2884:
3146:
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).
2197:. 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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if it is zero-dimensional, compact and
Hausdorff (equivalently, totally disconnected, compact and Hausdorff). These are precisely the spaces that are homeomorphic to the
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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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spaces where every open cover has finite subcover. Compact spaces are always Lindelöf and paracompact. Compact
Hausdorff spaces are therefore normal.
2295:
1352:
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.
1621:. Semi-local simple connectivity, a strictly weaker condition than local simple connectivity, is a necessary condition for the existence of a
1831:. Metrizable spaces are always Hausdorff and paracompact (and hence normal and Tychonoff), and first-countable. Moreover, a topological space
62:
of topological spaces which is closed under homeomorphisms. That is, a property of spaces is a topological property if whenever a space
3162:
2147:
if every open set is closed (hence clopen). The almost discrete spaces are precisely the finitely generated zero-dimensional spaces.
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17:
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possesses that property. Informally, a topological property is a property of the space that can be expressed using
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if it is not the union of a pair of disjoint non-empty open sets. Equivalently, a space is connected if the only
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1602:
765:; here, we are allowed to specify which point will be contained in the open set.) Equivalently, a space is T
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3148:
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271:
123:
3005:
883:
2990: – Numerical invariant that describes the linking of two closed curves in three-dimensional space
3008: – Physical quantities that take discrete values because of topological quantum physical effects
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2018:
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if every open cover has an open locally finite refinement. Paracompact
Hausdorff spaces are normal.
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base for its topology. Second-countable spaces are always separable, first-countable and Lindelöf.
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are open, or equivalently if arbitrary unions of closed sets are closed. These are precisely the
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is not the (possibly nondisjoint) union of two smaller closed non-empty subsets, then there is a
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itself. Non-empty ultra-connected compact spaces have a largest proper open subset called a
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if no two non-empty closed sets are disjoint. Every ultraconnected space is path-connected.
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homeomorphic, it is sufficient to find a topological property which is not shared by them.
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if the only open sets are the empty set and itself. Such a space is said to have the
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if no two non-empty open sets are disjoint. Every hyperconnected space is connected.
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2085:
Intuitively speaking, this means that the space looks the same at every point. All
1824:
1682:
1678:
1519:
1022:, so the terminology is consistent.) Tychonoff spaces are always regular Hausdorff.
701:
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if any two disjoint closed sets have disjoint neighbourhoods. Normal spaces admit
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Some of these terms are defined differently in older mathematical literature; see
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is not topological, it is sufficient to find two homeomorphic topological spaces
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1715:
1622:
1253:
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959:
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is a Baire space if the intersection of countably many dense open sets is dense.
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is homotopic to a constant map. Contractible spaces are always simply connected.
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1950:. A space is locally metrizable if every point has a metrizable neighbourhood.
1126:. A perfectly normal Hausdorff space must also be completely normal Hausdorff.
3156:
3127:
3119:
3071:
2367:{\displaystyle \Delta (X)=\min\{|G|:G\neq \varnothing ,G{\mbox{ is open}}\}.}
2154:
2129:
if it has a base of clopen sets. These are precisely the spaces with a small
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A common problem in topology is to decide whether two topological spaces are
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1941:
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1029:
78:
2984: – Function of a knot that takes the same value for equivalent knots
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1968:
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if all of its points are completely isolated, i.e. if any subset is open.
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1986:
1972:
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space. (A completely regular space is
Hausdorff if and only if it is T
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3149:
https://iopscience.iop.org/article/10.1088/0953-4075/46/10/104005/pdf
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3014: – Number of times a curve wraps around a point in the plane
1748:
if every continuous real-valued function on the space is bounded.
1084:. Completely normal Hausdorff spaces are always normal Hausdorff.
2548:, etc, which are not topological properties. To show a property
2511:
is said to be strongly discrete if every non-isolated point of
1080:. A completely normal space is Hausdorff if and only if it is T
2939: – Association of cohomology classes to principal bundles
2933: – Association of cohomology classes to principal bundles
712:
in the space, there is at least either an open set containing
3032:
Juhász, István; Soukup, Lajos; Szentmiklóssy, Zoltán (2008).
1270:
if every point has a local base consisting of connected sets.
944:
space. (A regular space is
Hausdorff if and only if it is T
832:
if every two distinct points have disjoint neighbourhoods. T
2491:
may be separated by pairwise disjoint neighborhoods. Space
2996: – List of concrete topologies and topological spaces
2765:{\displaystyle Y=(-{\tfrac {\pi }{2}},{\tfrac {\pi }{2}})}
1098:. A perfectly normal space must also be completely normal.
1944:
if it is metrizable with a separable and complete metric.
3104:. Reading, Mass.: Addison-Wesley Pub. Co. p. 369.
2941:
Pages displaying short descriptions of redirect targets
1280:
if it has no connected subset with more than one point.
515:
of the topology (the set of open subsets) of the space
423:
Cardinal function § Cardinal functions in topology
88:
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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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2823:{\displaystyle \operatorname {arctan} \colon X\to Y}
1738:
if every countable open cover has a finite subcover.
1049:. A normal space is Hausdorff if and only if it is T
66:
possesses that property every space homeomorphic to
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378:{\displaystyle \left(S,{\mathcal {T}}|_{S}\right)}
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233:{\displaystyle \left(S,{\mathcal {T}}|_{S}\right)}
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105:
2772:be metric spaces with the standard metric. Then,
1605:if every point has a local base of neighborhoods
1522:if it is path-connected and every continuous map
3154:
2948: – Characteristic classes of vector bundles
2314:
411:
1728:if every sequence has a convergent subsequence.
1053:. Normal Hausdorff spaces are always Tychonoff.
893:. Every completely Hausdorff space is Urysohn.
753:in the space, there is an open set containing
2539:
1342:. Path-connected spaces are always connected.
58:. Alternatively, a topological property is a
2954: – Topological invariant in mathematics
2677:For example, the metric space properties of
2358:
2317:
1989:if every subset is open or closed (or both).
502:
487:
444:
438:
3002: – Concept in mathematical knot theory
2107:if arbitrary intersections of open sets in
857:if every two distinct points have disjoint
1508:. Arc-connected spaces are path-connected.
1150:
3061:
3051:
2699:
2544:There are many examples of properties of
2451:is strongly discrete subset of the space
3118:
3097:
627:, the least cardinality of a subset of
14:
3155:
3034:"Resolvability and monotone normality"
1798:. In an ultra-connected compact space
1122:, if it is both perfectly normal and T
815:is the only point with this property.
745:if for every pair of distinct points
704:if for every pair of distinct points
416:
81:or not. To prove that two spaces are
2877:
2685:are not topological properties. Let
1094:if any two disjoint closed sets are
508:{\displaystyle \vert \tau (X)\vert }
89:Properties of topological properties
2850:is complete but not bounded, while
769:if all its singletons are closed. T
24:
2403:is called dispersion character of
2381:
2299:
2270:
1587:has a local base of neighborhoods
1554:{\displaystyle f\colon S^{1}\to X}
353:
297:{\displaystyle (X,{\mathcal {T}})}
286:
208:
149:{\displaystyle (X,{\mathcal {T}})}
138:
25:
3179:
2342:
2263:is maximally resolvable if it is
1710:. Some authors call these spaces
1096:precisely separated by a function
889:if every two distinct points are
268:, if for every topological space
120:, if for every topological space
3163:Properties of topological spaces
2882:
1954:
1814:
1243:
787:if every irreducible closed set
684:history of the separation axioms
27:Mathematical property of a space
1912:is identical with the topology
3025:
2897:format but may read better as
2814:
2759:
2726:
2706:{\displaystyle X=\mathbb {R} }
2535:of some strongly discrete set.
2390:
2384:
2329:
2321:
2308:
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2273:
2217:-resolvable then it is called
2117:category of topological spaces
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2057:
2040:{\displaystyle f\colon X\to X}
2031:
1899:
1893:
1883:such that the metric topology
1850:
1838:
1802:every open cover must contain
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3039:Israel Journal of Mathematics
2960: – Mathematical property
2870:is bounded but not complete.
1603:semi-locally simply connected
1595:Semi-locally simply connected
1431:{\displaystyle f\colon \to X}
1260:are the empty set and itself.
1067:have disjoint neighbourhoods.
923:have disjoint neighbourhoods.
671:
569:, the least cardinality of a
450:{\displaystyle \vert X\vert }
412:Common topological properties
326:{\displaystyle S\subseteq X,}
178:{\displaystyle S\subseteq X,}
3018:
1001:Completely regular Hausdorff
720:, or an open set containing
7:
2873:
1985:. A topological space is a
1827:if it is homeomorphic to a
1078:Completely normal Hausdorff
791:has a unique generic point
10:
3184:
3006:Topological quantum number
2540:Non-topological properties
2396:{\displaystyle \Delta (X)}
2285:{\displaystyle \Delta (X)}
1768:if every open cover has a
1211:if every open cover has a
1154:
1120:perfectly normal Hausdorff
1109:Perfectly normal Hausdorff
803:such that the closure of {
675:
420:
3098:Willard, Stephen (1970).
3063:10.1007/s11856-008-1017-y
2015:there is a homeomorphism
1975:in itself. Equivalently,
1591:that is simply connected.
1318:, i.e., a continuous map
1140:Number of isolated points
1014:is a completely regular T
2791:{\displaystyle X\cong Y}
2587:{\displaystyle X\cong Y}
1714:and reserve compact for
1577:locally simply connected
1569:Locally simply connected
1366:if for every two points
1294:if for every two points
948:, so the terminology is
2906:converting this article
2230:{\displaystyle \kappa }
2210:{\displaystyle \kappa }
2188:{\displaystyle \kappa }
2078:{\displaystyle f(x)=y.}
1993:Topological Homogeneity
1151:Countability conditions
1146:of a topological space.
984:separated by a function
891:separated by a function
2937:Characteristic numbers
2864:
2844:
2824:
2798:via the homeomorphism
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1857:
1796:Ultraconnected compact
1555:
1502:
1501:{\displaystyle p(1)=y}
1467:
1466:{\displaystyle p(0)=x}
1432:
1350:locally path-connected
1346:Locally path-connected
1228:if it is the union of
795:. In other words, if
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1858:
1856:{\displaystyle (X,T)}
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1503:
1468:
1433:
1183:if every point has a
1157:Axiom of countability
662:
642:
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588:
571:basis of the topology
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530:
510:
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403:
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328:
299:
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180:
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44:topological invariant
34:and related areas of
18:Topological invariant
2958:Fixed-point property
2952:Euler characteristic
2931:Characteristic class
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2241:Maximally resolvable
2221:
2201:
2179:
2170:Reidemeister torsion
2119:and continuous maps.
2051:
2019:
1940:. A space is called
1916:
1905:{\displaystyle T(d)}
1887:
1867:
1835:
1726:sequentially compact
1722:Sequentially compact
1526:
1477:
1442:
1398:
1322:: →
1278:totally disconnected
1274:Totally disconnected
966:is a closed set and
940:if it is a regular T
907:is a closed set and
880:completely Hausdorff
651:
631:
620:{\displaystyle d(X)}
602:
577:
562:{\displaystyle w(X)}
544:
519:
484:
461:
435:
389:
337:
308:
272:
244:
192:
160:
124:
97:
40:topological property
2292:-resolvable, where
2131:inductive dimension
1999:is (topologically)
1617:is contractible in
1034:partitions of unity
865:spaces are always T
836:spaces are always T
773:spaces are always T
46:is a property of a
2994:List of topologies
2908:, if appropriate.
2860:
2840:
2820:
2788:
2762:
2757:
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2703:
2664:
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2558:
2533:accumulation point
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2501:
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2441:
2419:{\displaystyle X.}
2416:
2393:
2364:
2356:
2282:
2253:
2227:
2207:
2185:
2113:finitely generated
2093:Finitely generated
2087:topological groups
2075:
2037:
1948:Locally metrizable
1928:{\displaystyle T.}
1925:
1902:
1873:
1853:
1565:to a constant map.
1551:
1498:
1463:
1428:
1378:, there is an arc
1306:, there is a path
970:is a point not in
960:completely regular
956:Completely regular
911:is a point not in
657:
637:
617:
583:
559:
525:
505:
467:
447:
417:Cardinal functions
401:{\displaystyle P.}
398:
375:
323:
304:and closed subset
294:
256:{\displaystyle P.}
253:
230:
175:
146:
103:
3120:Munkres, James R.
3000:Quantum invariant
2927:
2926:
2863:{\displaystyle Y}
2843:{\displaystyle X}
2756:
2741:
2667:{\displaystyle P}
2647:{\displaystyle Y}
2627:{\displaystyle P}
2607:{\displaystyle X}
2561:{\displaystyle P}
2524:{\displaystyle X}
2504:{\displaystyle X}
2484:{\displaystyle D}
2471:if the points in
2464:{\displaystyle X}
2444:{\displaystyle D}
2429:Strongly discrete
2355:
2256:{\displaystyle X}
1876:{\displaystyle X}
1736:countably compact
1732:Countably compact
1268:locally connected
1264:Locally connected
1061:completely normal
1057:Completely normal
938:regular Hausdorff
934:Regular Hausdorff
861:neighbourhoods. T
761:. (Compare with T
660:{\displaystyle X}
647:whose closure is
640:{\displaystyle X}
586:{\displaystyle X}
528:{\displaystyle X}
470:{\displaystyle X}
266:Weakly hereditary
106:{\displaystyle P}
48:topological space
16:(Redirected from
3175:
3141:
3115:
3101:General topology
3084:
3083:
3065:
3055:
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2977:Cohomotopy group
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2904:You can help by
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2192:
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2186:
2163:Boolean algebras
2127:zero-dimensional
2123:Zero-dimensional
2089:are homogeneous.
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1934:
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1911:
1909:
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1683:trivial topology
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1520:simply connected
1512:Simply connected
1507:
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1504:
1499:
1472:
1470:
1469:
1464:
1437:
1435:
1434:
1429:
1195:second-countable
1191:Second-countable
1142:. The number of
1092:perfectly normal
1088:Perfectly normal
1047:Normal Hausdorff
678:Separation axiom
666:
664:
663:
658:
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480:The cardinality
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369:
368:
363:
357:
356:
332:
330:
329:
324:
303:
301:
300:
295:
290:
289:
262:
260:
259:
254:
239:
237:
236:
231:
229:
225:
224:
223:
218:
212:
211:
184:
182:
181:
176:
155:
153:
152:
147:
142:
141:
112:
110:
109:
104:
21:
3183:
3182:
3178:
3177:
3176:
3174:
3173:
3172:
3153:
3152:
3144:
3138:
3112:
3093:
3088:
3087:
3030:
3026:
3021:
2940:
2923:
2917:
2914:
2903:
2887:
2883:
2876:
2855:
2852:
2851:
2835:
2832:
2831:
2803:
2800:
2799:
2777:
2774:
2773:
2747:
2732:
2718:
2715:
2714:
2698:
2690:
2687:
2686:
2659:
2656:
2655:
2639:
2636:
2635:
2619:
2616:
2615:
2599:
2596:
2595:
2573:
2570:
2569:
2553:
2550:
2549:
2542:
2516:
2513:
2512:
2496:
2493:
2492:
2476:
2473:
2472:
2456:
2453:
2452:
2436:
2433:
2432:
2408:
2405:
2404:
2379:
2376:
2375:
2351:
2328:
2320:
2297:
2294:
2293:
2268:
2265:
2264:
2248:
2245:
2244:
2222:
2219:
2218:
2202:
2199:
2198:
2180:
2177:
2176:
2145:almost discrete
2141:Almost discrete
2115:members of the
2052:
2049:
2048:
2020:
2017:
2016:
1957:
1917:
1914:
1913:
1888:
1885:
1884:
1868:
1865:
1864:
1836:
1833:
1832:
1817:
1790:locally compact
1786:Locally compact
1692:
1623:universal cover
1579:if every point
1539:
1535:
1527:
1524:
1523:
1478:
1475:
1474:
1443:
1440:
1439:
1399:
1396:
1395:
1394:continuous map
1246:
1181:first-countable
1177:First-countable
1159:
1153:
1144:isolated points
1125:
1116:
1105:
1083:
1074:
1052:
1043:
1021:
1017:
1012:Tychonoff space
1008:
993:
947:
943:
930:
887:
876:
868:
864:
847:
839:
835:
822:
776:
772:
768:
764:
735:
694:
680:
674:
652:
649:
648:
632:
629:
628:
603:
600:
599:
578:
575:
574:
545:
542:
541:
520:
517:
516:
485:
482:
481:
462:
459:
458:
436:
433:
432:
425:
419:
414:
390:
387:
386:
364:
359:
358:
352:
351:
344:
340:
338:
335:
334:
309:
306:
305:
285:
284:
273:
270:
269:
245:
242:
241:
219:
214:
213:
207:
206:
199:
195:
193:
190:
189:
161:
158:
157:
137:
136:
125:
122:
121:
98:
95:
94:
91:
28:
23:
22:
15:
12:
11:
5:
3181:
3171:
3170:
3168:Homeomorphisms
3165:
3143:
3142:
3136:
3116:
3110:
3094:
3092:
3089:
3086:
3085:
3023:
3022:
3020:
3017:
3016:
3015:
3012:Winding number
3009:
3003:
2997:
2991:
2988:Linking number
2985:
2982:Knot invariant
2979:
2973:Homotopy group
2970:
2961:
2955:
2949:
2943:
2934:
2925:
2924:
2890:
2888:
2881:
2875:
2872:
2859:
2839:
2819:
2816:
2813:
2810:
2807:
2787:
2784:
2781:
2761:
2755:
2752:
2746:
2740:
2737:
2731:
2728:
2725:
2722:
2701:
2697:
2694:
2663:
2654:does not have
2643:
2623:
2603:
2583:
2580:
2577:
2557:
2541:
2538:
2537:
2536:
2520:
2500:
2480:
2460:
2440:
2426:
2415:
2412:
2392:
2389:
2386:
2383:
2363:
2360:
2350:
2347:
2344:
2341:
2338:
2335:
2331:
2327:
2323:
2319:
2316:
2313:
2310:
2307:
2304:
2301:
2281:
2278:
2275:
2272:
2252:
2238:
2237:-irresolvable.
2226:
2206:
2184:
2173:
2166:
2148:
2138:
2120:
2090:
2074:
2071:
2068:
2065:
2062:
2059:
2056:
2036:
2033:
2030:
2027:
2024:
1990:
1980:
1956:
1953:
1952:
1951:
1945:
1935:
1924:
1921:
1901:
1898:
1895:
1892:
1872:
1852:
1849:
1846:
1843:
1840:
1816:
1813:
1812:
1811:
1793:
1783:
1773:
1759:
1749:
1739:
1729:
1719:
1691:
1688:
1687:
1686:
1668:
1665:ultraconnected
1661:Ultraconnected
1658:
1655:hyperconnected
1651:Hyperconnected
1648:
1626:
1592:
1566:
1550:
1547:
1542:
1538:
1534:
1531:
1509:
1497:
1494:
1491:
1488:
1485:
1482:
1462:
1459:
1456:
1453:
1450:
1447:
1427:
1424:
1421:
1418:
1415:
1412:
1409:
1406:
1403:
1353:
1343:
1292:path-connected
1284:Path-connected
1281:
1271:
1261:
1245:
1242:
1241:
1240:
1216:
1202:
1188:
1174:
1152:
1149:
1148:
1147:
1137:
1130:Discrete space
1127:
1123:
1114:
1103:
1099:
1085:
1081:
1072:
1068:
1065:separated sets
1054:
1050:
1041:
1037:
1023:
1019:
1015:
1006:
991:
987:
953:
945:
941:
928:
924:
894:
885:
882:. A space is
874:
870:
866:
862:
845:
841:
837:
833:
820:
816:
778:
774:
770:
766:
762:
733:
729:
692:
676:Main article:
673:
670:
669:
668:
656:
636:
616:
613:
610:
607:
594:
582:
558:
555:
552:
549:
536:
524:
504:
501:
498:
495:
492:
489:
478:
466:
446:
443:
440:
421:Main article:
418:
415:
413:
410:
409:
408:
397:
394:
373:
367:
362:
355:
350:
347:
343:
322:
319:
316:
313:
293:
288:
283:
280:
277:
263:
252:
249:
228:
222:
217:
210:
205:
202:
198:
174:
171:
168:
165:
145:
140:
135:
132:
129:
102:
90:
87:
56:homeomorphisms
26:
9:
6:
4:
3:
2:
3180:
3169:
3166:
3164:
3161:
3160:
3158:
3151:
3150:
3139:
3137:0-13-181629-2
3133:
3129:
3128:Prentice-Hall
3125:
3121:
3117:
3113:
3111:9780486434797
3107:
3103:
3102:
3096:
3095:
3081:
3077:
3073:
3069:
3064:
3059:
3054:
3049:
3045:
3041:
3040:
3035:
3028:
3024:
3013:
3010:
3007:
3004:
3001:
2998:
2995:
2992:
2989:
2986:
2983:
2980:
2978:
2974:
2971:
2969:
2965:
2962:
2959:
2956:
2953:
2950:
2947:
2944:
2938:
2935:
2932:
2929:
2928:
2921:
2912:is available.
2911:
2907:
2901:
2900:
2896:
2891:This article
2889:
2880:
2879:
2871:
2857:
2837:
2817:
2811:
2808:
2805:
2785:
2782:
2779:
2753:
2750:
2744:
2738:
2735:
2729:
2723:
2720:
2695:
2692:
2684:
2680:
2675:
2661:
2641:
2621:
2601:
2581:
2578:
2575:
2555:
2547:
2546:metric spaces
2534:
2518:
2498:
2478:
2458:
2438:
2430:
2427:
2413:
2410:
2387:
2361:
2354: is open
2348:
2345:
2339:
2336:
2333:
2325:
2311:
2305:
2276:
2250:
2242:
2239:
2224:
2204:
2196:
2182:
2174:
2172:
2171:
2167:
2164:
2160:
2156:
2153:. A space is
2152:
2149:
2146:
2143:. A space is
2142:
2139:
2136:
2132:
2128:
2125:. A space is
2124:
2121:
2118:
2114:
2110:
2106:
2102:
2098:
2094:
2091:
2088:
2072:
2069:
2066:
2060:
2054:
2034:
2028:
2025:
2022:
2014:
2010:
2006:
2003:if for every
2002:
1998:
1994:
1991:
1988:
1984:
1981:
1978:
1974:
1971:if it is not
1970:
1966:
1962:
1959:
1958:
1955:Miscellaneous
1949:
1946:
1943:
1939:
1936:
1922:
1919:
1896:
1890:
1870:
1847:
1844:
1841:
1830:
1826:
1823:. A space is
1822:
1819:
1818:
1815:Metrizability
1809:
1805:
1801:
1797:
1794:
1791:
1788:. A space is
1787:
1784:
1781:
1778:. A space is
1777:
1774:
1771:
1767:
1764:. A space is
1763:
1760:
1757:
1754:. A space is
1753:
1750:
1747:
1746:pseudocompact
1744:. A space is
1743:
1742:Pseudocompact
1740:
1737:
1734:. A space is
1733:
1730:
1727:
1724:. A space is
1723:
1720:
1717:
1713:
1709:
1706:has a finite
1705:
1701:
1698:. A space is
1697:
1694:
1693:
1684:
1680:
1677:. A space is
1676:
1672:
1669:
1666:
1663:. A space is
1662:
1659:
1656:
1653:. A space is
1652:
1649:
1646:
1642:
1638:
1634:
1630:
1627:
1624:
1620:
1616:
1612:
1608:
1604:
1600:
1596:
1593:
1590:
1586:
1582:
1578:
1574:
1570:
1567:
1564:
1548:
1540:
1536:
1532:
1529:
1521:
1517:
1513:
1510:
1495:
1492:
1486:
1480:
1460:
1457:
1451:
1445:
1425:
1416:
1413:
1410:
1404:
1401:
1393:
1389:
1385:
1381:
1377:
1373:
1369:
1365:
1364:arc-connected
1361:
1357:
1356:Arc-connected
1354:
1351:
1348:. A space is
1347:
1344:
1341:
1337:
1333:
1329:
1325:
1321:
1317:
1313:
1309:
1305:
1301:
1297:
1293:
1289:
1285:
1282:
1279:
1276:. A space is
1275:
1272:
1269:
1266:. A space is
1265:
1262:
1259:
1255:
1252:. A space is
1251:
1248:
1247:
1244:Connectedness
1238:
1235:
1231:
1227:
1224:. A space is
1223:
1221:
1217:
1214:
1210:
1207:. A space is
1206:
1203:
1200:
1196:
1193:. A space is
1192:
1189:
1186:
1182:
1179:. A space is
1178:
1175:
1173:dense subset.
1172:
1168:
1165:. A space is
1164:
1161:
1160:
1158:
1145:
1141:
1138:
1135:
1132:. A space is
1131:
1128:
1121:
1118:. A space is
1117:
1110:
1106:
1100:
1097:
1093:
1090:. A space is
1089:
1086:
1079:
1075:
1069:
1066:
1062:
1059:. A space is
1058:
1055:
1048:
1044:
1038:
1035:
1031:
1028:. A space is
1027:
1024:
1013:
1009:
1002:
998:
994:
988:
985:
981:
977:
973:
969:
965:
961:
958:. A space is
957:
954:
951:
939:
936:. A space is
935:
931:
925:
922:
918:
914:
910:
906:
902:
899:. A space is
898:
895:
892:
888:
881:
877:
871:
860:
856:
853:. A space is
852:
848:
842:
831:
828:. A space is
827:
823:
817:
814:
810:
806:
802:
798:
794:
790:
786:
783:. A space is
782:
779:
760:
756:
752:
748:
744:
741:. A space is
740:
736:
730:
727:
723:
719:
715:
711:
707:
703:
700:. A space is
699:
695:
689:
688:
687:
685:
679:
654:
634:
611:
605:
598:
595:
580:
573:of the space
572:
553:
547:
540:
537:
522:
496:
490:
479:
464:
457:of the space
441:
431:
427:
426:
424:
395:
392:
385:has property
371:
365:
348:
345:
341:
333:the subspace
320:
317:
314:
311:
281:
278:
267:
264:
250:
247:
240:has property
226:
220:
203:
200:
196:
188:
172:
169:
166:
163:
133:
130:
119:
116:
115:
114:
100:
86:
84:
80:
75:
73:
69:
65:
61:
57:
53:
49:
45:
41:
37:
33:
19:
3145:
3123:
3100:
3053:math/0609092
3043:
3037:
3027:
2915:
2910:Editing help
2892:
2683:completeness
2676:
2543:
2428:
2240:
2175:
2168:
2159:Stone spaces
2150:
2140:
2134:
2122:
2108:
2100:
2096:
2092:
2012:
2008:
2004:
1996:
1992:
1982:
1976:
1964:
1960:
1947:
1937:
1829:metric space
1820:
1807:
1803:
1799:
1795:
1785:
1775:
1761:
1751:
1741:
1731:
1721:
1712:quasicompact
1711:
1707:
1695:
1674:
1670:
1660:
1650:
1644:
1641:identity map
1637:contractible
1632:
1629:Contractible
1628:
1618:
1614:
1610:
1606:
1598:
1594:
1588:
1584:
1580:
1572:
1568:
1515:
1511:
1387:
1383:
1379:
1375:
1371:
1367:
1359:
1355:
1345:
1339:
1335:
1331:
1327:
1323:
1319:
1315:
1311:
1307:
1303:
1299:
1295:
1287:
1283:
1273:
1263:
1249:
1219:
1218:
1204:
1197:if it has a
1190:
1176:
1169:if it has a
1162:
1139:
1129:
1112:
1108:
1101:
1087:
1077:
1070:
1056:
1046:
1039:
1025:
1005:Completely T
1004:
1000:
996:
989:
979:
975:
971:
967:
963:
962:if whenever
955:
933:
926:
920:
916:
912:
908:
904:
903:if whenever
896:
884:completely T
879:
873:Completely T
872:
858:
850:
843:
825:
818:
812:
808:
804:
800:
796:
792:
788:
780:
758:
754:
750:
746:
738:
731:
725:
721:
717:
713:
709:
705:
697:
690:
681:
596:
538:
265:
117:
92:
82:
79:homeomorphic
76:
67:
63:
60:proper class
43:
39:
29:
3046:(1): 1–16.
2946:Chern class
2830:. However,
2679:boundedness
2195:-resolvable
2001:homogeneous
1969:Baire space
1961:Baire space
1780:paracompact
1776:Paracompact
1690:Compactness
1597:. A space
1571:. A space
1390:, i.e., an
1258:clopen sets
1187:local base.
1113:perfectly T
1063:if any two
430:cardinality
156:and subset
93:A property
36:mathematics
3157:Categories
3091:References
2968:cohomology
2918:March 2017
2594:such that
2105:Alexandrov
2099:. A space
2097:Alexandrov
2047:such that
1995:. A space
1987:door space
1983:Door space
1963:. A space
1825:metrizable
1821:Metrizable
1704:open cover
1679:indiscrete
1671:Indiscrete
1631:. A space
1609:such that
1514:. A space
1358:. A space
1286:. A space
1155:See also:
950:consistent
702:Kolmogorov
698:Kolmogorov
672:Separation
118:Hereditary
3072:0021-2172
3019:Citations
2815:→
2809::
2783:≅
2751:π
2736:π
2730:−
2579:≅
2382:Δ
2343:∅
2340:≠
2300:Δ
2271:Δ
2225:κ
2205:κ
2183:κ
2032:→
2026::
1772:subcover.
1770:countable
1756:σ-compact
1752:σ-compact
1716:Hausdorff
1702:if every
1563:homotopic
1546:→
1533::
1423:→
1405::
1392:injective
1254:connected
1250:Connected
1237:subspaces
1230:countably
1226:σ-compact
1215:subcover.
1213:countable
1199:countable
1185:countable
1171:countable
1167:separable
1163:Separable
997:Tychonoff
830:Hausdorff
826:Hausdorff
807:} equals
491:τ
315:⊆
167:⊆
72:open sets
52:invariant
3124:Topology
3122:(2000).
3080:14743623
2964:Homology
2874:See also
2243:. Space
1808:monolith
1766:Lindelöf
1762:Lindelöf
1708:subcover
1613:loop in
1222:-compact
1209:Lindelöf
1205:Lindelöf
1134:discrete
757:but not
724:but not
716:but not
187:subspace
50:that is
32:topology
2531:is the
2374:Number
2155:Boolean
2151:Boolean
1700:compact
1696:Compact
1675:trivial
1639:if the
1234:compact
974:, then
915:, then
901:regular
897:Regular
855:Urysohn
851:Urysohn
743:Fréchet
739:Fréchet
597:Density
3134:
3108:
3078:
3070:
2893:is in
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