
3083:
3078:
3066:
3064:
3063:
3058:
3046:
3044:
3043:
3038:
3016:
3014:
3013:
3008:
3006:
3002:
3001:
2987:
2986:
2969:
2968:
2943:
2941:
2940:
2935:
2927:
2919:
2918:
2902:
2900:
2899:
2894:
2882:
2880:
2879:
2874:
2871:
2866:
2855:
2851:
2850:
2833:
2832:
2796:
2794:
2793:
2788:
2769:
2767:
2766:
2761:
2749:
2747:
2746:
2741:
2711:
2709:
2708:
2703:
2701:
2700:
2678:
2676:
2675:
2670:
2659:
2658:
2642:
2640:
2639:
2634:
2622:
2620:
2619:
2614:
2599:
2597:
2596:
2591:
2578:
2577:
2568:
2567:
2561:is said to be a
2560:
2558:
2557:
2552:
2534:
2532:
2531:
2526:
2510:
2508:
2507:
2502:
2478:
2476:
2475:
2470:
2416:
2414:
2413:
2408:
2396:
2394:
2393:
2388:
2377:to mean a point
2365:
2363:
2362:
2357:
2352:
2351:
2335:
2333:
2332:
2327:
2315:
2313:
2312:
2307:
2305:
2304:
2288:
2286:
2285:
2280:
2256:
2254:
2253:
2248:
2236:
2234:
2233:
2228:
2217:
2216:
2200:
2198:
2197:
2192:
2190:
2189:
2167:
2165:
2164:
2159:
2154:
2146:
2145:
2129:
2127:
2126:
2121:
2109:
2107:
2106:
2101:
2089:
2087:
2086:
2081:
2070:
2069:
2053:
2051:
2050:
2045:
2043:
2025:
2023:
2022:
2017:
2002:
2000:
1999:
1994:
1979:
1977:
1976:
1971:
1968:
1963:
1952:
1948:
1947:
1930:
1929:
1913:
1912:
1905:
1904:
1899:is said to be a
1898:
1896:
1895:
1890:
1872:
1870:
1869:
1864:
1830:
1828:
1827:
1822:
1806:
1805:
1799:
1797:
1796:
1791:
1779:
1777:
1776:
1771:
1756:
1754:
1753:
1748:
1726:
1724:
1723:
1718:
1706:
1704:
1703:
1698:
1684:
1682:
1681:
1676:
1655:
1653:
1652:
1647:
1635:
1633:
1632:
1627:
1611:uncountably many
1608:
1606:
1605:
1600:
1586:
1584:
1583:
1578:
1562:
1561:
1555:
1553:
1552:
1547:
1535:
1533:
1532:
1527:
1512:
1510:
1509:
1504:
1485:
1483:
1482:
1477:
1458:
1456:
1455:
1450:
1435:
1433:
1432:
1427:
1408:
1406:
1405:
1400:
1372:
1370:
1369:
1364:
1352:
1350:
1349:
1344:
1314:
1312:
1311:
1306:
1291:
1289:
1288:
1283:
1281:
1280:
1264:
1262:
1261:
1256:
1241:
1239:
1238:
1233:
1221:
1219:
1218:
1213:
1201:
1199:
1198:
1193:
1169:
1167:
1166:
1161:
1159:
1158:
1141:
1139:
1138:
1133:
1114:
1112:
1111:
1106:
1094:
1092:
1091:
1086:
1074:
1072:
1071:
1066:
1050:
1048:
1047:
1042:
1030:
1029:
1015:
1013:
1012:
1007:
995:
993:
992:
987:
975:
973:
972:
967:
949:
947:
946:
941:
922:. Each positive
920:rational numbers
900:
898:
897:
892:
884:
883:
848:
846:
845:
840:
838:
836:
822:
820:
819:
798:
797:
755:
753:
752:
747:
745:
733:
731:
730:
727:{\displaystyle }
725:
701:
699:
698:
693:
677:
675:
674:
669:
667:
655:
653:
652:
647:
629:
627:
626:
621:
598:
596:
595:
590:
578:
576:
575:
570:
558:
556:
555:
550:
538:
536:
535:
530:
514:
512:
511:
506:
475:net converges to
439:
437:
436:
431:
420:
419:
403:
401:
400:
395:
383:
381:
380:
375:
360:
358:
357:
352:
340:
338:
337:
332:
320:
318:
317:
312:
297:
295:
294:
289:
287:
286:
285:
269:
268:
234:
232:
231:
226:
211:
209:
208:
203:
191:
189:
188:
183:
171:
169:
168:
163:
151:
149:
148:
143:
127:
125:
124:
119:
107:
105:
104:
99:
87:
85:
84:
79:
64:
62:
61:
56:
7034:
7033:
7029:
7028:
7027:
7025:
7024:
7023:
6999:
6998:
6997:
6992:
6923:
6905:
6901:Urysohn's lemma
6862:
6826:
6712:
6703:
6675:low-dimensional
6633:
6628:
6581:
6567:
6543:
6529:
6515:Dugundji, James
6513:
6499:
6471:
6468:
6463:
6462:
6454:
6450:
6441:
6439:
6430:
6429:
6425:
6416:
6415:
6411:
6402:
6401:
6397:
6389:
6385:
6377:
6373:
6368:
6323:
6314:
6301:
6296:
6274:
6273:
6254:
6253:
6234:
6233:
6211:
6210:
6179:
6178:
6150:
6149:
6124:
6123:
6104:
6103:
6081:
6080:
6061:
6060:
6041:
6040:
6039:is a subset of
6021:
6020:
5997:
5996:
5993:
5971:
5970:
5951:
5950:
5922:
5921:
5896:
5895:
5870:
5869:
5850:
5849:
5830:
5829:
5803:
5802:
5779:
5778:
5775:
5753:
5752:
5733:
5732:
5707:
5706:
5687:
5686:
5672:
5652:
5651:
5629:
5628:
5609:
5608:
5586:
5585:
5566:
5565:
5543:
5542:
5523:
5522:
5500:
5499:
5480:
5479:
5460:
5459:
5440:
5439:
5438:By assumption,
5417:
5416:
5397:
5396:
5377:
5376:
5357:
5356:
5334:
5333:
5314:
5313:
5291:
5290:
5271:
5270:
5248:
5247:
5228:
5227:
5205:
5204:
5185:
5184:
5162:
5161:
5142:
5141:
5122:
5121:
5094:
5093:
5065:
5064:
5063:if and only if
5024:
5023:
5004:
5003:
4984:
4983:
4954:
4953:
4950:
4927:
4926:
4907:
4906:
4887:
4886:
4867:
4866:
4847:
4846:
4815:
4814:
4795:
4794:
4772:
4771:
4752:
4751:
4729:
4728:
4709:
4708:
4686:
4685:
4666:
4665:
4632:
4631:
4612:
4611:
4592:
4591:
4572:
4571:
4549:
4548:
4526:
4525:
4506:
4505:
4483:
4482:
4463:
4462:
4440:
4439:
4420:
4419:
4397:
4396:
4377:
4376:
4330:
4329:
4310:
4309:
4290:
4289:
4270:
4269:
4247:
4246:
4218:
4217:
4214:
4180:
4179:
4160:
4159:
4158:if and only if
4125:
4124:
4105:
4104:
4082:
4081:
4062:
4061:
4042:
4041:
4019:
4018:
3999:
3998:
3958:
3957:
3928:
3927:
3902:
3901:
3795:
3794:
3766:
3765:
3737:
3736:
3713:
3712:
3681:
3680:
3666:
3641:
3640:
3621:
3620:
3601:
3600:
3581:
3580:
3551:
3550:
3526:
3525:
3506:
3505:
3486:
3485:
3458:
3441:
3440:
3419:
3414:
3413:
3394:
3393:
3371:
3370:
3345:
3344:
3317:
3306:
3305:
3286:
3285:
3257:
3246:
3245:
3224:
3213:
3212:
3193:
3192:
3159:
3148:
3147:
3125:
3124:
3100:
3089:
3088:
3069:
3068:
3049:
3048:
3023:
3022:
2978:
2977:
2973:
2960:
2949:
2948:
2910:
2905:
2904:
2885:
2884:
2842:
2838:
2824:
2819:
2818:
2817:Every sequence
2815:
2776:
2775:
2752:
2751:
2714:
2713:
2692:
2681:
2680:
2650:
2645:
2644:
2625:
2624:
2605:
2604:
2582:
2581:
2575:
2574:
2565:
2564:
2537:
2536:
2517:
2516:
2481:
2480:
2434:
2433:
2399:
2398:
2379:
2378:
2343:
2338:
2337:
2318:
2317:
2316:if and only if
2296:
2291:
2290:
2271:
2270:
2239:
2238:
2208:
2203:
2202:
2181:
2170:
2169:
2137:
2132:
2131:
2112:
2111:
2092:
2091:
2061:
2056:
2055:
2028:
2027:
2005:
2004:
1985:
1984:
1939:
1935:
1921:
1916:
1915:
1910:
1909:
1902:
1901:
1875:
1874:
1852:
1851:
1848:
1836:
1810:
1809:
1803:
1802:
1782:
1781:
1759:
1758:
1733:
1732:
1709:
1708:
1689:
1688:
1664:
1663:
1638:
1637:
1615:
1614:
1591:
1590:
1566:
1565:
1559:
1558:
1538:
1537:
1515:
1514:
1495:
1494:
1491:
1465:
1464:
1441:
1440:
1415:
1414:
1379:
1378:
1355:
1354:
1329:
1328:
1297:
1296:
1272:
1267:
1266:
1244:
1243:
1224:
1223:
1204:
1203:
1178:
1177:
1150:
1145:
1144:
1124:
1123:
1097:
1096:
1095:different from
1077:
1076:
1057:
1056:
1033:
1032:
1027:
1026:
998:
997:
978:
977:
955:
954:
932:
931:
912:
907:
875:
861:
860:
826:
811:
789:
784:
783:
736:
735:
704:
703:
684:
683:
658:
657:
632:
631:
612:
611:
610:. For example,
608:boundary points
581:
580:
561:
560:
541:
540:
521:
520:
497:
496:
486:adherent points
411:
406:
405:
386:
385:
363:
362:
343:
342:
323:
322:
303:
302:
270:
260:
252:
251:
214:
213:
194:
193:
174:
173:
154:
153:
134:
133:
110:
109:
90:
89:
70:
69:
47:
46:
28:
17:
12:
11:
5:
7032:
7030:
7022:
7021:
7016:
7011:
7001:
7000:
6994:
6993:
6991:
6990:
6980:
6979:
6978:
6973:
6968:
6953:
6943:
6933:
6921:
6910:
6907:
6906:
6904:
6903:
6898:
6893:
6888:
6883:
6878:
6872:
6870:
6864:
6863:
6861:
6860:
6855:
6850:
6848:Winding number
6845:
6840:
6834:
6832:
6828:
6827:
6825:
6824:
6819:
6814:
6809:
6804:
6799:
6794:
6789:
6788:
6787:
6782:
6780:homotopy group
6772:
6771:
6770:
6765:
6760:
6755:
6750:
6740:
6735:
6730:
6720:
6718:
6714:
6713:
6706:
6704:
6702:
6701:
6696:
6691:
6690:
6689:
6679:
6678:
6677:
6667:
6662:
6657:
6652:
6647:
6641:
6639:
6635:
6634:
6629:
6627:
6626:
6619:
6612:
6604:
6598:
6597:
6579:
6565:
6541:
6527:
6511:
6497:
6467:
6464:
6461:
6460:
6448:
6423:
6409:
6395:
6383:
6370:
6369:
6367:
6364:
6363:
6362:
6356:
6350:
6344:
6341:Isolated point
6338:
6332:
6326:
6317:
6308:
6305:Adherent point
6300:
6297:
6284:
6281:
6261:
6241:
6221:
6218:
6198:
6195:
6192:
6189:
6186:
6172:
6160:
6157:
6137:
6134:
6131:
6111:
6091:
6088:
6068:
6048:
6028:
6004:
5981:
5978:
5958:
5938:
5935:
5932:
5929:
5909:
5906:
5903:
5883:
5880:
5877:
5857:
5837:
5823:
5810:
5786:
5763:
5760:
5740:
5720:
5717:
5714:
5694:
5680:
5676:isolated point
5659:
5639:
5636:
5616:
5596:
5593:
5573:
5553:
5550:
5530:
5510:
5507:
5487:
5467:
5447:
5427:
5424:
5404:
5384:
5364:
5344:
5341:
5321:
5301:
5298:
5278:
5258:
5255:
5235:
5215:
5212:
5192:
5172:
5169:
5149:
5129:
5104:
5101:
5081:
5078:
5075:
5072:
5052:
5049:
5046:
5043:
5040:
5037:
5034:
5031:
5011:
4991:
4974:
4961:
4937:
4934:
4914:
4894:
4874:
4854:
4834:
4831:
4828:
4825:
4822:
4802:
4782:
4779:
4759:
4739:
4736:
4727:clearly meets
4716:
4696:
4693:
4673:
4651:
4648:
4645:
4642:
4639:
4619:
4599:
4579:
4559:
4556:
4536:
4533:
4513:
4493:
4490:
4470:
4450:
4447:
4427:
4407:
4404:
4384:
4370:
4363:
4349:
4346:
4343:
4340:
4337:
4317:
4297:
4277:
4257:
4254:
4234:
4231:
4228:
4225:
4202:
4199:
4196:
4193:
4190:
4187:
4167:
4147:
4144:
4141:
4138:
4135:
4132:
4112:
4092:
4089:
4069:
4049:
4029:
4026:
4006:
3992:
3980:
3977:
3974:
3971:
3968:
3965:
3941:
3938:
3935:
3915:
3912:
3909:
3887:
3884:
3881:
3878:
3875:
3872:
3869:
3866:
3863:
3860:
3857:
3854:
3844:
3841:
3838:
3835:
3832:
3829:
3826:
3823:
3820:
3817:
3814:
3811:
3808:
3805:
3802:
3782:
3779:
3776:
3773:
3753:
3750:
3747:
3744:
3733:disjoint union
3720:
3700:
3697:
3694:
3691:
3688:
3674:adherent point
3665:
3662:
3661:
3660:
3648:
3628:
3608:
3588:
3578:
3564:
3561:
3558:
3546:
3545:
3533:
3513:
3493:
3470:
3465:
3461:
3457:
3454:
3451:
3448:
3426:
3422:
3401:
3381:
3378:
3358:
3355:
3352:
3341:
3340:
3329:
3324:
3320:
3316:
3313:
3293:
3281:
3280:
3269:
3264:
3260:
3256:
3253:
3231:
3227:
3223:
3220:
3200:
3180:
3177:
3174:
3171:
3166:
3162:
3158:
3155:
3135:
3132:
3112:
3107:
3103:
3099:
3096:
3076:
3056:
3036:
3033:
3030:
3005:
3000:
2996:
2993:
2990:
2985:
2981:
2976:
2972:
2967:
2963:
2959:
2956:
2933:
2930:
2926:
2922:
2917:
2913:
2892:
2870:
2865:
2862:
2859:
2854:
2849:
2845:
2841:
2836:
2831:
2827:
2814:
2811:
2786:
2783:
2759:
2739:
2736:
2733:
2730:
2727:
2724:
2721:
2699:
2695:
2691:
2688:
2679:there is some
2668:
2665:
2662:
2657:
2653:
2632:
2612:
2600:if, for every
2589:
2550:
2547:
2544:
2524:
2500:
2497:
2494:
2491:
2488:
2468:
2465:
2462:
2459:
2456:
2453:
2450:
2447:
2444:
2441:
2420:
2406:
2386:
2355:
2350:
2346:
2325:
2303:
2299:
2278:
2246:
2226:
2223:
2220:
2215:
2211:
2188:
2184:
2180:
2177:
2168:there is some
2157:
2153:
2149:
2144:
2140:
2119:
2099:
2079:
2076:
2073:
2068:
2064:
2042:
2038:
2035:
2015:
2012:
1992:
1980:if, for every
1967:
1962:
1959:
1956:
1951:
1946:
1942:
1938:
1933:
1928:
1924:
1888:
1885:
1882:
1862:
1859:
1835:
1832:
1820:
1817:
1789:
1769:
1766:
1746:
1743:
1740:
1716:
1696:
1674:
1671:
1645:
1625:
1622:
1598:
1576:
1573:
1545:
1525:
1522:
1502:
1490:
1487:
1475:
1472:
1459:is called the
1448:
1425:
1422:
1398:
1395:
1392:
1389:
1386:
1362:
1342:
1339:
1336:
1304:
1279:
1275:
1254:
1251:
1231:
1211:
1191:
1188:
1185:
1157:
1153:
1131:
1104:
1084:
1064:
1040:
1005:
985:
965:
962:
939:
911:
908:
906:
903:
890:
887:
882:
878:
874:
871:
868:
858:
854:
835:
832:
829:
825:
818:
814:
810:
807:
804:
801:
796:
792:
744:
723:
720:
717:
714:
711:
691:
666:
645:
642:
639:
619:
601:isolated point
588:
568:
548:
528:
504:
494:
480:
456:
429:
426:
423:
418:
414:
393:
373:
370:
350:
330:
310:
284:
280:
277:
273:
267:
263:
259:
224:
221:
201:
181:
161:
141:
117:
97:
77:
54:
15:
13:
10:
9:
6:
4:
3:
2:
7031:
7020:
7017:
7015:
7012:
7010:
7007:
7006:
7004:
6989:
6981:
6977:
6974:
6972:
6969:
6967:
6964:
6963:
6962:
6954:
6952:
6948:
6944:
6942:
6938:
6934:
6932:
6927:
6922:
6920:
6912:
6911:
6908:
6902:
6899:
6897:
6894:
6892:
6889:
6887:
6884:
6882:
6879:
6877:
6874:
6873:
6871:
6869:
6865:
6859:
6858:Orientability
6856:
6854:
6851:
6849:
6846:
6844:
6841:
6839:
6836:
6835:
6833:
6829:
6823:
6820:
6818:
6815:
6813:
6810:
6808:
6805:
6803:
6800:
6798:
6795:
6793:
6790:
6786:
6783:
6781:
6778:
6777:
6776:
6773:
6769:
6766:
6764:
6761:
6759:
6756:
6754:
6751:
6749:
6746:
6745:
6744:
6741:
6739:
6736:
6734:
6731:
6729:
6725:
6722:
6721:
6719:
6715:
6710:
6700:
6697:
6695:
6694:Set-theoretic
6692:
6688:
6685:
6684:
6683:
6680:
6676:
6673:
6672:
6671:
6668:
6666:
6663:
6661:
6658:
6656:
6655:Combinatorial
6653:
6651:
6648:
6646:
6643:
6642:
6640:
6636:
6632:
6625:
6620:
6618:
6613:
6611:
6606:
6605:
6602:
6594:
6590:
6589:
6584:
6580:
6576:
6572:
6568:
6562:
6558:
6554:
6550:
6546:
6542:
6538:
6534:
6530:
6524:
6520:
6516:
6512:
6508:
6504:
6500:
6494:
6490:
6486:
6485:
6480:
6479:
6474:
6470:
6469:
6465:
6457:
6452:
6449:
6438:on 2021-04-21
6437:
6433:
6427:
6424:
6420:. 2021-01-13.
6419:
6413:
6410:
6406:. 2021-01-13.
6405:
6399:
6396:
6392:
6391:Bourbaki 1989
6387:
6384:
6380:
6379:Dugundji 1966
6375:
6372:
6365:
6360:
6357:
6354:
6351:
6348:
6345:
6342:
6339:
6336:
6333:
6330:
6327:
6321:
6318:
6312:
6309:
6306:
6303:
6302:
6298:
6295:
6282:
6279:
6259:
6239:
6219:
6216:
6193:
6184:
6171:
6158:
6155:
6135:
6129:
6109:
6089:
6086:
6066:
6046:
6026:
6018:
6002:
5992:
5979:
5976:
5956:
5936:
5933:
5930:
5927:
5904:
5878:
5855:
5835:
5822:
5808:
5800:
5784:
5774:
5761:
5758:
5738:
5715:
5692:
5679:
5677:
5671:
5657:
5637:
5634:
5614:
5594:
5591:
5571:
5551:
5548:
5528:
5508:
5505:
5485:
5465:
5445:
5425:
5422:
5402:
5382:
5362:
5342:
5339:
5319:
5299:
5296:
5276:
5256:
5253:
5233:
5213:
5210:
5190:
5170:
5167:
5147:
5127:
5119:
5115:
5102:
5099:
5076:
5070:
5047:
5041:
5038:
5035:
5032:
5029:
5009:
4989:
4981:
4973:
4959:
4949:
4935:
4932:
4912:
4892:
4872:
4852:
4832:
4826:
4820:
4800:
4780:
4777:
4757:
4737:
4734:
4714:
4694:
4691:
4671:
4662:
4649:
4643:
4637:
4617:
4597:
4577:
4557:
4554:
4534:
4531:
4511:
4491:
4488:
4468:
4448:
4445:
4425:
4405:
4402:
4382:
4369:
4367:
4361:
4347:
4341:
4335:
4315:
4295:
4275:
4255:
4252:
4229:
4223:
4213:
4200:
4194:
4185:
4165:
4145:
4139:
4130:
4110:
4090:
4087:
4067:
4047:
4027:
4024:
4004:
3991:
3978:
3972:
3963:
3955:
3939:
3936:
3933:
3913:
3910:
3907:
3898:
3885:
3879:
3873:
3867:
3864:
3858:
3852:
3839:
3833:
3830:
3824:
3818:
3815:
3809:
3803:
3800:
3777:
3771:
3748:
3742:
3734:
3718:
3695:
3689:
3686:
3677:
3675:
3671:
3663:
3646:
3626:
3606:
3586:
3576:
3562:
3559:
3556:
3548:
3547:
3531:
3511:
3491:
3483:
3482:
3481:
3468:
3463:
3459:
3455:
3452:
3449:
3446:
3424:
3420:
3399:
3379:
3376:
3356:
3353:
3350:
3327:
3322:
3318:
3314:
3311:
3291:
3283:
3282:
3267:
3262:
3258:
3254:
3251:
3229:
3225:
3221:
3218:
3198:
3175:
3169:
3164:
3160:
3156:
3153:
3133:
3130:
3110:
3105:
3101:
3097:
3094:
3074:
3054:
3034:
3031:
3028:
3020:
3019:
3018:
3003:
2994:
2991:
2988:
2983:
2979:
2974:
2970:
2965:
2961:
2957:
2954:
2947:
2931:
2920:
2915:
2911:
2890:
2863:
2860:
2857:
2852:
2847:
2843:
2839:
2834:
2829:
2825:
2812:
2810:
2808:
2804:
2800:
2784:
2781:
2773:
2757:
2737:
2734:
2731:
2725:
2719:
2697:
2693:
2689:
2686:
2666:
2663:
2660:
2655:
2651:
2630:
2610:
2603:
2602:neighbourhood
2587:
2580:
2579:
2570:
2569:
2566:cluster point
2548:
2545:
2542:
2522:
2514:
2495:
2492:
2489:
2466:
2463:
2454:
2451:
2448:
2442:
2439:
2431:
2427:
2422:
2418:
2404:
2384:
2376:
2371:
2369:
2353:
2348:
2344:
2323:
2301:
2297:
2276:
2268:
2264:
2260:
2244:
2224:
2221:
2218:
2213:
2209:
2186:
2182:
2178:
2175:
2155:
2147:
2142:
2138:
2117:
2097:
2077:
2074:
2071:
2066:
2062:
2036:
2033:
2013:
2010:
1990:
1983:
1982:neighbourhood
1960:
1957:
1954:
1949:
1944:
1940:
1936:
1931:
1926:
1922:
1914:
1906:
1903:cluster point
1886:
1883:
1880:
1860:
1857:
1847:
1843:
1838:
1833:
1831:
1818:
1815:
1807:
1787:
1767:
1764:
1744:
1741:
1738:
1730:
1714:
1694:
1685:
1672:
1669:
1661:
1660:
1643:
1623:
1620:
1612:
1596:
1587:
1574:
1571:
1563:
1543:
1523:
1520:
1500:
1488:
1486:
1473:
1470:
1462:
1446:
1437:
1423:
1420:
1412:
1393:
1384:
1377:of points in
1376:
1360:
1340:
1337:
1334:
1326:
1322:
1321:metric spaces
1318:
1302:
1293:
1277:
1273:
1252:
1249:
1229:
1209:
1189:
1186:
1183:
1175:
1171:
1155:
1151:
1129:
1120:
1116:
1102:
1082:
1062:
1054:
1053:neighbourhood
1038:
1031:
1023:
1022:cluster point
1019:
1003:
983:
963:
960:
953:
937:
925:
921:
916:
909:
904:
888:
880:
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681:
640:
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602:
586:
566:
546:
526:
518:
502:
493:
489:
488:(also called
487:
482:
478:
476:
472:
468:
464:
460:
455:
452:
449:
447:
443:
427:
424:
421:
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412:
391:
371:
368:
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328:
308:
301:
278:
275:
265:
261:
250:
246:
242:
241:cluster point
238:
222:
219:
199:
179:
159:
139:
131:
130:neighbourhood
115:
95:
75:
68:
52:
45:
41:
40:cluster point
37:
33:
26:
22:
6988:Publications
6853:Chern number
6843:Betti number
6726: /
6717:Key concepts
6665:Differential
6586:
6548:
6518:
6482:
6477:
6456:Munkres 2000
6451:
6440:. Retrieved
6436:the original
6426:
6412:
6398:
6386:
6374:
6252:is empty or
6176:
5994:
5827:
5776:
5684:
5673:
5117:
5116:
4979:
4978:
4951:
4885:(other than
4663:
4374:
4215:
3996:
3899:
3679:The closure
3678:
3667:
3342:
2944:so that its
2816:
2803:limit points
2573:
2563:
2513:directed set
2423:
2372:
2259:metric space
1908:
1900:
1849:
1837:
1801:
1686:
1657:
1588:
1557:
1492:
1438:
1294:
1174:metric space
1121:
1117:
1025:
1021:
1017:
929:
857:limit points
758:
605:
516:
483:
450:
244:
240:
39:
35:
31:
29:
6951:Wikiversity
6868:Key results
6177:As long as
5995:If a space
4080:other than
3793:; that is,
2419:limit point
1729:cardinality
1461:derived set
1327:are), then
1319:(which all
1172:(such as a
1018:limit point
924:real number
682:. However,
321:is a point
172:other than
88:is a point
32:limit point
7009:Limit sets
7003:Categories
6797:CW complex
6738:Continuity
6728:Closed set
6687:cohomology
6466:References
6442:2021-01-14
5203:is not in
4481:is not in
4362:definition
4216:If we use
3664:Properties
2799:Clustering
2712:such that
2643:and every
2201:such that
2130:and every
2054:such that
1840:See also:
1613:points of
905:Definition
765:closed set
490:points of
404:such that
6976:geometric
6971:algebraic
6822:Cobordism
6758:Hausdorff
6753:connected
6670:Geometric
6660:Continuum
6650:Algebraic
6593:EMS Press
6537:395340485
6475:(1989) .
6366:Citations
6188:∖
6133:∖
5931:≠
5670:is open.
5039:∪
4189:∖
4134:∖
3967:∖
3937:⊆
3911:∈
3883:∅
3865:∩
3831:∪
3804:
3711:of a set
3690:
3587:ω
3560:∈
3492:ω
3464:∙
3456:
3425:∙
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3323:∙
3315:
3292:ω
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3032:∈
2995:∈
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2929:→
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2869:∞
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2732:∈
2690:≥
2661:∈
2546:∈
2496:≤
2461:→
2455:≤
2368:limit set
2349:∙
2302:∙
2219:∈
2179:≥
2148:∈
2072:∈
2037:∈
1966:∞
1927:∙
1884:∈
1742:∩
1609:contains
1388:∖
1338:∈
1265:In fact,
1187:∈
1115:itself.
1051:if every
806:−
519:point of
515:contains
422:∈
279:∈
237:sequences
7014:Topology
6941:Wikibook
6919:Category
6807:Manifold
6775:Homotopy
6733:Interior
6724:Open set
6682:Homology
6631:Topology
6575:42683260
6549:Topology
6547:(2000).
6519:Topology
6517:(1966).
6507:18588129
6299:See also
6015:has the
5799:discrete
5777:A space
3900:A point
3575:that is
3549:A point
3146:we have
2430:sequence
2269:), then
1873:a point
1375:sequence
1176:), then
1119:point.
976:A point
249:sequence
6966:general
6768:uniform
6748:compact
6699:Digital
6595:, 2001
6487:].
5949:and so
5521:and so
5120:2: Let
4366:closure
3954:closure
2807:filters
849:has no
492:closure
446:filters
6961:Topics
6763:metric
6638:Fields
6573:
6563:
6535:
6525:
6505:
6495:
5269:Since
4905:), so
4813:is in
4684:is in
4630:is in
4438:is in
3668:Every
2772:subnet
2770:has a
2479:where
1844:, and
1409:whose
473:, the
6743:Space
6481:[
6174:Proof
5825:Proof
5682:Proof
5118:Proof
4980:Proof
4976:Proof
4372:Proof
3994:Proof
3731:is a
3670:limit
2946:image
2511:is a
2261:or a
2257:is a
1780:then
1636:then
1536:then
1411:limit
1315:is a
1170:space
1142:is a
1016:is a
852:limit
761:limit
678:with
298:in a
247:of a
65:in a
42:of a
38:, or
6571:OCLC
6561:ISBN
6533:OCLC
6523:ISBN
6503:OCLC
6493:ISBN
6019:and
4610:and
4328:and
3484:Any
3191:and
2801:and
2515:and
1323:and
930:Let
767:and
517:some
461:, a
444:and
442:nets
239:. A
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6102:If
5828:If
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2110:of
2003:of
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1024:or
1020:or
996:in
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690:0.5
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6569:.
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