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2657:. In other words, if you are "outside" a closed set, you may move a small amount in any direction and still stay outside the set. This is also true if the boundary is the empty set, e.g. in the metric space of rational numbers, for the set of numbers of which the square is less than
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Hausdorff space into a compact
Hausdorff space, may be described as adjoining limits of certain nonconvergent nets to the space.
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Furthermore, every closed subset of a compact space is compact, and every compact subspace of a
Hausdorff space is closed.
4164:
682:, which are more general than topological spaces. Notice that this characterization also depends on the surrounding space
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678:, instead of all nets. One value of this characterization is that it may be used as a definition in the context of
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is an unusual closed set in the sense that it consists entirely of boundary points and is nowhere dense.
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of any family of closed sets is closed (this includes intersections of infinitely many closed sets)
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310:. Yet another equivalent definition is that a set is closed if and only if it contains all of its
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2006:{\displaystyle f\left(\operatorname {cl} _{X}A\right)\subseteq \operatorname {cl} _{Y}(f(A))}
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3497: – A function that sends open (resp. closed) subsets to open (resp. closed) subsets
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Closed sets also give a useful characterization of compactness: a topological space
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Whether a set is closed depends on the space in which it is embedded. However, the
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with empty intersection admits a finite subcollection with empty intersection.
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a subset is closed if and only if it contains every point that is close to it.
3852:
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17:
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2702:
1125:
this terminology allows for a plain
English description of closed subsets:
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3512:
3357:
3342:
3013:
can be constructed as the intersection of all of these closed supersets.
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is compact if and only if every collection of nonempty closed subsets of
2334:, as well as for other spaces that carry topological structures, such as
2327:
675:
556:
70:
50:
46:
31:
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have the properties listed above, then there exists a unique topology
306:
Equivalently, a set is closed if and only if it contains all of its
674:(such as a metric space), it is enough to consider only convergent
3248:
Some sets are neither open nor closed, for instance the half-open
3083:
for an explanation of the bracket and parenthesis set notation.)
555:
An alternative characterization of closed sets is available via
3900:
1848:
for some (or equivalently, for every) topological super-space
77:, a closed set can be defined as a set which contains all its
2364:", in the sense that, if you embed a compact Hausdorff space
3381:
is an infinite and unbounded closed set in the real numbers.
3205:(inclusive) is closed in the space of rational numbers, but
3896:
3121:
is closed in the metric space of real numbers, and the set
3436:
is continuous if and only if preimages of closed sets in
2899:
The intersection property also allows one to define the
3529:
Pages displaying short descriptions of redirect targets
3508:
Pages displaying short descriptions of redirect targets
3499:
Pages displaying short descriptions of redirect targets
979:{\displaystyle x\in \operatorname {cl} _{A\cup \{x\}}A}
3341:
Singleton points (and thus finite sets) are closed in
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The notion of closed set is defined above in terms of
705:
because whether or not a sequence or net converges in
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3527: – Connected open subset of a topological space
3506: – Connected open subset of a topological space
464:{\displaystyle A\subseteq \operatorname {cl} _{X}A.}
4163:
4127:
4013:
3934:
3777:. New Jersey: World Scientific Publishing Company.
3707:
3674:
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3286:Some sets are both open and closed and are called
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2946:which is defined as the smallest closed subset of
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1436:be closed in the "larger" surrounding super-space
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1179:if and only if there exists some net (valued) in
2528:if there exist disjoint, nonempty, open subsets
2447:; the "surrounding space" does not matter here.
1841:{\displaystyle A=X\cap \operatorname {cl} _{Y}A}
93:operation. This should not be confused with a
3416:is a function between topological spaces then
2063:is continuous if and only if for every subset
3912:
3627:and not on the whole surrounding space (e.g.
3016:Sets that can be constructed as the union of
2753:{\displaystyle \mathbb {F} \neq \varnothing }
1893:Closed sets can also be used to characterize
850:{\displaystyle x\in \operatorname {cl} _{X}A}
8:
3669:
3663:
3614:
3608:
3491: – Subset which is both open and closed
1386:), which is how it is possible for a subset
1005:
999:
965:
959:
919:
913:
3515: – Basic subset of a topological space
1549:is always a (potentially proper) subset of
546:{\displaystyle A=\operatorname {cl} _{X}A.}
34:. For a set closed under an operation, see
30:This article is about the complement of an
4280:
4253:
3919:
3905:
3897:
1700:{\displaystyle A=\operatorname {cl} _{X}A}
3655:
3632:
3600:
3577:
3557:
3521: – Open set containing a given point
3461:
3441:
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3389:
3367:
3366:
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3301:
3256:
3231:
3230:
3210:
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3170:
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1976:
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1756:{\displaystyle \operatorname {cl} _{Y}A.}
1738:
1732:
1712:
1707:), it is nevertheless still possible for
1685:
1673:
1653:
1633:
1610:
1590:
1578:{\displaystyle \operatorname {cl} _{Y}A,}
1560:
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396:{\displaystyle \operatorname {cl} _{X}A;}
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3835:Handbook of Analysis and Its Foundations
3695:
3545:
2747:
1330:
234:
176:
3029:sets. These sets need not be closed.
2874:are exactly those sets that belong to
725:depends on what points are present in
252:{\displaystyle X\setminus A\in \tau .}
2183:is continuous at a fixed given point
1133:In terms of net convergence, a point
7:
3775:Convergence Foundations Of Topology
3710:Principles of Mathematical Analysis
3315:
3245:is not closed in the real numbers.
2427:will always be a closed subset of
279:if and only if it is equal to its
25:
3837:. San Diego, CA: Academic Press.
3238:{\displaystyle \cap \mathbb {Q} }
3154:{\displaystyle \cap \mathbb {Q} }
2330:, a concept that makes sense for
1082:is thus the set of all points in
85:, a closed set is a set which is
4279:
4252:
4242:
4232:
4221:
4211:
4210:
4004:
2842:such that the closed subsets of
2384:in an arbitrary Hausdorff space
1246:of some other topological space
3650:or any other space containing
3552:In particular, whether or not
3400:
3318:
3303:
3270:
3258:
3224:
3212:
3140:
3128:
3108:
3096:
3060:
3048:
2861:
2849:
2653:A closed set contains its own
2303:
2297:
2274:
2268:
2147:
2141:
2111:maps points that are close to
2000:
1997:
1991:
1985:
1913:
1668:(which happens if and only if
211:
199:
147:
135:
1:
3020:many closed sets are denoted
2993:Specifically, the closure of
2892:{\displaystyle \mathbb {F} .}
2251:{\displaystyle A\subseteq X,}
2085:{\displaystyle A\subseteq X,}
1585:which denotes the closure of
3374:{\displaystyle \mathbb {Z} }
3324:{\displaystyle [1,+\infty )}
2795:{\displaystyle \mathbb {F} }
2131:to points that are close to
2032:{\displaystyle A\subseteq X}
1478:{\displaystyle A\subseteq X}
1405:{\displaystyle A\subseteq X}
1366:(although not an element of
1339:{\displaystyle Y\setminus X}
1042:). Because the closure of
928:{\displaystyle A\cup \{x\},}
811:{\displaystyle A\subseteq X}
627:of every net of elements of
422:{\displaystyle A\subseteq X}
333:{\displaystyle A\subseteq X}
185:{\displaystyle X\setminus A}
27:Complement of an open subset
3807:. Boston: Allyn and Bacon.
3773:; Mynard, Frédéric (2016).
3682:as a topological subspace).
3675:{\displaystyle A\cup \{x\}}
3620:{\displaystyle A\cup \{x\}}
2639:consisting of closed sets.
2449:Stone–Čech compactification
1509:topological super-space of
1011:{\displaystyle A\cup \{x\}}
340:is always contained in its
4322:
4173:Banach fixed-point theorem
3860:Willard, Stephen (2004) .
2780:such that the elements of
2646:
2039:; this can be reworded in
877:belongs to the closure of
53:, and related branches of
29:
4206:
4002:
2867:{\displaystyle (X,\tau )}
2649:Kuratowski closure axioms
2451:, a process that turns a
1727:to be a proper subset of
217:{\displaystyle (X,\tau )}
153:{\displaystyle (X,\tau )}
3409:{\displaystyle f:X\to Y}
2712:In fact, if given a set
2708:The whole set is closed.
2340:differentiable manifolds
2209:if and only if whenever
1922:{\displaystyle f:X\to Y}
105:By definition, a subset
1291:topological super-space
583:of a topological space
40:Closed (disambiguation)
4228:Mathematics portal
4128:Metrics and properties
4114:Second-countable space
3676:
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3277:
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3080:Interval (mathematics)
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2987:
2960:
2940:
2917:
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2836:
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2796:
2774:
2754:
2726:
2698:closed sets is closed.
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2605:
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2475:
2441:
2421:
2401:
2378:
2322:More about closed sets
2313:
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2203:
2202:{\displaystyle x\in X}
2177:
2157:
2125:
2105:
2086:
2057:
2033:
2007:
1923:
1885:
1862:
1842:
1797:
1783:is a closed subset of
1777:
1757:
1721:
1701:
1662:
1648:is a closed subset of
1642:
1622:
1599:
1579:
1543:
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1406:
1380:
1360:
1340:
1310:
1283:
1263:
1236:
1216:
1193:
1173:
1153:
1152:{\displaystyle x\in X}
1119:
1096:
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1012:
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871:
851:
812:
782:
762:
742:
719:
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664:
641:
617:
597:
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505:
491:is a closed subset of
485:
465:
423:
397:
361:
334:
300:
273:
253:
218:
186:
154:
119:
101:Equivalent definitions
38:. For other uses, see
3677:
3645:
3622:
3587:
3567:
3474:
3451:
3431:
3411:
3376:
3326:
3278:
3276:{\displaystyle [0,1)}
3240:
3200:
3180:
3156:
3116:
3068:
3008:
2988:
2961:
2941:
2918:
2894:
2869:
2837:
2817:
2815:{\displaystyle \tau }
2797:
2775:
2755:
2727:
2672:
2626:
2606:
2583:
2563:
2543:
2519:
2496:
2476:
2442:
2422:
2402:
2379:
2314:
2312:{\displaystyle f(A).}
2282:
2253:
2229:is close to a subset
2224:
2204:
2178:
2158:
2156:{\displaystyle f(A).}
2126:
2106:
2087:
2058:
2034:
2008:
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1407:
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1284:
1264:
1237:
1217:
1194:
1174:
1159:is close to a subset
1154:
1120:
1097:
1077:
1057:
1037:
1013:
981:
930:
892:
872:
857:(or equivalently, if
852:
813:
783:
763:
743:
720:
700:
672:first-countable space
665:
642:
623:if and only if every
618:
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578:
548:
506:
486:
466:
424:
398:
362:
342:(topological) closure
335:
301:
274:
254:
219:
192:is an open subset of
187:
155:
120:
83:complete metric space
36:closure (mathematics)
4183:Invariance of domain
4135:Euler characteristic
4109:Bundle (mathematics)
3654:
3631:
3599:
3592:depends only on the
3576:
3556:
3525:Region (mathematics)
3460:
3440:
3420:
3388:
3363:
3300:
3283:in the real numbers.
3255:
3209:
3189:
3169:
3125:
3093:
3045:
2997:
2974:
2950:
2927:
2907:
2878:
2846:
2826:
2806:
2784:
2764:
2736:
2716:
2661:
2633:totally disconnected
2615:
2592:
2572:
2552:
2532:
2508:
2504:A topological space
2485:
2465:
2431:
2411:
2388:
2368:
2291:
2280:{\displaystyle f(x)}
2262:
2233:
2213:
2187:
2167:
2135:
2115:
2095:
2067:
2047:
2017:
1937:
1901:
1895:continuous functions
1872:
1852:
1807:
1787:
1767:
1731:
1711:
1672:
1652:
1632:
1609:
1589:
1553:
1533:
1513:
1489:
1463:
1440:
1416:
1390:
1370:
1350:
1324:
1320:exist some point in
1297:
1273:
1250:
1244:topological subspace
1226:
1203:
1183:
1163:
1137:
1106:
1086:
1066:
1046:
1026:
1018:is endowed with the
990:
939:
904:
899:topological subspace
881:
861:
822:
796:
772:
752:
729:
709:
686:
651:
631:
607:
587:
567:
515:
495:
475:
433:
407:
371:
367:which is denoted by
348:
318:
287:
263:
228:
196:
170:
132:
109:
4193:Tychonoff's theorem
4188:Poincaré conjecture
3942:General (point-set)
259:A set is closed in
4178:De Rham cohomology
4099:Polyhedral complex
4089:Simplicial complex
3872:Dover Publications
3672:
3643:{\displaystyle X,}
3640:
3617:
3582:
3562:
3534:Regular closed set
3472:{\displaystyle X.}
3469:
3446:
3426:
3406:
3371:
3321:
3273:
3235:
3195:
3175:
3151:
3111:
3063:
3003:
2986:{\displaystyle A.}
2983:
2956:
2939:{\displaystyle X,}
2936:
2913:
2889:
2864:
2832:
2812:
2792:
2770:
2750:
2722:
2670:{\displaystyle 2.}
2667:
2621:
2604:{\displaystyle X.}
2601:
2578:
2558:
2538:
2514:
2491:
2471:
2453:completely regular
2437:
2417:
2400:{\displaystyle X,}
2397:
2374:
2332:topological spaces
2309:
2277:
2248:
2219:
2199:
2173:
2153:
2121:
2101:
2082:
2053:
2029:
2003:
1919:
1884:{\displaystyle X.}
1881:
1858:
1838:
1793:
1773:
1753:
1717:
1697:
1658:
1638:
1621:{\displaystyle Y;}
1618:
1595:
1575:
1539:
1519:
1495:
1475:
1452:{\displaystyle Y.}
1449:
1422:
1402:
1376:
1356:
1336:
1309:{\displaystyle X,}
1306:
1279:
1262:{\displaystyle Y,}
1259:
1232:
1215:{\displaystyle x.}
1212:
1199:that converges to
1189:
1169:
1149:
1118:{\displaystyle A,}
1115:
1102:that are close to
1092:
1072:
1052:
1032:
1008:
976:
925:
887:
867:
847:
808:
778:
758:
741:{\displaystyle X.}
738:
715:
698:{\displaystyle X,}
695:
680:convergence spaces
663:{\displaystyle A.}
660:
637:
613:
593:
573:
543:
501:
481:
461:
419:
393:
360:{\displaystyle X,}
357:
330:
299:{\displaystyle X.}
296:
269:
249:
214:
182:
166:if its complement
150:
115:
4293:
4292:
4082:fundamental group
3881:978-0-486-43479-7
3844:978-0-12-622760-4
3814:978-0-697-06889-7
3784:978-981-4571-52-4
3740:Munkres, James R.
3585:{\displaystyle A}
3565:{\displaystyle x}
3449:{\displaystyle Y}
3429:{\displaystyle f}
3198:{\displaystyle 1}
3178:{\displaystyle 0}
3006:{\displaystyle X}
2959:{\displaystyle X}
2916:{\displaystyle A}
2835:{\displaystyle X}
2773:{\displaystyle X}
2732:and a collection
2725:{\displaystyle X}
2624:{\displaystyle X}
2581:{\displaystyle X}
2561:{\displaystyle B}
2541:{\displaystyle A}
2517:{\displaystyle X}
2494:{\displaystyle X}
2474:{\displaystyle X}
2440:{\displaystyle X}
2420:{\displaystyle D}
2377:{\displaystyle D}
2362:absolutely closed
2222:{\displaystyle x}
2176:{\displaystyle f}
2124:{\displaystyle A}
2104:{\displaystyle f}
2056:{\displaystyle f}
2013:for every subset
1861:{\displaystyle Y}
1796:{\displaystyle X}
1776:{\displaystyle A}
1720:{\displaystyle A}
1661:{\displaystyle X}
1641:{\displaystyle A}
1598:{\displaystyle A}
1542:{\displaystyle A}
1522:{\displaystyle X}
1498:{\displaystyle Y}
1425:{\displaystyle X}
1379:{\displaystyle X}
1359:{\displaystyle A}
1346:that is close to
1282:{\displaystyle Y}
1235:{\displaystyle X}
1192:{\displaystyle A}
1172:{\displaystyle A}
1095:{\displaystyle X}
1075:{\displaystyle X}
1055:{\displaystyle A}
1035:{\displaystyle X}
1022:induced on it by
1020:subspace topology
890:{\displaystyle A}
870:{\displaystyle x}
781:{\displaystyle X}
761:{\displaystyle x}
718:{\displaystyle X}
640:{\displaystyle A}
616:{\displaystyle X}
596:{\displaystyle X}
576:{\displaystyle A}
504:{\displaystyle X}
484:{\displaystyle A}
272:{\displaystyle X}
127:topological space
118:{\displaystyle A}
75:topological space
16:(Redirected from
4313:
4306:General topology
4283:
4282:
4256:
4255:
4246:
4236:
4226:
4225:
4214:
4213:
4008:
3921:
3914:
3907:
3898:
3893:
3863:General Topology
3856:
3826:
3796:
3762:
3761:
3746:(2nd ed.).
3736:
3730:
3729:
3713:
3700:
3683:
3681:
3679:
3678:
3673:
3649:
3647:
3646:
3641:
3626:
3624:
3623:
3618:
3591:
3589:
3588:
3583:
3571:
3569:
3568:
3563:
3550:
3530:
3509:
3500:
3478:
3476:
3475:
3470:
3455:
3453:
3452:
3447:
3435:
3433:
3432:
3427:
3415:
3413:
3412:
3407:
3380:
3378:
3377:
3372:
3370:
3351:Hausdorff spaces
3330:
3328:
3327:
3322:
3282:
3280:
3279:
3274:
3244:
3242:
3241:
3236:
3234:
3204:
3202:
3201:
3196:
3184:
3182:
3181:
3176:
3163:rational numbers
3160:
3158:
3157:
3152:
3150:
3120:
3118:
3117:
3114:{\displaystyle }
3112:
3077:is closed. (See
3072:
3070:
3069:
3066:{\displaystyle }
3064:
3012:
3010:
3009:
3004:
2992:
2990:
2989:
2984:
2965:
2963:
2962:
2957:
2945:
2943:
2942:
2937:
2922:
2920:
2919:
2914:
2898:
2896:
2895:
2890:
2885:
2873:
2871:
2870:
2865:
2841:
2839:
2838:
2833:
2821:
2819:
2818:
2813:
2801:
2799:
2798:
2793:
2791:
2779:
2777:
2776:
2771:
2759:
2757:
2756:
2751:
2743:
2731:
2729:
2728:
2723:
2676:
2674:
2673:
2668:
2630:
2628:
2627:
2622:
2610:
2608:
2607:
2602:
2587:
2585:
2584:
2579:
2567:
2565:
2564:
2559:
2547:
2545:
2544:
2539:
2523:
2521:
2520:
2515:
2500:
2498:
2497:
2492:
2480:
2478:
2477:
2472:
2446:
2444:
2443:
2438:
2426:
2424:
2423:
2418:
2406:
2404:
2403:
2398:
2383:
2381:
2380:
2375:
2358:Hausdorff spaces
2318:
2316:
2315:
2310:
2286:
2284:
2283:
2278:
2257:
2255:
2254:
2249:
2228:
2226:
2225:
2220:
2208:
2206:
2205:
2200:
2182:
2180:
2179:
2174:
2162:
2160:
2159:
2154:
2130:
2128:
2127:
2122:
2110:
2108:
2107:
2102:
2091:
2089:
2088:
2083:
2062:
2060:
2059:
2054:
2038:
2036:
2035:
2030:
2012:
2010:
2009:
2004:
1981:
1980:
1968:
1964:
1957:
1956:
1928:
1926:
1925:
1920:
1890:
1888:
1887:
1882:
1867:
1865:
1864:
1859:
1847:
1845:
1844:
1839:
1831:
1830:
1802:
1800:
1799:
1794:
1782:
1780:
1779:
1774:
1762:
1760:
1759:
1754:
1743:
1742:
1726:
1724:
1723:
1718:
1706:
1704:
1703:
1698:
1690:
1689:
1667:
1665:
1664:
1659:
1647:
1645:
1644:
1639:
1628:indeed, even if
1627:
1625:
1624:
1619:
1604:
1602:
1601:
1596:
1584:
1582:
1581:
1576:
1565:
1564:
1548:
1546:
1545:
1540:
1528:
1526:
1525:
1520:
1504:
1502:
1501:
1496:
1484:
1482:
1481:
1476:
1458:
1456:
1455:
1450:
1431:
1429:
1428:
1423:
1412:to be closed in
1411:
1409:
1408:
1403:
1385:
1383:
1382:
1377:
1365:
1363:
1362:
1357:
1345:
1343:
1342:
1337:
1315:
1313:
1312:
1307:
1288:
1286:
1285:
1280:
1268:
1266:
1265:
1260:
1241:
1239:
1238:
1233:
1221:
1219:
1218:
1213:
1198:
1196:
1195:
1190:
1178:
1176:
1175:
1170:
1158:
1156:
1155:
1150:
1124:
1122:
1121:
1116:
1101:
1099:
1098:
1093:
1081:
1079:
1078:
1073:
1061:
1059:
1058:
1053:
1041:
1039:
1038:
1033:
1017:
1015:
1014:
1009:
985:
983:
982:
977:
969:
968:
934:
932:
931:
926:
896:
894:
893:
888:
876:
874:
873:
868:
856:
854:
853:
848:
840:
839:
817:
815:
814:
809:
787:
785:
784:
779:
767:
765:
764:
759:
747:
745:
744:
739:
724:
722:
721:
716:
704:
702:
701:
696:
669:
667:
666:
661:
647:also belongs to
646:
644:
643:
638:
622:
620:
619:
614:
602:
600:
599:
594:
582:
580:
579:
574:
552:
550:
549:
544:
533:
532:
510:
508:
507:
502:
490:
488:
487:
482:
470:
468:
467:
462:
451:
450:
428:
426:
425:
420:
402:
400:
399:
394:
383:
382:
366:
364:
363:
358:
339:
337:
336:
331:
314:. Every subset
305:
303:
302:
297:
278:
276:
275:
270:
258:
256:
255:
250:
223:
221:
220:
215:
191:
189:
188:
183:
159:
157:
156:
151:
124:
122:
121:
116:
21:
4321:
4320:
4316:
4315:
4314:
4312:
4311:
4310:
4296:
4295:
4294:
4289:
4220:
4202:
4198:Urysohn's lemma
4159:
4123:
4009:
4000:
3972:low-dimensional
3930:
3925:
3882:
3859:
3845:
3831:Schechter, Eric
3829:
3815:
3801:Dugundji, James
3799:
3785:
3771:Dolecki, Szymon
3769:
3766:
3765:
3758:
3738:
3737:
3733:
3726:
3702:
3701:
3697:
3692:
3687:
3686:
3652:
3651:
3629:
3628:
3597:
3596:
3574:
3573:
3554:
3553:
3551:
3547:
3542:
3528:
3507:
3498:
3485:
3458:
3457:
3438:
3437:
3418:
3417:
3386:
3385:
3361:
3360:
3346:
3298:
3297:
3253:
3252:
3207:
3206:
3187:
3186:
3167:
3166:
3123:
3122:
3091:
3090:
3043:
3042:
3035:
3026:
2995:
2994:
2972:
2971:
2948:
2947:
2925:
2924:
2905:
2904:
2876:
2875:
2844:
2843:
2824:
2823:
2804:
2803:
2782:
2781:
2762:
2761:
2734:
2733:
2714:
2713:
2659:
2658:
2651:
2645:
2613:
2612:
2590:
2589:
2588:whose union is
2570:
2569:
2550:
2549:
2530:
2529:
2506:
2505:
2483:
2482:
2463:
2462:
2429:
2428:
2409:
2408:
2386:
2385:
2366:
2365:
2324:
2289:
2288:
2260:
2259:
2231:
2230:
2211:
2210:
2185:
2184:
2165:
2164:
2133:
2132:
2113:
2112:
2093:
2092:
2065:
2064:
2045:
2044:
2015:
2014:
1972:
1948:
1947:
1943:
1935:
1934:
1933:if and only if
1899:
1898:
1870:
1869:
1850:
1849:
1822:
1805:
1804:
1803:if and only if
1785:
1784:
1765:
1764:
1734:
1729:
1728:
1709:
1708:
1681:
1670:
1669:
1650:
1649:
1630:
1629:
1607:
1606:
1587:
1586:
1556:
1551:
1550:
1531:
1530:
1511:
1510:
1487:
1486:
1461:
1460:
1438:
1437:
1414:
1413:
1388:
1387:
1368:
1367:
1348:
1347:
1322:
1321:
1295:
1294:
1271:
1270:
1248:
1247:
1224:
1223:
1201:
1200:
1181:
1180:
1161:
1160:
1135:
1134:
1104:
1103:
1084:
1083:
1064:
1063:
1044:
1043:
1024:
1023:
988:
987:
948:
937:
936:
902:
901:
879:
878:
859:
858:
831:
820:
819:
794:
793:
770:
769:
750:
749:
727:
726:
707:
706:
684:
683:
649:
648:
629:
628:
605:
604:
585:
584:
565:
564:
524:
513:
512:
511:if and only if
493:
492:
473:
472:
442:
431:
430:
405:
404:
374:
369:
368:
346:
345:
316:
315:
312:boundary points
285:
284:
261:
260:
226:
225:
194:
193:
168:
167:
130:
129:
107:
106:
103:
95:closed manifold
43:
28:
23:
22:
15:
12:
11:
5:
4319:
4317:
4309:
4308:
4298:
4297:
4291:
4290:
4288:
4287:
4277:
4276:
4275:
4270:
4265:
4250:
4240:
4230:
4218:
4207:
4204:
4203:
4201:
4200:
4195:
4190:
4185:
4180:
4175:
4169:
4167:
4161:
4160:
4158:
4157:
4152:
4147:
4145:Winding number
4142:
4137:
4131:
4129:
4125:
4124:
4122:
4121:
4116:
4111:
4106:
4101:
4096:
4091:
4086:
4085:
4084:
4079:
4077:homotopy group
4069:
4068:
4067:
4062:
4057:
4052:
4047:
4037:
4032:
4027:
4017:
4015:
4011:
4010:
4003:
4001:
3999:
3998:
3993:
3988:
3987:
3986:
3976:
3975:
3974:
3964:
3959:
3954:
3949:
3944:
3938:
3936:
3932:
3931:
3926:
3924:
3923:
3916:
3909:
3901:
3895:
3894:
3880:
3857:
3843:
3827:
3813:
3797:
3783:
3764:
3763:
3756:
3731:
3724:
3694:
3693:
3691:
3688:
3685:
3684:
3671:
3668:
3665:
3662:
3659:
3639:
3636:
3616:
3613:
3610:
3607:
3604:
3581:
3561:
3544:
3543:
3541:
3538:
3537:
3536:
3531:
3522:
3516:
3510:
3501:
3492:
3484:
3481:
3480:
3479:
3468:
3465:
3456:are closed in
3445:
3425:
3405:
3402:
3399:
3396:
3393:
3382:
3369:
3354:
3344:
3339:
3332:
3320:
3317:
3314:
3311:
3308:
3305:
3291:
3284:
3272:
3269:
3266:
3263:
3260:
3246:
3233:
3229:
3226:
3223:
3220:
3217:
3214:
3194:
3174:
3149:
3145:
3142:
3139:
3136:
3133:
3130:
3110:
3107:
3104:
3101:
3098:
3084:
3082:
3062:
3059:
3056:
3053:
3050:
3034:
3031:
3024:
3002:
2982:
2979:
2955:
2935:
2932:
2912:
2888:
2884:
2863:
2860:
2857:
2854:
2851:
2831:
2811:
2790:
2769:
2760:of subsets of
2749:
2746:
2742:
2721:
2710:
2709:
2706:
2699:
2697:
2685:
2666:
2644:
2641:
2620:
2600:
2597:
2577:
2557:
2537:
2513:
2490:
2470:
2436:
2416:
2396:
2393:
2373:
2344:uniform spaces
2323:
2320:
2308:
2305:
2302:
2299:
2296:
2276:
2273:
2270:
2267:
2247:
2244:
2241:
2238:
2218:
2198:
2195:
2192:
2172:
2152:
2149:
2146:
2143:
2140:
2120:
2100:
2081:
2078:
2075:
2072:
2052:
2028:
2025:
2022:
2002:
1999:
1996:
1993:
1990:
1987:
1984:
1979:
1975:
1971:
1967:
1963:
1960:
1955:
1951:
1946:
1942:
1918:
1915:
1912:
1909:
1906:
1880:
1877:
1857:
1837:
1834:
1829:
1825:
1821:
1818:
1815:
1812:
1792:
1772:
1752:
1749:
1746:
1741:
1737:
1716:
1696:
1693:
1688:
1684:
1680:
1677:
1657:
1637:
1617:
1614:
1594:
1574:
1571:
1568:
1563:
1559:
1538:
1518:
1508:
1494:
1474:
1471:
1468:
1448:
1445:
1435:
1421:
1401:
1398:
1395:
1375:
1355:
1335:
1332:
1329:
1319:
1305:
1302:
1292:
1278:
1269:in which case
1258:
1255:
1231:
1211:
1208:
1188:
1168:
1148:
1145:
1142:
1131:
1130:
1114:
1111:
1091:
1071:
1051:
1031:
1007:
1004:
1001:
998:
995:
975:
972:
967:
964:
961:
958:
955:
951:
947:
944:
924:
921:
918:
915:
912:
909:
886:
866:
846:
843:
838:
834:
830:
827:
807:
804:
801:
791:
788:is said to be
777:
757:
737:
734:
714:
694:
691:
659:
656:
636:
612:
592:
572:
542:
539:
536:
531:
527:
523:
520:
500:
480:
460:
457:
454:
449:
445:
441:
438:
418:
415:
412:
392:
389:
386:
381:
377:
356:
353:
329:
326:
323:
295:
292:
268:
248:
245:
242:
239:
236:
233:
224:; that is, if
213:
210:
207:
204:
201:
181:
178:
175:
164:
149:
146:
143:
140:
137:
114:
102:
99:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
4318:
4307:
4304:
4303:
4301:
4286:
4278:
4274:
4271:
4269:
4266:
4264:
4261:
4260:
4259:
4251:
4249:
4245:
4241:
4239:
4235:
4231:
4229:
4224:
4219:
4217:
4209:
4208:
4205:
4199:
4196:
4194:
4191:
4189:
4186:
4184:
4181:
4179:
4176:
4174:
4171:
4170:
4168:
4166:
4162:
4156:
4155:Orientability
4153:
4151:
4148:
4146:
4143:
4141:
4138:
4136:
4133:
4132:
4130:
4126:
4120:
4117:
4115:
4112:
4110:
4107:
4105:
4102:
4100:
4097:
4095:
4092:
4090:
4087:
4083:
4080:
4078:
4075:
4074:
4073:
4070:
4066:
4063:
4061:
4058:
4056:
4053:
4051:
4048:
4046:
4043:
4042:
4041:
4038:
4036:
4033:
4031:
4028:
4026:
4022:
4019:
4018:
4016:
4012:
4007:
3997:
3994:
3992:
3991:Set-theoretic
3989:
3985:
3982:
3981:
3980:
3977:
3973:
3970:
3969:
3968:
3965:
3963:
3960:
3958:
3955:
3953:
3952:Combinatorial
3950:
3948:
3945:
3943:
3940:
3939:
3937:
3933:
3929:
3922:
3917:
3915:
3910:
3908:
3903:
3902:
3899:
3891:
3887:
3883:
3877:
3873:
3869:
3868:Mineola, N.Y.
3865:
3864:
3858:
3854:
3850:
3846:
3840:
3836:
3832:
3828:
3824:
3820:
3816:
3810:
3806:
3802:
3798:
3794:
3790:
3786:
3780:
3776:
3772:
3768:
3767:
3759:
3757:0-13-181629-2
3753:
3749:
3748:Prentice Hall
3745:
3741:
3735:
3732:
3727:
3725:0-07-054235-X
3721:
3717:
3712:
3711:
3705:
3704:Rudin, Walter
3699:
3696:
3689:
3666:
3660:
3657:
3637:
3634:
3611:
3605:
3602:
3595:
3579:
3559:
3549:
3546:
3539:
3535:
3532:
3526:
3523:
3520:
3519:Neighbourhood
3517:
3514:
3511:
3505:
3504:Closed region
3502:
3496:
3493:
3490:
3487:
3486:
3482:
3466:
3463:
3443:
3423:
3403:
3397:
3394:
3391:
3383:
3359:
3355:
3352:
3348:
3340:
3337:
3333:
3312:
3309:
3306:
3296:
3292:
3289:
3285:
3267:
3264:
3261:
3251:
3247:
3227:
3221:
3218:
3215:
3192:
3172:
3164:
3143:
3137:
3134:
3131:
3105:
3102:
3099:
3089:
3088:unit interval
3085:
3081:
3078:
3076:
3057:
3054:
3051:
3041:
3037:
3036:
3032:
3030:
3028:
3027:
3019:
3014:
3000:
2980:
2977:
2969:
2953:
2933:
2930:
2910:
2902:
2886:
2858:
2855:
2852:
2829:
2809:
2767:
2744:
2719:
2707:
2704:
2700:
2695:
2692:
2690:
2686:
2683:
2679:
2678:
2677:
2664:
2656:
2650:
2642:
2640:
2638:
2635:if it has an
2634:
2618:
2611:Furthermore,
2598:
2595:
2575:
2555:
2535:
2527:
2511:
2502:
2488:
2468:
2459:
2456:
2454:
2450:
2434:
2414:
2394:
2391:
2371:
2363:
2359:
2356:
2351:
2349:
2345:
2341:
2337:
2336:metric spaces
2333:
2329:
2321:
2319:
2306:
2300:
2294:
2271:
2265:
2245:
2242:
2239:
2236:
2216:
2196:
2193:
2190:
2170:
2150:
2144:
2138:
2118:
2098:
2079:
2076:
2073:
2070:
2050:
2042:
2041:plain English
2026:
2023:
2020:
1994:
1988:
1982:
1977:
1973:
1969:
1965:
1961:
1958:
1953:
1949:
1944:
1940:
1932:
1916:
1910:
1907:
1904:
1896:
1891:
1878:
1875:
1855:
1835:
1832:
1827:
1823:
1819:
1816:
1813:
1810:
1790:
1770:
1750:
1747:
1744:
1739:
1735:
1714:
1694:
1691:
1686:
1682:
1678:
1675:
1655:
1635:
1615:
1612:
1592:
1572:
1569:
1566:
1561:
1557:
1536:
1516:
1506:
1492:
1472:
1469:
1466:
1446:
1443:
1433:
1419:
1399:
1396:
1393:
1373:
1353:
1333:
1327:
1317:
1303:
1300:
1290:
1276:
1256:
1253:
1245:
1229:
1209:
1206:
1186:
1166:
1146:
1143:
1140:
1128:
1127:
1126:
1112:
1109:
1089:
1069:
1049:
1029:
1021:
1002:
996:
993:
973:
970:
962:
956:
953:
949:
945:
942:
922:
916:
910:
907:
900:
884:
864:
844:
841:
836:
832:
828:
825:
805:
802:
799:
789:
775:
755:
735:
732:
712:
692:
689:
681:
677:
673:
657:
654:
634:
626:
610:
603:is closed in
590:
570:
562:
558:
553:
540:
537:
534:
529:
525:
521:
518:
498:
478:
458:
455:
452:
447:
443:
439:
436:
416:
413:
410:
390:
387:
384:
379:
375:
354:
351:
343:
327:
324:
321:
313:
309:
293:
290:
282:
266:
246:
243:
240:
237:
231:
208:
205:
202:
179:
173:
165:
162:
144:
141:
138:
128:
112:
100:
98:
96:
92:
88:
84:
80:
76:
72:
68:
64:
60:
56:
52:
48:
41:
37:
33:
19:
18:Closed subset
4285:Publications
4150:Chern number
4140:Betti number
4024:
4023: /
4014:Key concepts
3962:Differential
3862:
3834:
3804:
3774:
3743:
3734:
3709:
3698:
3572:is close to
3548:
3075:real numbers
3021:
3015:
2711:
2682:intersection
2652:
2526:disconnected
2503:
2460:
2457:
2352:
2348:gauge spaces
2325:
2287:is close to
1892:
1289:is called a
1132:
563:. A subset
554:
403:that is, if
308:limit points
161:
104:
79:limit points
58:
44:
4248:Wikiversity
4165:Key results
3716:McGraw-Hill
3356:The set of
3288:clopen sets
3038:The closed
2923:in a space
2163:Similarly,
1316:then there
55:mathematics
4094:CW complex
4035:Continuity
4025:Closed set
3984:cohomology
3690:References
3495:Closed map
3489:Clopen set
3336:Cantor set
3331:is closed.
2966:that is a
2705:is closed.
2647:See also:
2643:Properties
2637:open basis
1931:continuous
471:Moreover,
160:is called
89:under the
67:complement
59:closed set
4273:geometric
4268:algebraic
4119:Cobordism
4055:Hausdorff
4050:connected
3967:Geometric
3957:Continuum
3947:Algebraic
3853:175294365
3823:395340485
3793:945169917
3661:∪
3606:∪
3401:→
3316:∞
3228:∩
3144:∩
3018:countably
2903:of a set
2859:τ
2810:τ
2748:∅
2745:≠
2703:empty set
2328:open sets
2240:⊆
2194:∈
2074:⊆
2024:⊆
1983:
1970:⊆
1959:
1914:→
1833:
1820:∩
1763:However,
1745:
1692:
1567:
1470:⊆
1397:⊆
1331:∖
1144:∈
997:∪
971:
957:∪
946:∈
911:∪
842:
829:∈
803:⊆
792:a subset
676:sequences
557:sequences
535:
453:
440:⊆
414:⊆
385:
325:⊆
244:τ
241:∈
235:∖
209:τ
177:∖
145:τ
4300:Category
4238:Wikibook
4216:Category
4104:Manifold
4072:Homotopy
4030:Interior
4021:Open set
3979:Homology
3928:Topology
3833:(1996).
3805:Topology
3803:(1966).
3744:Topology
3742:(2000).
3706:(1976).
3594:subspace
3513:Open set
3483:See also
3358:integers
3250:interval
3165:between
3040:interval
3033:Examples
2968:superset
2694:finitely
2655:boundary
1897:: a map
935:meaning
790:close to
748:A point
71:open set
51:topology
47:geometry
32:open set
4263:general
4065:uniform
4045:compact
3996:Digital
2901:closure
2355:compact
1485:and if
1432:but to
897:in the
281:closure
81:. In a
73:. In a
4258:Topics
4060:metric
3935:Fields
3890:115240
3888:
3878:
3851:
3841:
3821:
3811:
3791:
3781:
3754:
3722:
3347:spaces
2346:, and
986:where
163:closed
87:closed
69:is an
65:whose
4040:Space
3540:Notes
2689:union
2407:then
2360:are "
2258:then
1529:then
1318:might
1242:is a
670:In a
625:limit
429:then
125:of a
91:limit
61:is a
3886:OCLC
3876:ISBN
3849:OCLC
3839:ISBN
3819:OCLC
3809:ISBN
3789:OCLC
3779:ISBN
3752:ISBN
3720:ISBN
3349:and
3334:The
3293:The
3185:and
3086:The
2701:The
2696:many
2687:The
2680:Any
2548:and
2043:as:
561:nets
559:and
57:, a
3384:If
3295:ray
3161:of
3073:of
2970:of
2822:on
2691:of
2631:is
2568:of
2524:is
2350:.
1929:is
1868:of
1605:in
1507:any
1505:is
1459:If
1434:not
1293:of
1222:If
1062:in
818:if
768:in
344:in
283:in
63:set
45:In
4302::
3884:.
3874:.
3870::
3866:.
3847:.
3817:.
3787:.
3750:.
3718:.
3714:.
2665:2.
2342:,
2338:,
1974:cl
1950:cl
1824:cl
1736:cl
1683:cl
1558:cl
950:cl
833:cl
526:cl
444:cl
376:cl
97:.
49:,
3920:e
3913:t
3906:v
3892:.
3855:.
3825:.
3795:.
3760:.
3728:.
3670:}
3667:x
3664:{
3658:A
3638:,
3635:X
3615:}
3612:x
3609:{
3603:A
3580:A
3560:x
3467:.
3464:X
3444:Y
3424:f
3404:Y
3398:X
3395::
3392:f
3368:Z
3353:.
3345:1
3343:T
3319:)
3313:+
3310:,
3307:1
3304:[
3290:.
3271:)
3268:1
3265:,
3262:0
3259:[
3232:Q
3225:]
3222:1
3219:,
3216:0
3213:[
3193:1
3173:0
3148:Q
3141:]
3138:1
3135:,
3132:0
3129:[
3109:]
3106:1
3103:,
3100:0
3097:[
3061:]
3058:b
3055:,
3052:a
3049:[
3025:σ
3023:F
3001:X
2981:.
2978:A
2954:X
2934:,
2931:X
2911:A
2887:.
2883:F
2862:)
2856:,
2853:X
2850:(
2830:X
2789:F
2768:X
2741:F
2720:X
2619:X
2599:.
2596:X
2576:X
2556:B
2536:A
2512:X
2489:X
2469:X
2435:X
2415:D
2395:,
2392:X
2372:D
2307:.
2304:)
2301:A
2298:(
2295:f
2275:)
2272:x
2269:(
2266:f
2246:,
2243:X
2237:A
2217:x
2197:X
2191:x
2171:f
2151:.
2148:)
2145:A
2142:(
2139:f
2119:A
2099:f
2080:,
2077:X
2071:A
2051:f
2027:X
2021:A
2001:)
1998:)
1995:A
1992:(
1989:f
1986:(
1978:Y
1966:)
1962:A
1954:X
1945:(
1941:f
1917:Y
1911:X
1908::
1905:f
1879:.
1876:X
1856:Y
1836:A
1828:Y
1817:X
1814:=
1811:A
1791:X
1771:A
1751:.
1748:A
1740:Y
1715:A
1695:A
1687:X
1679:=
1676:A
1656:X
1636:A
1616:;
1613:Y
1593:A
1573:,
1570:A
1562:Y
1537:A
1517:X
1493:Y
1473:X
1467:A
1447:.
1444:Y
1420:X
1400:X
1394:A
1374:X
1354:A
1334:X
1328:Y
1304:,
1301:X
1277:Y
1257:,
1254:Y
1230:X
1210:.
1207:x
1187:A
1167:A
1147:X
1141:x
1113:,
1110:A
1090:X
1070:X
1050:A
1030:X
1006:}
1003:x
1000:{
994:A
974:A
966:}
963:x
960:{
954:A
943:x
923:,
920:}
917:x
914:{
908:A
885:A
865:x
845:A
837:X
826:x
806:X
800:A
776:X
756:x
736:.
733:X
713:X
693:,
690:X
658:.
655:A
635:A
611:X
591:X
571:A
541:.
538:A
530:X
522:=
519:A
499:X
479:A
459:.
456:A
448:X
437:A
417:X
411:A
391:;
388:A
380:X
355:,
352:X
328:X
322:A
294:.
291:X
267:X
247:.
238:A
232:X
212:)
206:,
203:X
200:(
180:A
174:X
148:)
142:,
139:X
136:(
113:A
42:.
20:)
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