2873:
3135:
Simon
Moulieras, Maciej Lewenstein and Graciana Puentes, Entanglement engineering and topological protection by discrete-time quantum walks, Journal of Physics B: Atomic, Molecular and Optical Physics 46 (10), 104005 (2013).
2186:. A space is said to be κ-resolvable (respectively: almost κ-resolvable) if it contains κ dense sets that are pairwise disjoint (respectively: almost disjoint over the ideal of nowhere dense subsets). If the space is not
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372:
227:
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if it is zero-dimensional, compact and
Hausdorff (equivalently, totally disconnected, compact and Hausdorff). These are precisely the spaces that are homeomorphic to the
1781:
if every point has a local base consisting of compact neighbourhoods. Slightly different definitions are also used. Locally compact
Hausdorff spaces are always Tychonoff.
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464:
100:
2133:
1707:
spaces where every open cover has finite subcover. Compact spaces are always Lindelöf and paracompact. Compact
Hausdorff spaces are therefore normal.
2284:
1341:
if every point has a local base consisting of path-connected sets. A locally path-connected space is connected if and only if it is path-connected.
1610:. Semi-local simple connectivity, a strictly weaker condition than local simple connectivity, is a necessary condition for the existence of a
1820:. Metrizable spaces are always Hausdorff and paracompact (and hence normal and Tychonoff), and first-countable. Moreover, a topological space
51:
of topological spaces which is closed under homeomorphisms. That is, a property of spaces is a topological property if whenever a space
3151:
2136:
if every open set is closed (hence clopen). The almost discrete spaces are precisely the finitely generated zero-dimensional spaces.
843:
2705:
2790:
325:
180:
3124:
3098:
1084:
411:
1108:
672:
59:
possesses that property. Informally, a topological property is a property of the space that can be expressed using
1245:
if it is not the union of a pair of disjoint non-empty open sets. Equivalently, a space is connected if the only
2105:
1565:
2667:
3027:
1591:
754:; here, we are allowed to specify which point will be contained in the open set.) Equivalently, a space is T
472:
2151:
3137:
1514:
260:
112:
2994:
872:
2979: – Numerical invariant that describes the linking of two closed curves in three-dimensional space
2997: – Physical quantities that take discrete values because of topological quantum physical effects
2101:
2677:
2007:
1771:
if every open cover has an open locally finite refinement. Paracompact
Hausdorff spaces are normal.
1386:
423:
296:
148:
1190:
base for its topology. Second-countable spaces are always separable, first-countable and Lindelöf.
3156:
2100:
are open, or equivalently if arbitrary unions of closed sets are closed. These are precisely the
972:
926:
879:
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is not the (possibly nondisjoint) union of two smaller closed non-empty subsets, then there is a
40:
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itself. Non-empty ultra-connected compact spaces have a largest proper open subset called a
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2158:
1823:
1714:
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if no two non-empty closed sets are disjoint. Every ultraconnected space is path-connected.
1266:
1155:
74:
homeomorphic, it is sufficient to find a topological property which is not shared by them.
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590:
532:
8:
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48:
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1989:
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if the only open sets are the empty set and itself. Such a space is said to have the
1629:
1256:
1225:
1049:
36:
3068:
1754:
1646:
if no two non-empty open sets are disjoint. Every hyperconnected space is connected.
3046:
2965:
2115:
2074:
Intuitively speaking, this means that the space looks the same at every point. All
1813:
1671:
1667:
1508:
1011:, so the terminology is consistent.) Tychonoff spaces are always regular Hausdorff.
690:
666:
559:
1021:
if any two disjoint closed sets have disjoint neighbourhoods. Normal spaces admit
671:
Some of these terms are defined differently in older mathematical literature; see
3088:
2557:
is not topological, it is sufficient to find two homeomorphic topological spaces
1778:
1704:
1611:
1242:
1000:
948:
818:
1968:
is a Baire space if the intersection of countably many dense open sets is dense.
1636:
is homotopic to a constant map. Contractible spaces are always simply connected.
3000:
2976:
2970:
2961:
1653:
1643:
1280:
1132:
1122:
1053:
3051:
3022:
1939:. A space is locally metrizable if every point has a metrizable neighbourhood.
1115:. A perfectly normal Hausdorff space must also be completely normal Hausdorff.
3145:
3116:
3108:
3060:
2356:{\displaystyle \Delta (X)=\min\{|G|:G\neq \varnothing ,G{\mbox{ is open}}\}.}
2143:
2118:
if it has a base of clopen sets. These are precisely the spaces with a small
1734:
1688:
1352:
1222:
1218:
889:
66:
A common problem in topology is to decide whether two topological spaces are
44:
2898:
2534:
1930:
1817:
1018:
67:
2973: – Function of a knot that takes the same value for equivalent knots
2934:
2521:
2147:
1957:
1768:
1125:
if all of its points are completely isolated, i.e. if any subset is open.
773:
418:
24:
3020:
2956:
1975:
1961:
1692:
1246:
1007:
space. (A completely regular space is
Hausdorff if and only if it is T
938:
3138:
https://iopscience.iop.org/article/10.1088/0953-4075/46/10/104005/pdf
3041:
2887:
1758:
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1380:
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1187:
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1159:
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731:
60:
20:
3003: – Number of times a curve wraps around a point in the plane
1737:
if every continuous real-valued function on the space is bounded.
1073:. Completely normal Hausdorff spaces are always normal Hausdorff.
2537:, etc, which are not topological properties. To show a property
2500:
is said to be strongly discrete if every non-isolated point of
1069:. A completely normal space is Hausdorff if and only if it is T
2928: – Association of cohomology classes to principal bundles
2922: – Association of cohomology classes to principal bundles
701:
in the space, there is at least either an open set containing
3021:
Juhász, István; Soukup, Lajos; Szentmiklóssy, Zoltán (2008).
1259:
if every point has a local base consisting of connected sets.
933:
space. (A regular space is
Hausdorff if and only if it is T
821:
if every two distinct points have disjoint neighbourhoods. T
2480:
may be separated by pairwise disjoint neighborhoods. Space
2985: – List of concrete topologies and topological spaces
2754:{\displaystyle Y=(-{\tfrac {\pi }{2}},{\tfrac {\pi }{2}})}
1087:. A perfectly normal space must also be completely normal.
1933:
if it is metrizable with a separable and complete metric.
3093:. Reading, Mass.: Addison-Wesley Pub. Co. p. 369.
2930:
Pages displaying short descriptions of redirect targets
1269:
if it has no connected subset with more than one point.
504:
of the topology (the set of open subsets) of the space
412:
Cardinal function § Cardinal functions in topology
77:
2737:
2722:
2341:
1852:
is said to be metrizable if there exists a metric for
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if it is the union of countably many compact subsets.
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263:
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183:
151:
115:
88:
2812:{\displaystyle \operatorname {arctan} \colon X\to Y}
1727:
if every countable open cover has a finite subcover.
1038:. A normal space is Hausdorff if and only if it is T
55:
possesses that property every space homeomorphic to
2851:
2831:
2811:
2779:
2753:
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367:{\displaystyle \left(S,{\mathcal {T}}|_{S}\right)}
366:
314:
285:
244:
222:{\displaystyle \left(S,{\mathcal {T}}|_{S}\right)}
221:
166:
137:
94:
2761:be metric spaces with the standard metric. Then,
1594:if every point has a local base of neighborhoods
1511:if it is path-connected and every continuous map
3143:
2937: – Characteristic classes of vector bundles
2303:
400:
1717:if every sequence has a convergent subsequence.
1042:. Normal Hausdorff spaces are always Tychonoff.
882:. Every completely Hausdorff space is Urysohn.
742:in the space, there is an open set containing
2528:
1331:. Path-connected spaces are always connected.
47:. Alternatively, a topological property is a
2943: – Topological invariant in mathematics
2666:For example, the metric space properties of
2347:
2306:
1978:if every subset is open or closed (or both).
491:
476:
433:
427:
2991: – Concept in mathematical knot theory
2096:if arbitrary intersections of open sets in
846:if every two distinct points have disjoint
1497:. Arc-connected spaces are path-connected.
1139:
3050:
3040:
2688:
2533:There are many examples of properties of
2440:is strongly discrete subset of the space
3107:
3086:
616:, the least cardinality of a subset of
3144:
3023:"Resolvability and monotone normality"
1787:. In an ultra-connected compact space
1111:, if it is both perfectly normal and T
804:is the only point with this property.
734:if for every pair of distinct points
693:if for every pair of distinct points
405:
70:or not. To prove that two spaces are
2866:
2674:are not topological properties. Let
1083:if any two disjoint closed sets are
497:{\displaystyle \vert \tau (X)\vert }
78:Properties of topological properties
2839:is complete but not bounded, while
758:if all its singletons are closed. T
13:
2392:is called dispersion character of
2370:
2288:
2259:
1576:has a local base of neighborhoods
1543:{\displaystyle f\colon S^{1}\to X}
342:
286:{\displaystyle (X,{\mathcal {T}})}
275:
197:
138:{\displaystyle (X,{\mathcal {T}})}
127:
14:
3168:
2331:
2252:is maximally resolvable if it is
1699:. Some authors call these spaces
1085:precisely separated by a function
878:if every two distinct points are
257:, if for every topological space
109:, if for every topological space
3152:Properties of topological spaces
2871:
1943:
1803:
1232:
776:if every irreducible closed set
673:history of the separation axioms
16:Mathematical property of a space
1901:is identical with the topology
3014:
2886:format but may read better as
2803:
2748:
2715:
2695:{\displaystyle X=\mathbb {R} }
2524:of some strongly discrete set.
2379:
2373:
2318:
2310:
2297:
2291:
2268:
2262:
2206:-resolvable then it is called
2106:category of topological spaces
2052:
2046:
2029:{\displaystyle f\colon X\to X}
2020:
1888:
1882:
1872:such that the metric topology
1839:
1827:
1791:every open cover must contain
1678:
1534:
1478:
1472:
1443:
1437:
1411:
1408:
1396:
603:
597:
545:
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482:
349:
280:
264:
204:
132:
116:
1:
3079:
3028:Israel Journal of Mathematics
2949: – Mathematical property
2859:is bounded but not complete.
1592:semi-locally simply connected
1584:Semi-locally simply connected
1420:{\displaystyle f\colon \to X}
1249:are the empty set and itself.
1056:have disjoint neighbourhoods.
912:have disjoint neighbourhoods.
660:
558:, the least cardinality of a
439:{\displaystyle \vert X\vert }
401:Common topological properties
315:{\displaystyle S\subseteq X,}
167:{\displaystyle S\subseteq X,}
3007:
990:Completely regular Hausdorff
709:, or an open set containing
7:
2862:
1974:. A topological space is a
1816:if it is homeomorphic to a
1067:Completely normal Hausdorff
780:has a unique generic point
10:
3173:
2995:Topological quantum number
2529:Non-topological properties
2385:{\displaystyle \Delta (X)}
2274:{\displaystyle \Delta (X)}
1757:if every open cover has a
1200:if every open cover has a
1143:
1109:perfectly normal Hausdorff
1098:Perfectly normal Hausdorff
792:such that the closure of {
664:
409:
3087:Willard, Stephen (1970).
3052:10.1007/s11856-008-1017-y
2004:there is a homeomorphism
1964:in itself. Equivalently,
1580:that is simply connected.
1307:, i.e., a continuous map
1129:Number of isolated points
1003:is a completely regular T
2780:{\displaystyle X\cong Y}
2576:{\displaystyle X\cong Y}
1703:and reserve compact for
1566:locally simply connected
1558:Locally simply connected
1355:if for every two points
1283:if for every two points
937:, so the terminology is
2895:converting this article
2219:{\displaystyle \kappa }
2199:{\displaystyle \kappa }
2177:{\displaystyle \kappa }
2067:{\displaystyle f(x)=y.}
1982:Topological Homogeneity
1140:Countability conditions
1135:of a topological space.
973:separated by a function
880:separated by a function
2926:Characteristic numbers
2853:
2833:
2813:
2787:via the homeomorphism
2781:
2755:
2696:
2657:
2637:
2617:
2597:
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2474:
2454:
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2409:
2386:
2357:
2275:
2246:
2220:
2200:
2178:
2068:
2030:
1918:
1895:
1866:
1846:
1785:Ultraconnected compact
1544:
1491:
1490:{\displaystyle p(1)=y}
1456:
1455:{\displaystyle p(0)=x}
1421:
1339:locally path-connected
1335:Locally path-connected
1217:if it is the union of
784:. In other words, if
650:
630:
610:
576:
552:
518:
498:
460:
440:
391:
368:
316:
287:
246:
223:
168:
139:
96:
2854:
2834:
2814:
2782:
2756:
2697:
2658:
2638:
2618:
2598:
2578:
2552:
2515:
2495:
2475:
2455:
2435:
2410:
2387:
2358:
2276:
2247:
2221:
2201:
2179:
2069:
2031:
1919:
1896:
1867:
1847:
1845:{\displaystyle (X,T)}
1545:
1492:
1457:
1422:
1172:if every point has a
1146:Axiom of countability
651:
631:
611:
577:
560:basis of the topology
553:
519:
499:
461:
441:
392:
369:
317:
288:
247:
224:
169:
140:
97:
33:topological invariant
23:and related areas of
2947:Fixed-point property
2941:Euler characteristic
2920:Characteristic class
2843:
2823:
2791:
2765:
2706:
2678:
2647:
2627:
2607:
2587:
2561:
2541:
2504:
2484:
2464:
2444:
2424:
2396:
2367:
2285:
2256:
2236:
2230:Maximally resolvable
2210:
2190:
2168:
2159:Reidemeister torsion
2108:and continuous maps.
2040:
2008:
1929:. A space is called
1905:
1894:{\displaystyle T(d)}
1876:
1856:
1824:
1715:sequentially compact
1711:Sequentially compact
1515:
1466:
1431:
1387:
1311:: →
1267:totally disconnected
1263:Totally disconnected
955:is a closed set and
929:if it is a regular T
896:is a closed set and
869:completely Hausdorff
640:
620:
609:{\displaystyle d(X)}
591:
566:
551:{\displaystyle w(X)}
533:
508:
473:
450:
424:
378:
326:
297:
261:
233:
181:
149:
113:
86:
29:topological property
2281:-resolvable, where
2120:inductive dimension
1988:is (topologically)
1606:is contractible in
1023:partitions of unity
854:spaces are always T
825:spaces are always T
762:spaces are always T
35:is a property of a
2983:List of topologies
2897:, if appropriate.
2849:
2829:
2809:
2777:
2751:
2746:
2731:
2692:
2653:
2633:
2613:
2593:
2573:
2547:
2522:accumulation point
2510:
2490:
2470:
2450:
2430:
2408:{\displaystyle X.}
2405:
2382:
2353:
2345:
2271:
2242:
2216:
2196:
2174:
2102:finitely generated
2082:Finitely generated
2076:topological groups
2064:
2026:
1937:Locally metrizable
1917:{\displaystyle T.}
1914:
1891:
1862:
1842:
1554:to a constant map.
1540:
1487:
1452:
1417:
1367:, there is an arc
1295:, there is a path
959:is a point not in
949:completely regular
945:Completely regular
900:is a point not in
646:
626:
606:
572:
548:
514:
494:
456:
436:
406:Cardinal functions
390:{\displaystyle P.}
387:
364:
312:
293:and closed subset
283:
245:{\displaystyle P.}
242:
219:
164:
135:
92:
3109:Munkres, James R.
2989:Quantum invariant
2916:
2915:
2852:{\displaystyle Y}
2832:{\displaystyle X}
2745:
2730:
2656:{\displaystyle P}
2636:{\displaystyle Y}
2616:{\displaystyle P}
2596:{\displaystyle X}
2550:{\displaystyle P}
2513:{\displaystyle X}
2493:{\displaystyle X}
2473:{\displaystyle D}
2460:if the points in
2453:{\displaystyle X}
2433:{\displaystyle D}
2418:Strongly discrete
2344:
2245:{\displaystyle X}
1865:{\displaystyle X}
1725:countably compact
1721:Countably compact
1257:locally connected
1253:Locally connected
1050:completely normal
1046:Completely normal
927:regular Hausdorff
923:Regular Hausdorff
850:neighbourhoods. T
750:. (Compare with T
649:{\displaystyle X}
636:whose closure is
629:{\displaystyle X}
575:{\displaystyle X}
517:{\displaystyle X}
459:{\displaystyle X}
255:Weakly hereditary
95:{\displaystyle P}
37:topological space
3164:
3130:
3104:
3090:General topology
3073:
3072:
3054:
3044:
3018:
2966:Cohomotopy group
2931:
2911:
2908:
2902:
2893:You can help by
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2815:
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2784:
2783:
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2757:
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2747:
2738:
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2698:
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2519:
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2414:
2412:
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2406:
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2223:
2222:
2217:
2205:
2203:
2202:
2197:
2183:
2181:
2180:
2175:
2152:Boolean algebras
2116:zero-dimensional
2112:Zero-dimensional
2078:are homogeneous.
2073:
2071:
2070:
2065:
2035:
2033:
2032:
2027:
1923:
1921:
1920:
1915:
1900:
1898:
1897:
1892:
1871:
1869:
1868:
1863:
1851:
1849:
1848:
1843:
1672:trivial topology
1549:
1547:
1546:
1541:
1533:
1532:
1509:simply connected
1501:Simply connected
1496:
1494:
1493:
1488:
1461:
1459:
1458:
1453:
1426:
1424:
1423:
1418:
1184:second-countable
1180:Second-countable
1131:. The number of
1081:perfectly normal
1077:Perfectly normal
1036:Normal Hausdorff
667:Separation axiom
655:
653:
652:
647:
635:
633:
632:
627:
615:
613:
612:
607:
581:
579:
578:
573:
557:
555:
554:
549:
523:
521:
520:
515:
503:
501:
500:
495:
469:The cardinality
465:
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457:
445:
443:
442:
437:
396:
394:
393:
388:
373:
371:
370:
365:
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357:
352:
346:
345:
321:
319:
318:
313:
292:
290:
289:
284:
279:
278:
251:
249:
248:
243:
228:
226:
225:
220:
218:
214:
213:
212:
207:
201:
200:
173:
171:
170:
165:
144:
142:
141:
136:
131:
130:
101:
99:
98:
93:
3172:
3171:
3167:
3166:
3165:
3163:
3162:
3161:
3142:
3141:
3133:
3127:
3101:
3082:
3077:
3076:
3019:
3015:
3010:
2929:
2912:
2906:
2903:
2892:
2876:
2872:
2865:
2844:
2841:
2840:
2824:
2821:
2820:
2792:
2789:
2788:
2766:
2763:
2762:
2736:
2721:
2707:
2704:
2703:
2687:
2679:
2676:
2675:
2648:
2645:
2644:
2628:
2625:
2624:
2608:
2605:
2604:
2588:
2585:
2584:
2562:
2559:
2558:
2542:
2539:
2538:
2531:
2505:
2502:
2501:
2485:
2482:
2481:
2465:
2462:
2461:
2445:
2442:
2441:
2425:
2422:
2421:
2397:
2394:
2393:
2368:
2365:
2364:
2340:
2317:
2309:
2286:
2283:
2282:
2257:
2254:
2253:
2237:
2234:
2233:
2211:
2208:
2207:
2191:
2188:
2187:
2169:
2166:
2165:
2134:almost discrete
2130:Almost discrete
2104:members of the
2041:
2038:
2037:
2009:
2006:
2005:
1946:
1906:
1903:
1902:
1877:
1874:
1873:
1857:
1854:
1853:
1825:
1822:
1821:
1806:
1779:locally compact
1775:Locally compact
1681:
1612:universal cover
1568:if every point
1528:
1524:
1516:
1513:
1512:
1467:
1464:
1463:
1432:
1429:
1428:
1388:
1385:
1384:
1383:continuous map
1235:
1170:first-countable
1166:First-countable
1148:
1142:
1133:isolated points
1114:
1105:
1094:
1072:
1063:
1041:
1032:
1010:
1006:
1001:Tychonoff space
997:
982:
936:
932:
919:
876:
865:
857:
853:
836:
828:
824:
811:
765:
761:
757:
753:
724:
683:
669:
663:
641:
638:
637:
621:
618:
617:
592:
589:
588:
567:
564:
563:
534:
531:
530:
509:
506:
505:
474:
471:
470:
451:
448:
447:
425:
422:
421:
414:
408:
403:
379:
376:
375:
353:
348:
347:
341:
340:
333:
329:
327:
324:
323:
298:
295:
294:
274:
273:
262:
259:
258:
234:
231:
230:
208:
203:
202:
196:
195:
188:
184:
182:
179:
178:
150:
147:
146:
126:
125:
114:
111:
110:
87:
84:
83:
80:
17:
12:
11:
5:
3170:
3160:
3159:
3157:Homeomorphisms
3154:
3132:
3131:
3125:
3105:
3099:
3083:
3081:
3078:
3075:
3074:
3012:
3011:
3009:
3006:
3005:
3004:
3001:Winding number
2998:
2992:
2986:
2980:
2977:Linking number
2974:
2971:Knot invariant
2968:
2962:Homotopy group
2959:
2950:
2944:
2938:
2932:
2923:
2914:
2913:
2879:
2877:
2870:
2864:
2861:
2848:
2828:
2808:
2805:
2802:
2799:
2796:
2776:
2773:
2770:
2750:
2744:
2741:
2735:
2729:
2726:
2720:
2717:
2714:
2711:
2690:
2686:
2683:
2652:
2643:does not have
2632:
2612:
2592:
2572:
2569:
2566:
2546:
2530:
2527:
2526:
2525:
2509:
2489:
2469:
2449:
2429:
2415:
2404:
2401:
2381:
2378:
2375:
2372:
2352:
2349:
2339:
2336:
2333:
2330:
2327:
2324:
2320:
2316:
2312:
2308:
2305:
2302:
2299:
2296:
2293:
2290:
2270:
2267:
2264:
2261:
2241:
2227:
2226:-irresolvable.
2215:
2195:
2173:
2162:
2155:
2137:
2127:
2109:
2079:
2063:
2060:
2057:
2054:
2051:
2048:
2045:
2025:
2022:
2019:
2016:
2013:
1979:
1969:
1945:
1942:
1941:
1940:
1934:
1924:
1913:
1910:
1890:
1887:
1884:
1881:
1861:
1841:
1838:
1835:
1832:
1829:
1805:
1802:
1801:
1800:
1782:
1772:
1762:
1748:
1738:
1728:
1718:
1708:
1680:
1677:
1676:
1675:
1657:
1654:ultraconnected
1650:Ultraconnected
1647:
1644:hyperconnected
1640:Hyperconnected
1637:
1615:
1581:
1555:
1539:
1536:
1531:
1527:
1523:
1520:
1498:
1486:
1483:
1480:
1477:
1474:
1471:
1451:
1448:
1445:
1442:
1439:
1436:
1416:
1413:
1410:
1407:
1404:
1401:
1398:
1395:
1392:
1342:
1332:
1281:path-connected
1273:Path-connected
1270:
1260:
1250:
1234:
1231:
1230:
1229:
1205:
1191:
1177:
1163:
1141:
1138:
1137:
1136:
1126:
1119:Discrete space
1116:
1112:
1103:
1092:
1088:
1074:
1070:
1061:
1057:
1054:separated sets
1043:
1039:
1030:
1026:
1012:
1008:
1004:
995:
980:
976:
942:
934:
930:
917:
913:
883:
874:
871:. A space is
863:
859:
855:
851:
834:
830:
826:
822:
809:
805:
767:
763:
759:
755:
751:
722:
718:
681:
665:Main article:
662:
659:
658:
657:
645:
625:
605:
602:
599:
596:
583:
571:
547:
544:
541:
538:
525:
513:
493:
490:
487:
484:
481:
478:
467:
455:
435:
432:
429:
410:Main article:
407:
404:
402:
399:
398:
397:
386:
383:
362:
356:
351:
344:
339:
336:
332:
311:
308:
305:
302:
282:
277:
272:
269:
266:
252:
241:
238:
217:
211:
206:
199:
194:
191:
187:
163:
160:
157:
154:
134:
129:
124:
121:
118:
91:
79:
76:
45:homeomorphisms
15:
9:
6:
4:
3:
2:
3169:
3158:
3155:
3153:
3150:
3149:
3147:
3140:
3139:
3128:
3126:0-13-181629-2
3122:
3118:
3117:Prentice-Hall
3114:
3110:
3106:
3102:
3100:9780486434797
3096:
3092:
3091:
3085:
3084:
3070:
3066:
3062:
3058:
3053:
3048:
3043:
3038:
3034:
3030:
3029:
3024:
3017:
3013:
3002:
2999:
2996:
2993:
2990:
2987:
2984:
2981:
2978:
2975:
2972:
2969:
2967:
2963:
2960:
2958:
2954:
2951:
2948:
2945:
2942:
2939:
2936:
2933:
2927:
2924:
2921:
2918:
2917:
2910:
2901:is available.
2900:
2896:
2890:
2889:
2885:
2880:This article
2878:
2869:
2868:
2860:
2846:
2826:
2806:
2800:
2797:
2794:
2774:
2771:
2768:
2742:
2739:
2733:
2727:
2724:
2718:
2712:
2709:
2684:
2681:
2673:
2669:
2664:
2650:
2630:
2610:
2590:
2570:
2567:
2564:
2544:
2536:
2535:metric spaces
2523:
2507:
2487:
2467:
2447:
2427:
2419:
2416:
2402:
2399:
2376:
2350:
2343: is open
2337:
2334:
2328:
2325:
2322:
2314:
2300:
2294:
2265:
2239:
2231:
2228:
2213:
2193:
2185:
2171:
2163:
2161:
2160:
2156:
2153:
2149:
2145:
2142:. A space is
2141:
2138:
2135:
2132:. A space is
2131:
2128:
2125:
2121:
2117:
2114:. A space is
2113:
2110:
2107:
2103:
2099:
2095:
2091:
2087:
2083:
2080:
2077:
2061:
2058:
2055:
2049:
2043:
2023:
2017:
2014:
2011:
2003:
1999:
1995:
1992:if for every
1991:
1987:
1983:
1980:
1977:
1973:
1970:
1967:
1963:
1960:if it is not
1959:
1955:
1951:
1948:
1947:
1944:Miscellaneous
1938:
1935:
1932:
1928:
1925:
1911:
1908:
1885:
1879:
1859:
1836:
1833:
1830:
1819:
1815:
1812:. A space is
1811:
1808:
1807:
1804:Metrizability
1798:
1794:
1790:
1786:
1783:
1780:
1777:. A space is
1776:
1773:
1770:
1767:. A space is
1766:
1763:
1760:
1756:
1753:. A space is
1752:
1749:
1746:
1743:. A space is
1742:
1739:
1736:
1735:pseudocompact
1733:. A space is
1732:
1731:Pseudocompact
1729:
1726:
1723:. A space is
1722:
1719:
1716:
1713:. A space is
1712:
1709:
1706:
1702:
1698:
1695:has a finite
1694:
1690:
1687:. A space is
1686:
1683:
1682:
1673:
1669:
1666:. A space is
1665:
1661:
1658:
1655:
1652:. A space is
1651:
1648:
1645:
1642:. A space is
1641:
1638:
1635:
1631:
1627:
1623:
1619:
1616:
1613:
1609:
1605:
1601:
1597:
1593:
1589:
1585:
1582:
1579:
1575:
1571:
1567:
1563:
1559:
1556:
1553:
1537:
1529:
1525:
1521:
1518:
1510:
1506:
1502:
1499:
1484:
1481:
1475:
1469:
1449:
1446:
1440:
1434:
1414:
1405:
1402:
1399:
1393:
1390:
1382:
1378:
1374:
1370:
1366:
1362:
1358:
1354:
1353:arc-connected
1350:
1346:
1345:Arc-connected
1343:
1340:
1337:. A space is
1336:
1333:
1330:
1326:
1322:
1318:
1314:
1310:
1306:
1302:
1298:
1294:
1290:
1286:
1282:
1278:
1274:
1271:
1268:
1265:. A space is
1264:
1261:
1258:
1255:. A space is
1254:
1251:
1248:
1244:
1241:. A space is
1240:
1237:
1236:
1233:Connectedness
1227:
1224:
1220:
1216:
1213:. A space is
1212:
1210:
1206:
1203:
1199:
1196:. A space is
1195:
1192:
1189:
1185:
1182:. A space is
1181:
1178:
1175:
1171:
1168:. A space is
1167:
1164:
1162:dense subset.
1161:
1157:
1154:. A space is
1153:
1150:
1149:
1147:
1134:
1130:
1127:
1124:
1121:. A space is
1120:
1117:
1110:
1107:. A space is
1106:
1099:
1095:
1089:
1086:
1082:
1079:. A space is
1078:
1075:
1068:
1064:
1058:
1055:
1051:
1048:. A space is
1047:
1044:
1037:
1033:
1027:
1024:
1020:
1017:. A space is
1016:
1013:
1002:
998:
991:
987:
983:
977:
974:
970:
966:
962:
958:
954:
950:
947:. A space is
946:
943:
940:
928:
925:. A space is
924:
920:
914:
911:
907:
903:
899:
895:
891:
888:. A space is
887:
884:
881:
877:
870:
866:
860:
849:
845:
842:. A space is
841:
837:
831:
820:
817:. A space is
816:
812:
806:
803:
799:
795:
791:
787:
783:
779:
775:
772:. A space is
771:
768:
749:
745:
741:
737:
733:
730:. A space is
729:
725:
719:
716:
712:
708:
704:
700:
696:
692:
689:. A space is
688:
684:
678:
677:
676:
674:
668:
643:
623:
600:
594:
587:
584:
569:
562:of the space
561:
542:
536:
529:
526:
511:
485:
479:
468:
453:
446:of the space
430:
420:
416:
415:
413:
384:
381:
374:has property
360:
354:
337:
334:
330:
322:the subspace
309:
306:
303:
300:
270:
267:
256:
253:
239:
236:
229:has property
215:
209:
192:
189:
185:
177:
161:
158:
155:
152:
122:
119:
108:
105:
104:
103:
89:
75:
73:
69:
64:
62:
58:
54:
50:
46:
42:
38:
34:
30:
26:
22:
3134:
3112:
3089:
3042:math/0609092
3032:
3026:
3016:
2904:
2899:Editing help
2881:
2672:completeness
2665:
2532:
2417:
2229:
2164:
2157:
2148:Stone spaces
2139:
2129:
2123:
2111:
2097:
2089:
2085:
2081:
2001:
1997:
1993:
1985:
1981:
1971:
1965:
1953:
1949:
1936:
1926:
1818:metric space
1809:
1796:
1792:
1788:
1784:
1774:
1764:
1750:
1740:
1730:
1720:
1710:
1701:quasicompact
1700:
1696:
1684:
1663:
1659:
1649:
1639:
1633:
1630:identity map
1626:contractible
1621:
1618:Contractible
1617:
1607:
1603:
1599:
1595:
1587:
1583:
1577:
1573:
1569:
1561:
1557:
1504:
1500:
1376:
1372:
1368:
1364:
1360:
1356:
1348:
1344:
1334:
1328:
1324:
1320:
1316:
1312:
1308:
1304:
1300:
1296:
1292:
1288:
1284:
1276:
1272:
1262:
1252:
1238:
1208:
1207:
1193:
1186:if it has a
1179:
1165:
1158:if it has a
1151:
1128:
1118:
1101:
1097:
1090:
1076:
1066:
1059:
1045:
1035:
1028:
1014:
994:Completely T
993:
989:
985:
978:
968:
964:
960:
956:
952:
951:if whenever
944:
922:
915:
909:
905:
901:
897:
893:
892:if whenever
885:
873:completely T
868:
862:Completely T
861:
847:
839:
832:
814:
807:
801:
797:
793:
789:
785:
781:
777:
769:
747:
743:
739:
735:
727:
720:
714:
710:
706:
702:
698:
694:
686:
679:
670:
585:
527:
254:
106:
81:
71:
68:homeomorphic
65:
56:
52:
49:proper class
32:
28:
18:
3035:(1): 1–16.
2935:Chern class
2819:. However,
2668:boundedness
2184:-resolvable
1990:homogeneous
1958:Baire space
1950:Baire space
1769:paracompact
1765:Paracompact
1679:Compactness
1586:. A space
1560:. A space
1379:, i.e., an
1247:clopen sets
1176:local base.
1102:perfectly T
1052:if any two
419:cardinality
145:and subset
82:A property
25:mathematics
3146:Categories
3080:References
2957:cohomology
2907:March 2017
2583:such that
2094:Alexandrov
2088:. A space
2086:Alexandrov
2036:such that
1984:. A space
1976:door space
1972:Door space
1952:. A space
1814:metrizable
1810:Metrizable
1693:open cover
1668:indiscrete
1660:Indiscrete
1620:. A space
1598:such that
1503:. A space
1347:. A space
1275:. A space
1144:See also:
939:consistent
691:Kolmogorov
687:Kolmogorov
661:Separation
107:Hereditary
3061:0021-2172
3008:Citations
2804:→
2798::
2772:≅
2740:π
2725:π
2719:−
2568:≅
2371:Δ
2332:∅
2329:≠
2289:Δ
2260:Δ
2214:κ
2194:κ
2172:κ
2021:→
2015::
1761:subcover.
1759:countable
1745:σ-compact
1741:σ-compact
1705:Hausdorff
1691:if every
1552:homotopic
1535:→
1522::
1412:→
1394::
1381:injective
1243:connected
1239:Connected
1226:subspaces
1219:countably
1215:σ-compact
1204:subcover.
1202:countable
1188:countable
1174:countable
1160:countable
1156:separable
1152:Separable
986:Tychonoff
819:Hausdorff
815:Hausdorff
796:} equals
480:τ
304:⊆
156:⊆
61:open sets
41:invariant
3113:Topology
3111:(2000).
3069:14743623
2953:Homology
2863:See also
2232:. Space
1797:monolith
1755:Lindelöf
1751:Lindelöf
1697:subcover
1602:loop in
1211:-compact
1198:Lindelöf
1194:Lindelöf
1123:discrete
746:but not
713:but not
705:but not
176:subspace
39:that is
21:topology
2520:is the
2363:Number
2144:Boolean
2140:Boolean
1689:compact
1685:Compact
1664:trivial
1628:if the
1223:compact
963:, then
904:, then
890:regular
886:Regular
844:Urysohn
840:Urysohn
732:Fréchet
728:Fréchet
586:Density
3123:
3097:
3067:
3059:
2882:is in
2795:arctan
2623:, but
2420:. Set
1962:meagre
1931:Polish
1927:Polish
1327:(1) =
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