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only closed sets are capable of being precisely separated by a function, but just because two sets are closed and separated by a function does not mean that they are automatically precisely separated by a function (even a different function).
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closed by including the point 1 in it, but you cannot make them both closed while keeping them disjoint. Note that if any two sets are separated by closed neighbourhoods, then certainly they are
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are various conditions that are sometimes imposed upon topological spaces, many of which can be described in terms of the various types of separated sets. As an example we will define the T
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that are related to each other in a certain way: roughly speaking, neither overlapping nor touching. The notion of when two sets are separated or not is important both to the notion of
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must be topologically distinguishable. Thus for singletons, topological distinguishability is a condition in between disjointness and separatedness.
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The properties below are presented in increasing order of specificity, each being a stronger notion than the preceding one.
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and again it makes no difference.) Note that if any two sets are precisely separated by a function, then they are
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can be considered to be separated. A most basic way in which two sets can be separated is if they are
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Note that if any two sets are separated by neighbourhoods, then certainly they are separated. If
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are separated when they are disjoint and each is disjoint from the other's derived set, that is,
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even though the point 1 belongs to both of their closures. A more general example is that in any
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axiom, which is the condition imposed on separated spaces. Specifically, a topological space is
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at the point 1. If two sets are separated by a continuous function, then they are also
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are open and disjoint, then they must be separated by neighbourhoods; just take
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neighbourhoods, but this makes no difference in the end.) For the example of
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if these are the only two possibilities. Conversely, if a nonempty subset
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For this reason, separatedness is often used with closed sets (as in the
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in this definition, but this makes no difference.) In our example,
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The property of being separated can also be expressed in terms of
602:. This property has nothing to do with topology as such, but only
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744:{\displaystyle A\cap {\bar {B}}=\varnothing ={\bar {A}}\cap B.}
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2958:, but there may be other possibilities. A topological space
516:(defined below), which are somewhat related but different.
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3150: – Axioms in topology defining notions of "separation"
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that one point belongs to but the other point does not. If
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are disjoint. (Sometimes you will see the requirement that
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separated by closed neighbourhoods. You could make either
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2754:(Again, you may also see the unit interval in place of
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is separated from its own complement, and if the only
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have to be disjoint from each other; for example, the
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are again a completely different topological concept.
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Type of relation for subsets of a topological space
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3061:Relation to topologically distinguishable points
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318:but its sources remain unclear because it lacks
1013:{\displaystyle B_{s}(q)=\{x\in X:d(q,x)<s\}}
932:{\displaystyle B_{r}(p)=\{x\in X:d(p,x)<r\}}
2978:to share this property is the empty set, then
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3099:are topologically distinguishable, then the
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2623:precisely separated by a continuous function
1164:{\textstyle A'\cap B=\varnothing =B'\cap A.}
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528:There are various ways in which two subsets
3057:is an open-connected component of itself.)
512:Separated sets should not be confused with
284:Learn how and when to remove these messages
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3187:(2 ed.). Prentice Hall. p. 211.
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349:Learn how and when to remove this message
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2627:if there exists a continuous function
29:
755:Hausdorff−Lennes Separation Condition
680:if each is disjoint from the other's
7:
2910:Separated spaces are usually called
2194:{\displaystyle B\subseteq f^{-1}(1)}
2149:{\displaystyle A\subseteq f^{-1}(0)}
389:adding citations to reliable sources
2907:} are separated by neighbourhoods.
2654:{\displaystyle f:X\to \mathbb {R} }
2062:{\displaystyle f:X\to \mathbb {R} }
3044:
3024:
3004:
2379:separated by closed neighbourhoods
2028:separated by a continuous function
1743:separated by closed neighbourhoods
25:
3139: – Type of topological space
3119:} are separated, then the points
1074:(indicated by the prime symbol):
265:This article has multiple issues.
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3017:, authorities differ on whether
2990:. (In the degenerate case where
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3264:Foundations of General Topology
1063:{\displaystyle d(p,q)\geq r+s.}
376:needs additional citations for
273:or discuss these issues on the
3067:Topological distinguishability
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753:This property is known as the
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1:
3085:topologically distinguishable
2853:{\displaystyle \mathbb {R} ,}
2772:{\displaystyle \mathbb {R} ,}
844:{\displaystyle \mathbb {R} ,}
2950:. This is certainly true if
2928:Relation to connected spaces
2747:{\displaystyle B=f^{-1}(1).}
2291:{\displaystyle \mathbb {R} }
2104:{\displaystyle \mathbb {R} }
1871:are disjoint. Our examples,
3262:Pervin, William J. (1964),
2699:{\displaystyle A=f^{-1}(0)}
1979:separated by neighbourhoods
1791:and a closed neighbourhood
1268:separated by neighbourhoods
18:Separated by neighbourhoods
3702:
3548:Banach fixed-point theorem
3071:Given a topological space
3064:
3050:{\displaystyle \emptyset }
3030:{\displaystyle \emptyset }
3010:{\displaystyle \emptyset }
2938:Given a topological space
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2868:
3581:
3377:
3239:Willard, Stephen (2004).
3037:is connected and whether
2946:to be separated from its
2518:{\displaystyle V=f^{-1},}
2241:map to 1. (Sometimes the
235:
2984:open-connected component
2452:{\displaystyle U=f^{-1}}
2221:map to 0 and members of
1590:{\displaystyle V=(1,3).}
1549:{\displaystyle U=(-1,1)}
1508:{\displaystyle B=(1,2],}
509:for topological spaces.
485:and related branches of
304:This article includes a
3143:Locally Hausdorff space
2781:separated by a function
1694:normal separation axiom
1472:{\displaystyle A=[0,1)}
1020:are separated whenever
568:of a topological space
333:more precise citations.
3603:Mathematics portal
3503:Metrics and properties
3489:Second-countable space
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2899:, the singleton sets {
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1926:{\displaystyle (1,2],}
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138:(completely Hausdorff)
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2828:{\displaystyle \{1\}}
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2802:{\displaystyle \{0\}}
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2350:{\displaystyle (1,2]}
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816:{\displaystyle (1,2]}
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789:{\displaystyle [0,1)}
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3558:Invariance of domain
3510:Euler characteristic
3484:Bundle (mathematics)
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2570:{\displaystyle 1/2.}
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2547:positive real number
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2276:is used in place of
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1685:{\displaystyle V=B.}
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385:improve this article
3568:Tychonoff's theorem
3563:Poincaré conjecture
3317:General (point-set)
3087:if there exists an
2034:continuous function
1656:{\displaystyle U=A}
156:(regular Hausdorff)
3553:De Rham cohomology
3474:Polyhedral complex
3464:Simplicial complex
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2887:if, given any two
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1214:{\displaystyle B'}
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306:list of references
209:(completely normal
191:(normal Hausdorff)
39:topological spaces
3681:Separation axioms
3668:
3667:
3457:fundamental group
3214:Munkres, James R.
3181:Munkres, James R.
2877:separation axioms
2612:{\displaystyle B}
2592:{\displaystyle A}
2538:{\displaystyle c}
2398:{\displaystyle f}
2370:{\displaystyle f}
2234:{\displaystyle B}
2214:{\displaystyle A}
2089:to the real line
2082:{\displaystyle X}
2017:{\displaystyle B}
1997:{\displaystyle A}
1970:{\displaystyle V}
1950:{\displaystyle U}
1864:{\displaystyle V}
1844:{\displaystyle U}
1824:{\displaystyle B}
1804:{\displaystyle V}
1784:{\displaystyle A}
1764:{\displaystyle U}
1732:{\displaystyle B}
1712:{\displaystyle A}
1630:{\displaystyle B}
1610:{\displaystyle A}
1428:{\displaystyle V}
1408:{\displaystyle U}
1388:{\displaystyle V}
1368:{\displaystyle U}
1348:{\displaystyle B}
1328:{\displaystyle V}
1308:{\displaystyle A}
1288:{\displaystyle U}
1257:{\displaystyle B}
1237:{\displaystyle A}
1107:{\displaystyle B}
1087:{\displaystyle A}
729:
708:
673:{\displaystyle X}
645:{\displaystyle B}
625:{\displaystyle A}
581:{\displaystyle X}
561:{\displaystyle B}
541:{\displaystyle A}
507:separation axioms
499:topological space
479:
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229:(perfectly normal
34:Separation axioms
16:(Redirected from
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3266:, Academic Press
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3241:General Topology
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3148:Separation axiom
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2913:Hausdorff spaces
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2458:
2456:
2455:
2450:
2430:
2429:
2404:
2402:
2401:
2396:
2376:
2374:
2373:
2368:
2356:
2354:
2353:
2348:
2327:
2326:
2321:
2297:
2295:
2294:
2289:
2287:
2275:
2273:
2272:
2269:{\displaystyle }
2267:
2240:
2238:
2237:
2232:
2220:
2218:
2217:
2212:
2200:
2198:
2197:
2192:
2181:
2180:
2155:
2153:
2152:
2147:
2136:
2135:
2110:
2108:
2107:
2102:
2100:
2088:
2086:
2085:
2080:
2068:
2066:
2065:
2060:
2058:
2030:
2029:
2023:
2021:
2020:
2015:
2003:
2001:
2000:
1995:
1976:
1974:
1973:
1968:
1956:
1954:
1953:
1948:
1932:
1930:
1929:
1924:
1900:
1899:
1894:
1870:
1868:
1867:
1862:
1850:
1848:
1847:
1842:
1830:
1828:
1827:
1822:
1810:
1808:
1807:
1802:
1790:
1788:
1787:
1782:
1770:
1768:
1767:
1762:
1745:
1744:
1738:
1736:
1735:
1730:
1718:
1716:
1715:
1710:
1691:
1689:
1688:
1683:
1662:
1660:
1659:
1654:
1636:
1634:
1633:
1628:
1616:
1614:
1613:
1608:
1596:
1594:
1593:
1588:
1555:
1553:
1552:
1547:
1514:
1512:
1511:
1506:
1476:
1475:
1470:
1434:
1432:
1431:
1426:
1414:
1412:
1411:
1406:
1394:
1392:
1391:
1386:
1374:
1372:
1371:
1366:
1354:
1352:
1351:
1346:
1334:
1332:
1331:
1326:
1314:
1312:
1311:
1306:
1294:
1292:
1291:
1286:
1270:
1269:
1263:
1261:
1260:
1255:
1243:
1241:
1240:
1235:
1220:
1218:
1217:
1212:
1210:
1195:
1193:
1192:
1187:
1185:
1170:
1168:
1167:
1162:
1151:
1128:
1113:
1111:
1110:
1105:
1093:
1091:
1090:
1085:
1069:
1067:
1066:
1061:
1019:
1017:
1016:
1011:
955:
954:
938:
936:
935:
930:
874:
873:
850:
848:
847:
842:
837:
822:
820:
819:
814:
793:
792:
787:
750:
748:
747:
742:
731:
730:
722:
710:
709:
701:
679:
677:
676:
671:
658:
657:
651:
649:
648:
643:
631:
629:
628:
623:
587:
585:
584:
579:
567:
565:
564:
559:
547:
545:
544:
539:
518:Separable spaces
514:separated spaces
503:connected spaces
474:
467:
456:
449:
445:
442:
436:
434:
400:"Separated sets"
393:
369:
361:
354:
347:
343:
340:
334:
329:this article by
320:inline citations
299:
298:
291:
280:
258:
257:
250:
231: Hausdorff)
226:
221:
211: Hausdorff)
206:
201:
188:
183:
168:
167:
153:
148:
135:
130:
115:
114:
99:
94:
81:
76:
63:
58:
30:
21:
3701:
3700:
3696:
3695:
3694:
3692:
3691:
3690:
3671:
3670:
3669:
3664:
3595:
3577:
3573:Urysohn's lemma
3534:
3498:
3384:
3375:
3347:low-dimensional
3305:
3300:
3270:
3261:
3255:
3238:
3232:
3212:
3208:
3203:
3202:
3195:
3179:
3178:
3174:
3166:
3162:
3157:
3137:Hausdorff space
3133:
3069:
3063:
3039:
3038:
3019:
3018:
2999:
2998:
2936:
2934:Connected space
2930:
2921:
2882:
2873:
2867:
2837:
2836:
2811:
2810:
2785:
2784:
2756:
2755:
2719:
2708:
2707:
2674:
2663:
2662:
2629:
2628:
2622:
2621:
2601:
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2581:
2580:
2551:
2550:
2527:
2526:
2472:
2461:
2460:
2418:
2407:
2406:
2387:
2386:
2359:
2358:
2300:
2299:
2278:
2277:
2246:
2245:
2223:
2222:
2203:
2202:
2169:
2158:
2157:
2124:
2113:
2112:
2091:
2090:
2071:
2070:
2069:from the space
2037:
2036:
2027:
2026:
2006:
2005:
1986:
1985:
1959:
1958:
1939:
1938:
1873:
1872:
1853:
1852:
1833:
1832:
1813:
1812:
1793:
1792:
1773:
1772:
1753:
1752:
1742:
1741:
1721:
1720:
1701:
1700:
1665:
1664:
1639:
1638:
1619:
1618:
1599:
1598:
1558:
1557:
1517:
1516:
1515:you could take
1443:
1442:
1417:
1416:
1397:
1396:
1377:
1376:
1357:
1356:
1337:
1336:
1317:
1316:
1297:
1296:
1277:
1276:
1267:
1266:
1246:
1245:
1226:
1225:
1203:
1198:
1197:
1178:
1173:
1172:
1144:
1121:
1116:
1115:
1096:
1095:
1076:
1075:
1022:
1021:
946:
941:
940:
865:
860:
859:
828:
827:
766:
765:
688:
687:
662:
661:
655:
654:
634:
633:
614:
613:
570:
569:
550:
549:
530:
529:
526:
475:
464:
463:
462:
457:
446:
440:
437:
394:
392:
382:
370:
355:
344:
338:
335:
324:
310:related reading
300:
296:
259:
255:
244:
230:
224:
222:
219:
210:
204:
202:
199:
186:
184:
181:
169:
165:
164:
151:
149:
146:
133:
131:
128:
116:
112:
110:
97:
95:
92:
79:
77:
74:
61:
59:
56:
36:
28:
23:
22:
15:
12:
11:
5:
3699:
3697:
3689:
3688:
3683:
3673:
3672:
3666:
3665:
3663:
3662:
3652:
3651:
3650:
3645:
3640:
3625:
3615:
3605:
3593:
3582:
3579:
3578:
3576:
3575:
3570:
3565:
3560:
3555:
3550:
3544:
3542:
3536:
3535:
3533:
3532:
3527:
3522:
3520:Winding number
3517:
3512:
3506:
3504:
3500:
3499:
3497:
3496:
3491:
3486:
3481:
3476:
3471:
3466:
3461:
3460:
3459:
3454:
3452:homotopy group
3444:
3443:
3442:
3437:
3432:
3427:
3422:
3412:
3407:
3402:
3392:
3390:
3386:
3385:
3378:
3376:
3374:
3373:
3368:
3363:
3362:
3361:
3351:
3350:
3349:
3339:
3334:
3329:
3324:
3319:
3313:
3311:
3307:
3306:
3301:
3299:
3298:
3291:
3284:
3276:
3269:
3268:
3259:
3253:
3245:Addison-Wesley
3236:
3230:
3209:
3207:
3204:
3201:
3200:
3193:
3172:
3159:
3158:
3156:
3153:
3152:
3151:
3145:
3140:
3132:
3129:
3101:singleton sets
3065:Main article:
3062:
3059:
3046:
3026:
3006:
2994:is itself the
2932:Main article:
2929:
2926:
2919:
2880:
2869:Main article:
2866:
2863:
2849:
2845:
2835:are closed in
2824:
2821:
2818:
2798:
2795:
2792:
2768:
2764:
2743:
2740:
2737:
2734:
2729:
2726:
2722:
2718:
2715:
2695:
2692:
2689:
2684:
2681:
2677:
2673:
2670:
2649:
2645:
2642:
2639:
2636:
2608:
2588:
2566:
2562:
2558:
2534:
2514:
2511:
2508:
2505:
2502:
2499:
2496:
2493:
2490:
2487:
2482:
2479:
2475:
2471:
2468:
2448:
2445:
2442:
2439:
2436:
2433:
2428:
2425:
2421:
2417:
2414:
2394:
2366:
2346:
2343:
2340:
2337:
2334:
2319:
2316:
2313:
2310:
2307:
2286:
2265:
2262:
2259:
2256:
2253:
2230:
2210:
2190:
2187:
2184:
2179:
2176:
2172:
2168:
2165:
2145:
2142:
2139:
2134:
2131:
2127:
2123:
2120:
2099:
2078:
2057:
2053:
2050:
2047:
2044:
2013:
1993:
1966:
1946:
1936:
1922:
1919:
1916:
1913:
1910:
1907:
1892:
1889:
1886:
1883:
1880:
1860:
1840:
1820:
1800:
1780:
1760:
1751:neighbourhood
1747:if there is a
1728:
1708:
1681:
1678:
1675:
1672:
1652:
1649:
1646:
1626:
1606:
1586:
1583:
1580:
1577:
1574:
1571:
1568:
1565:
1545:
1542:
1539:
1536:
1533:
1530:
1527:
1524:
1504:
1501:
1498:
1495:
1492:
1489:
1486:
1483:
1468:
1465:
1462:
1459:
1456:
1453:
1450:
1424:
1404:
1384:
1364:
1344:
1324:
1304:
1284:
1274:neighbourhoods
1253:
1233:
1209:
1206:
1184:
1181:
1160:
1157:
1154:
1150:
1147:
1143:
1140:
1137:
1134:
1131:
1127:
1124:
1103:
1083:
1059:
1056:
1053:
1050:
1047:
1044:
1041:
1038:
1035:
1032:
1029:
1009:
1006:
1003:
1000:
997:
994:
991:
988:
985:
982:
979:
976:
973:
970:
967:
964:
961:
958:
953:
949:
928:
925:
922:
919:
916:
913:
910:
907:
904:
901:
898:
895:
892:
889:
886:
883:
880:
877:
872:
868:
840:
836:
812:
809:
806:
803:
800:
785:
782:
779:
776:
773:
756:
740:
737:
734:
728:
725:
719:
716:
713:
707:
704:
698:
695:
669:
641:
621:
577:
557:
537:
525:
522:
491:separated sets
477:
476:
459:
458:
373:
371:
364:
357:
356:
314:external links
303:
301:
294:
289:
263:
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127:
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109:
104:
103:
100:
91:
86:
85:
82:
73:
68:
67:
64:
55:
50:
49:
48:classification
42:
41:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
3698:
3687:
3684:
3682:
3679:
3678:
3676:
3661:
3653:
3649:
3646:
3644:
3641:
3639:
3636:
3635:
3634:
3626:
3624:
3620:
3616:
3614:
3610:
3606:
3604:
3599:
3594:
3592:
3584:
3583:
3580:
3574:
3571:
3569:
3566:
3564:
3561:
3559:
3556:
3554:
3551:
3549:
3546:
3545:
3543:
3541:
3537:
3531:
3530:Orientability
3528:
3526:
3523:
3521:
3518:
3516:
3513:
3511:
3508:
3507:
3505:
3501:
3495:
3492:
3490:
3487:
3485:
3482:
3480:
3477:
3475:
3472:
3470:
3467:
3465:
3462:
3458:
3455:
3453:
3450:
3449:
3448:
3445:
3441:
3438:
3436:
3433:
3431:
3428:
3426:
3423:
3421:
3418:
3417:
3416:
3413:
3411:
3408:
3406:
3403:
3401:
3397:
3394:
3393:
3391:
3387:
3382:
3372:
3369:
3367:
3366:Set-theoretic
3364:
3360:
3357:
3356:
3355:
3352:
3348:
3345:
3344:
3343:
3340:
3338:
3335:
3333:
3330:
3328:
3327:Combinatorial
3325:
3323:
3320:
3318:
3315:
3314:
3312:
3308:
3304:
3297:
3292:
3290:
3285:
3283:
3278:
3277:
3274:
3265:
3260:
3256:
3254:0-486-43479-6
3250:
3246:
3242:
3237:
3233:
3231:0-13-181629-2
3227:
3223:
3222:Prentice-Hall
3219:
3215:
3211:
3210:
3205:
3196:
3194:0-13-181629-2
3190:
3186:
3182:
3176:
3173:
3169:
3164:
3161:
3154:
3149:
3146:
3144:
3141:
3138:
3135:
3134:
3130:
3128:
3126:
3122:
3118:
3114:
3110:
3106:
3102:
3098:
3094:
3090:
3086:
3082:
3078:
3075:, two points
3074:
3068:
3060:
3058:
2997:
2993:
2989:
2985:
2981:
2977:
2973:
2969:
2965:
2961:
2957:
2953:
2949:
2945:
2941:
2935:
2927:
2925:
2923:
2915:
2914:
2908:
2906:
2902:
2898:
2894:
2890:
2886:
2878:
2872:
2864:
2862:
2847:
2819:
2793:
2782:
2766:
2741:
2735:
2727:
2724:
2720:
2716:
2713:
2690:
2682:
2679:
2675:
2671:
2668:
2640:
2637:
2634:
2626:
2606:
2586:
2577:
2564:
2560:
2556:
2548:
2532:
2512:
2506:
2503:
2500:
2497:
2494:
2491:
2488:
2480:
2477:
2473:
2469:
2466:
2443:
2440:
2437:
2434:
2426:
2423:
2419:
2415:
2412:
2392:
2384:
2380:
2364:
2341:
2338:
2335:
2314:
2311:
2308:
2260:
2257:
2254:
2244:
2243:unit interval
2228:
2208:
2185:
2177:
2174:
2170:
2166:
2163:
2140:
2132:
2129:
2125:
2121:
2118:
2076:
2048:
2045:
2042:
2035:
2031:
2011:
1991:
1982:
1980:
1964:
1944:
1934:
1920:
1914:
1911:
1908:
1887:
1884:
1881:
1858:
1838:
1818:
1798:
1778:
1758:
1750:
1746:
1726:
1706:
1697:
1695:
1679:
1676:
1673:
1670:
1650:
1647:
1644:
1624:
1604:
1584:
1578:
1575:
1572:
1566:
1563:
1540:
1537:
1534:
1531:
1525:
1522:
1502:
1496:
1493:
1490:
1484:
1481:
1463:
1460:
1457:
1451:
1448:
1440:
1439:
1422:
1402:
1382:
1362:
1342:
1322:
1302:
1282:
1275:
1272:if there are
1271:
1251:
1231:
1222:
1207:
1204:
1182:
1179:
1158:
1155:
1152:
1148:
1145:
1141:
1135:
1132:
1129:
1125:
1122:
1101:
1081:
1073:
1057:
1054:
1051:
1048:
1045:
1039:
1036:
1033:
1027:
1004:
1001:
995:
992:
989:
983:
980:
977:
974:
971:
965:
959:
951:
947:
923:
920:
914:
911:
908:
902:
899:
896:
893:
890:
884:
878:
870:
866:
858:
854:
838:
826:
807:
804:
801:
780:
777:
774:
764:
760:
754:
751:
738:
735:
732:
723:
717:
711:
702:
696:
693:
685:
683:
667:
659:
639:
619:
610:
607:
605:
601:
597:
593:
592:
575:
555:
535:
523:
521:
519:
515:
510:
508:
504:
500:
496:
493:are pairs of
492:
488:
484:
473:
470:
455:
452:
444:
433:
430:
426:
423:
419:
416:
412:
409:
405:
402: –
401:
397:
396:Find sources:
390:
386:
380:
379:
374:This article
372:
368:
363:
362:
353:
350:
342:
332:
328:
322:
321:
315:
311:
307:
302:
293:
292:
287:
285:
278:
277:
272:
271:
266:
261:
252:
251:
241:
238:
237:
234:
228:
223:
214:
208:
203:
194:
190:
185:
176:
172:
170:
159:
155:
150:
141:
137:
132:
123:
119:
117:
105:
101:
96:
87:
83:
78:
69:
65:
60:
51:
47:
43:
40:
35:
31:
19:
3660:Publications
3525:Chern number
3515:Betti number
3398: /
3389:Key concepts
3337:Differential
3263:
3240:
3217:
3184:
3175:
3163:
3124:
3120:
3116:
3112:
3108:
3104:
3096:
3092:
3084:
3080:
3076:
3072:
3070:
2991:
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2578:
2025:
1983:
1740:
1698:
1436:
1265:
1223:
853:metric space
758:
752:
686:
653:
611:
608:
596:intersection
589:
527:
511:
490:
480:
465:
447:
441:January 2018
438:
428:
421:
414:
407:
395:
383:Please help
378:verification
375:
345:
339:January 2018
336:
325:Please help
317:
281:
274:
268:
267:Please help
264:
126:completely T
66:(Kolmogorov)
3623:Wikiversity
3540:Key results
3168:Pervin 1964
1072:derived set
524:Definitions
497:of a given
487:mathematics
331:introducing
173:(Tychonoff)
102:(Hausdorff)
3675:Categories
3469:CW complex
3410:Continuity
3400:Closed set
3359:cohomology
2948:complement
2661:such that
2549:less than
2111:such that
1831:such that
1355:such that
857:open balls
604:set theory
411:newspapers
270:improve it
46:Kolmogorov
3648:geometric
3643:algebraic
3494:Cobordism
3430:Hausdorff
3425:connected
3342:Geometric
3332:Continuum
3322:Algebraic
3155:Citations
3045:∅
3025:∅
3005:∅
2996:empty set
2964:connected
2885:separated
2725:−
2680:−
2644:→
2579:The sets
2492:−
2478:−
2435:−
2424:−
2175:−
2167:⊆
2130:−
2122:⊆
2052:→
1984:The sets
1699:The sets
1532:−
1224:The sets
1153:∩
1139:∅
1130:∩
1046:≥
975:∈
894:∈
825:real line
763:intervals
733:∩
727:¯
715:∅
706:¯
697:∩
656:separated
612:The sets
600:empty set
276:talk page
120:(Urysohn)
84:(Fréchet)
3686:Topology
3613:Wikibook
3591:Category
3479:Manifold
3447:Homotopy
3405:Interior
3396:Open set
3354:Homology
3303:Topology
3218:Topology
3216:(2000).
3185:Topology
3183:(2000).
3131:See also
3089:open set
2889:distinct
2783:. Since
2383:preimage
1208:′
1183:′
1149:′
1126:′
591:disjoint
483:topology
3638:general
3440:uniform
3420:compact
3371:Digital
3206:Sources
3170:, p. 51
3115:} and {
3107:} and {
2903:} and {
2891:points
2545:is any
682:closure
598:is the
495:subsets
425:scholar
327:improve
240:History
166:3½
3633:Topics
3435:metric
3310:Fields
3251:
3228:
3191:
2982:is an
2972:subset
2922:spaces
2525:where
1749:closed
855:, two
427:
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413:
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398:
225:
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113:½
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3415:Space
432:JSTOR
418:books
312:, or
3249:ISBN
3226:ISBN
3189:ISBN
3123:and
3095:and
3083:are
3079:and
2895:and
2875:The
2809:and
2706:and
2619:are
2599:and
2459:and
2156:and
2024:are
2004:and
1933:are
1851:and
1739:are
1719:and
1663:and
1617:and
1556:and
1438:open
1415:and
1375:and
1315:and
1264:are
1244:and
1196:and
1094:and
1002:<
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652:are
632:and
548:and
404:news
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2962:is
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2405:as
2385:of
1957:or
1935:not
1811:of
1771:of
1696:).
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1335:of
1295:of
759:not
660:in
481:In
387:by
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