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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
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733:{\displaystyle A\cap {\bar {B}}=\varnothing ={\bar {A}}\cap B.}
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505:(defined below), which are somewhat related but different.
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3139: – 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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2743:(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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307:but its sources remain unclear because it lacks
1002:{\displaystyle B_{s}(q)=\{x\in X:d(q,x)<s\}}
921:{\displaystyle B_{r}(p)=\{x\in X:d(p,x)<r\}}
2967:to share this property is the empty set, then
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2612:precisely separated by a continuous function
1153:{\textstyle A'\cap B=\varnothing =B'\cap A.}
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517:There are various ways in which two subsets
3046:is an open-connected component of itself.)
501:Separated sets should not be confused with
273:Learn how and when to remove these messages
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3176:(2 ed.). Prentice Hall. p. 211.
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2616:if there exists a continuous function
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744:Hausdorff−Lennes Separation Condition
669:if each is disjoint from the other's
7:
2899:Separated spaces are usually called
2183:{\displaystyle B\subseteq f^{-1}(1)}
2138:{\displaystyle A\subseteq f^{-1}(0)}
378:adding citations to reliable sources
2896:} are separated by neighbourhoods.
2643:{\displaystyle f:X\to \mathbb {R} }
2051:{\displaystyle f:X\to \mathbb {R} }
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3013:
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2368:separated by closed neighbourhoods
2017:separated by a continuous function
1732:separated by closed neighbourhoods
14:
3128: – Type of topological space
3108:} are separated, then the points
1063:(indicated by the prime symbol):
254:This article has multiple issues.
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3006:, authorities differ on whether
2979:. (In the degenerate case where
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3253:Foundations of General Topology
1052:{\displaystyle d(p,q)\geq r+s.}
365:needs additional citations for
262:or discuss these issues on the
3056:Topological distinguishability
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742:This property is known as the
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1:
3074:topologically distinguishable
2842:{\displaystyle \mathbb {R} ,}
2761:{\displaystyle \mathbb {R} ,}
833:{\displaystyle \mathbb {R} ,}
2939:. This is certainly true if
2917:Relation to connected spaces
2736:{\displaystyle B=f^{-1}(1).}
2280:{\displaystyle \mathbb {R} }
2093:{\displaystyle \mathbb {R} }
1860:are disjoint. Our examples,
3251:Pervin, William J. (1964),
2688:{\displaystyle A=f^{-1}(0)}
1968:separated by neighbourhoods
1780:and a closed neighbourhood
1257:separated by neighbourhoods
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3537:Banach fixed-point theorem
3060:Given a topological space
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3039:{\displaystyle \emptyset }
3019:{\displaystyle \emptyset }
2999:{\displaystyle \emptyset }
2927:Given a topological space
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3228:Willard, Stephen (2004).
3026:is connected and whether
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2507:{\displaystyle V=f^{-1},}
2230:map to 1. (Sometimes the
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2973:open-connected component
2441:{\displaystyle U=f^{-1}}
2210:map to 0 and members of
1579:{\displaystyle V=(1,3).}
1538:{\displaystyle U=(-1,1)}
1497:{\displaystyle B=(1,2],}
498:for topological spaces.
474:and related branches of
293:This article includes a
3132:Locally Hausdorff space
2770:separated by a function
1683:normal separation axiom
1461:{\displaystyle A=[0,1)}
1009:are separated whenever
557:of a topological space
322:more precise citations.
3592:Mathematics portal
3492:Metrics and properties
3478:Second-countable space
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2817:{\displaystyle \{1\}}
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778:{\displaystyle [0,1)}
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3547:Invariance of domain
3499:Euler characteristic
3473:Bundle (mathematics)
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2559:{\displaystyle 1/2.}
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2536:positive real number
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1674:{\displaystyle V=B.}
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374:improve this article
3557:Tychonoff's theorem
3552:Poincaré conjecture
3306:General (point-set)
3076:if there exists an
2023:continuous function
1645:{\displaystyle U=A}
145:(regular Hausdorff)
3542:De Rham cohomology
3463:Polyhedral complex
3453:Simplicial complex
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2876:if, given any two
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1203:{\displaystyle B'}
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295:list of references
198:(completely normal
180:(normal Hausdorff)
28:topological spaces
3670:Separation axioms
3657:
3656:
3446:fundamental group
3203:Munkres, James R.
3170:Munkres, James R.
2866:separation axioms
2601:{\displaystyle B}
2581:{\displaystyle A}
2527:{\displaystyle c}
2387:{\displaystyle f}
2359:{\displaystyle f}
2223:{\displaystyle B}
2203:{\displaystyle A}
2078:to the real line
2071:{\displaystyle X}
2006:{\displaystyle B}
1986:{\displaystyle A}
1959:{\displaystyle V}
1939:{\displaystyle U}
1853:{\displaystyle V}
1833:{\displaystyle U}
1813:{\displaystyle B}
1793:{\displaystyle V}
1773:{\displaystyle A}
1753:{\displaystyle U}
1721:{\displaystyle B}
1701:{\displaystyle A}
1619:{\displaystyle B}
1599:{\displaystyle A}
1417:{\displaystyle V}
1397:{\displaystyle U}
1377:{\displaystyle V}
1357:{\displaystyle U}
1337:{\displaystyle B}
1317:{\displaystyle V}
1297:{\displaystyle A}
1277:{\displaystyle U}
1246:{\displaystyle B}
1226:{\displaystyle A}
1096:{\displaystyle B}
1076:{\displaystyle A}
718:
697:
662:{\displaystyle X}
634:{\displaystyle B}
614:{\displaystyle A}
570:{\displaystyle X}
550:{\displaystyle B}
530:{\displaystyle A}
496:separation axioms
488:topological space
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23:Separation axioms
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2418:
2393:
2391:
2390:
2385:
2365:
2363:
2362:
2357:
2345:
2343:
2342:
2337:
2316:
2315:
2310:
2286:
2284:
2283:
2278:
2276:
2264:
2262:
2261:
2258:{\displaystyle }
2256:
2229:
2227:
2226:
2221:
2209:
2207:
2206:
2201:
2189:
2187:
2186:
2181:
2170:
2169:
2144:
2142:
2141:
2136:
2125:
2124:
2099:
2097:
2096:
2091:
2089:
2077:
2075:
2074:
2069:
2057:
2055:
2054:
2049:
2047:
2019:
2018:
2012:
2010:
2009:
2004:
1992:
1990:
1989:
1984:
1965:
1963:
1962:
1957:
1945:
1943:
1942:
1937:
1921:
1919:
1918:
1913:
1889:
1888:
1883:
1859:
1857:
1856:
1851:
1839:
1837:
1836:
1831:
1819:
1817:
1816:
1811:
1799:
1797:
1796:
1791:
1779:
1777:
1776:
1771:
1759:
1757:
1756:
1751:
1734:
1733:
1727:
1725:
1724:
1719:
1707:
1705:
1704:
1699:
1680:
1678:
1677:
1672:
1651:
1649:
1648:
1643:
1625:
1623:
1622:
1617:
1605:
1603:
1602:
1597:
1585:
1583:
1582:
1577:
1544:
1542:
1541:
1536:
1503:
1501:
1500:
1495:
1465:
1464:
1459:
1423:
1421:
1420:
1415:
1403:
1401:
1400:
1395:
1383:
1381:
1380:
1375:
1363:
1361:
1360:
1355:
1343:
1341:
1340:
1335:
1323:
1321:
1320:
1315:
1303:
1301:
1300:
1295:
1283:
1281:
1280:
1275:
1259:
1258:
1252:
1250:
1249:
1244:
1232:
1230:
1229:
1224:
1209:
1207:
1206:
1201:
1199:
1184:
1182:
1181:
1176:
1174:
1159:
1157:
1156:
1151:
1140:
1117:
1102:
1100:
1099:
1094:
1082:
1080:
1079:
1074:
1058:
1056:
1055:
1050:
1008:
1006:
1005:
1000:
944:
943:
927:
925:
924:
919:
863:
862:
839:
837:
836:
831:
826:
811:
809:
808:
803:
782:
781:
776:
739:
737:
736:
731:
720:
719:
711:
699:
698:
690:
668:
666:
665:
660:
647:
646:
640:
638:
637:
632:
620:
618:
617:
612:
576:
574:
573:
568:
556:
554:
553:
548:
536:
534:
533:
528:
507:Separable spaces
503:separated spaces
492:connected spaces
463:
456:
445:
438:
434:
431:
425:
423:
389:"Separated sets"
382:
358:
350:
343:
336:
332:
329:
323:
318:this article by
309:inline citations
288:
287:
280:
269:
247:
246:
239:
220: Hausdorff)
215:
210:
200: Hausdorff)
195:
190:
177:
172:
157:
156:
142:
137:
124:
119:
104:
103:
88:
83:
70:
65:
52:
47:
19:
3690:
3689:
3685:
3684:
3683:
3681:
3680:
3679:
3660:
3659:
3658:
3653:
3584:
3566:
3562:Urysohn's lemma
3523:
3487:
3373:
3364:
3336:low-dimensional
3294:
3289:
3259:
3250:
3244:
3227:
3221:
3201:
3197:
3192:
3191:
3184:
3168:
3167:
3163:
3155:
3151:
3146:
3126:Hausdorff space
3122:
3058:
3052:
3028:
3027:
3008:
3007:
2988:
2987:
2925:
2923:Connected space
2919:
2910:
2871:
2862:
2856:
2826:
2825:
2800:
2799:
2774:
2773:
2745:
2744:
2708:
2697:
2696:
2663:
2652:
2651:
2618:
2617:
2611:
2610:
2590:
2589:
2570:
2569:
2540:
2539:
2516:
2515:
2461:
2450:
2449:
2407:
2396:
2395:
2376:
2375:
2348:
2347:
2289:
2288:
2267:
2266:
2235:
2234:
2212:
2211:
2192:
2191:
2158:
2147:
2146:
2113:
2102:
2101:
2080:
2079:
2060:
2059:
2058:from the space
2026:
2025:
2016:
2015:
1995:
1994:
1975:
1974:
1948:
1947:
1928:
1927:
1862:
1861:
1842:
1841:
1822:
1821:
1802:
1801:
1782:
1781:
1762:
1761:
1742:
1741:
1731:
1730:
1710:
1709:
1690:
1689:
1654:
1653:
1628:
1627:
1608:
1607:
1588:
1587:
1547:
1546:
1506:
1505:
1504:you could take
1432:
1431:
1406:
1405:
1386:
1385:
1366:
1365:
1346:
1345:
1326:
1325:
1306:
1305:
1286:
1285:
1266:
1265:
1256:
1255:
1235:
1234:
1215:
1214:
1192:
1187:
1186:
1167:
1162:
1161:
1133:
1110:
1105:
1104:
1085:
1084:
1065:
1064:
1011:
1010:
935:
930:
929:
854:
849:
848:
817:
816:
755:
754:
677:
676:
651:
650:
644:
643:
623:
622:
603:
602:
559:
558:
539:
538:
519:
518:
515:
464:
453:
452:
451:
446:
435:
429:
426:
383:
381:
371:
359:
344:
333:
327:
324:
313:
299:related reading
289:
285:
248:
244:
233:
219:
213:
211:
208:
199:
193:
191:
188:
175:
173:
170:
158:
154:
153:
140:
138:
135:
122:
120:
117:
105:
101:
99:
86:
84:
81:
68:
66:
63:
50:
48:
45:
25:
17:
12:
11:
5:
3688:
3686:
3678:
3677:
3672:
3662:
3661:
3655:
3654:
3652:
3651:
3641:
3640:
3639:
3634:
3629:
3614:
3604:
3594:
3582:
3571:
3568:
3567:
3565:
3564:
3559:
3554:
3549:
3544:
3539:
3533:
3531:
3525:
3524:
3522:
3521:
3516:
3511:
3509:Winding number
3506:
3501:
3495:
3493:
3489:
3488:
3486:
3485:
3480:
3475:
3470:
3465:
3460:
3455:
3450:
3449:
3448:
3443:
3441:homotopy group
3433:
3432:
3431:
3426:
3421:
3416:
3411:
3401:
3396:
3391:
3381:
3379:
3375:
3374:
3367:
3365:
3363:
3362:
3357:
3352:
3351:
3350:
3340:
3339:
3338:
3328:
3323:
3318:
3313:
3308:
3302:
3300:
3296:
3295:
3290:
3288:
3287:
3280:
3273:
3265:
3258:
3257:
3248:
3242:
3234:Addison-Wesley
3225:
3219:
3198:
3196:
3193:
3190:
3189:
3182:
3161:
3148:
3147:
3145:
3142:
3141:
3140:
3134:
3129:
3121:
3118:
3090:singleton sets
3054:Main article:
3051:
3048:
3035:
3015:
2995:
2983:is itself the
2921:Main article:
2918:
2915:
2908:
2869:
2858:Main article:
2855:
2852:
2838:
2834:
2824:are closed in
2813:
2810:
2807:
2787:
2784:
2781:
2757:
2753:
2732:
2729:
2726:
2723:
2718:
2715:
2711:
2707:
2704:
2684:
2681:
2678:
2673:
2670:
2666:
2662:
2659:
2638:
2634:
2631:
2628:
2625:
2597:
2577:
2555:
2551:
2547:
2523:
2503:
2500:
2497:
2494:
2491:
2488:
2485:
2482:
2479:
2476:
2471:
2468:
2464:
2460:
2457:
2437:
2434:
2431:
2428:
2425:
2422:
2417:
2414:
2410:
2406:
2403:
2383:
2355:
2335:
2332:
2329:
2326:
2323:
2308:
2305:
2302:
2299:
2296:
2275:
2254:
2251:
2248:
2245:
2242:
2219:
2199:
2179:
2176:
2173:
2168:
2165:
2161:
2157:
2154:
2134:
2131:
2128:
2123:
2120:
2116:
2112:
2109:
2088:
2067:
2046:
2042:
2039:
2036:
2033:
2002:
1982:
1955:
1935:
1925:
1911:
1908:
1905:
1902:
1899:
1896:
1881:
1878:
1875:
1872:
1869:
1849:
1829:
1809:
1789:
1769:
1749:
1740:neighbourhood
1736:if there is a
1717:
1697:
1670:
1667:
1664:
1661:
1641:
1638:
1635:
1615:
1595:
1575:
1572:
1569:
1566:
1563:
1560:
1557:
1554:
1534:
1531:
1528:
1525:
1522:
1519:
1516:
1513:
1493:
1490:
1487:
1484:
1481:
1478:
1475:
1472:
1457:
1454:
1451:
1448:
1445:
1442:
1439:
1413:
1393:
1373:
1353:
1333:
1313:
1293:
1273:
1263:neighbourhoods
1242:
1222:
1198:
1195:
1173:
1170:
1149:
1146:
1143:
1139:
1136:
1132:
1129:
1126:
1123:
1120:
1116:
1113:
1092:
1072:
1048:
1045:
1042:
1039:
1036:
1033:
1030:
1027:
1024:
1021:
1018:
998:
995:
992:
989:
986:
983:
980:
977:
974:
971:
968:
965:
962:
959:
956:
953:
950:
947:
942:
938:
917:
914:
911:
908:
905:
902:
899:
896:
893:
890:
887:
884:
881:
878:
875:
872:
869:
866:
861:
857:
829:
825:
801:
798:
795:
792:
789:
774:
771:
768:
765:
762:
745:
729:
726:
723:
717:
714:
708:
705:
702:
696:
693:
687:
684:
658:
630:
610:
566:
546:
526:
514:
511:
480:separated sets
466:
465:
448:
447:
362:
360:
353:
346:
345:
303:external links
292:
290:
283:
278:
252:
251:
249:
242:
235:
234:
232:
231:
225:
222:
221:
216:
207:
202:
201:
196:
187:
182:
181:
178:
169:
164:
163:
160:
152:
147:
146:
143:
134:
129:
128:
125:
116:
111:
110:
107:
98:
93:
92:
89:
80:
75:
74:
71:
62:
57:
56:
53:
44:
39:
38:
37:classification
31:
30:
15:
13:
10:
9:
6:
4:
3:
2:
3687:
3676:
3673:
3671:
3668:
3667:
3665:
3650:
3642:
3638:
3635:
3633:
3630:
3628:
3625:
3624:
3623:
3615:
3613:
3609:
3605:
3603:
3599:
3595:
3593:
3588:
3583:
3581:
3573:
3572:
3569:
3563:
3560:
3558:
3555:
3553:
3550:
3548:
3545:
3543:
3540:
3538:
3535:
3534:
3532:
3530:
3526:
3520:
3519:Orientability
3517:
3515:
3512:
3510:
3507:
3505:
3502:
3500:
3497:
3496:
3494:
3490:
3484:
3481:
3479:
3476:
3474:
3471:
3469:
3466:
3464:
3461:
3459:
3456:
3454:
3451:
3447:
3444:
3442:
3439:
3438:
3437:
3434:
3430:
3427:
3425:
3422:
3420:
3417:
3415:
3412:
3410:
3407:
3406:
3405:
3402:
3400:
3397:
3395:
3392:
3390:
3386:
3383:
3382:
3380:
3376:
3371:
3361:
3358:
3356:
3355:Set-theoretic
3353:
3349:
3346:
3345:
3344:
3341:
3337:
3334:
3333:
3332:
3329:
3327:
3324:
3322:
3319:
3317:
3316:Combinatorial
3314:
3312:
3309:
3307:
3304:
3303:
3301:
3297:
3293:
3286:
3281:
3279:
3274:
3272:
3267:
3266:
3263:
3254:
3249:
3245:
3243:0-486-43479-6
3239:
3235:
3231:
3226:
3222:
3220:0-13-181629-2
3216:
3212:
3211:Prentice-Hall
3208:
3204:
3200:
3199:
3194:
3185:
3183:0-13-181629-2
3179:
3175:
3171:
3165:
3162:
3158:
3153:
3150:
3143:
3138:
3135:
3133:
3130:
3127:
3124:
3123:
3119:
3117:
3115:
3111:
3107:
3103:
3099:
3095:
3091:
3087:
3083:
3079:
3075:
3071:
3067:
3064:, two points
3063:
3057:
3049:
3047:
2986:
2982:
2978:
2974:
2970:
2966:
2962:
2958:
2954:
2950:
2946:
2942:
2938:
2934:
2930:
2924:
2916:
2914:
2912:
2904:
2903:
2897:
2895:
2891:
2887:
2883:
2879:
2875:
2867:
2861:
2853:
2851:
2836:
2808:
2782:
2771:
2755:
2730:
2724:
2716:
2713:
2709:
2705:
2702:
2679:
2671:
2668:
2664:
2660:
2657:
2629:
2626:
2623:
2615:
2595:
2575:
2566:
2553:
2549:
2545:
2537:
2521:
2501:
2495:
2492:
2489:
2486:
2483:
2480:
2477:
2469:
2466:
2462:
2458:
2455:
2432:
2429:
2426:
2423:
2415:
2412:
2408:
2404:
2401:
2381:
2373:
2369:
2353:
2330:
2327:
2324:
2303:
2300:
2297:
2249:
2246:
2243:
2233:
2232:unit interval
2217:
2197:
2174:
2166:
2163:
2159:
2155:
2152:
2129:
2121:
2118:
2114:
2110:
2107:
2065:
2037:
2034:
2031:
2024:
2020:
2000:
1980:
1971:
1969:
1953:
1933:
1923:
1909:
1903:
1900:
1897:
1876:
1873:
1870:
1847:
1827:
1807:
1787:
1767:
1747:
1739:
1735:
1715:
1695:
1686:
1684:
1668:
1665:
1662:
1659:
1639:
1636:
1633:
1613:
1593:
1573:
1567:
1564:
1561:
1555:
1552:
1529:
1526:
1523:
1520:
1514:
1511:
1491:
1485:
1482:
1479:
1473:
1470:
1452:
1449:
1446:
1440:
1437:
1429:
1428:
1411:
1391:
1371:
1351:
1331:
1311:
1291:
1271:
1264:
1261:if there are
1260:
1240:
1220:
1211:
1196:
1193:
1171:
1168:
1147:
1144:
1141:
1137:
1134:
1130:
1124:
1121:
1118:
1114:
1111:
1090:
1070:
1062:
1046:
1043:
1040:
1037:
1034:
1028:
1025:
1022:
1016:
993:
990:
984:
981:
978:
972:
969:
966:
963:
960:
954:
948:
940:
936:
912:
909:
903:
900:
897:
891:
888:
885:
882:
879:
873:
867:
859:
855:
847:
843:
827:
815:
796:
793:
790:
769:
766:
763:
753:
749:
743:
740:
727:
724:
721:
712:
706:
700:
691:
685:
682:
674:
672:
656:
648:
628:
608:
599:
596:
594:
590:
586:
582:
581:
564:
544:
524:
512:
510:
508:
504:
499:
497:
493:
489:
485:
482:are pairs of
481:
477:
473:
462:
459:
444:
441:
433:
422:
419:
415:
412:
408:
405:
401:
398:
394:
391: –
390:
386:
385:Find sources:
379:
375:
369:
368:
363:This article
361:
357:
352:
351:
342:
339:
331:
321:
317:
311:
310:
304:
300:
296:
291:
282:
281:
276:
274:
267:
266:
261:
260:
255:
250:
241:
240:
230:
227:
226:
223:
217:
212:
203:
197:
192:
183:
179:
174:
165:
161:
159:
148:
144:
139:
130:
126:
121:
112:
108:
106:
94:
90:
85:
76:
72:
67:
58:
54:
49:
40:
36:
32:
29:
24:
20:
3649:Publications
3514:Chern number
3504:Betti number
3387: /
3378:Key concepts
3326:Differential
3252:
3229:
3206:
3173:
3164:
3152:
3113:
3109:
3105:
3101:
3097:
3093:
3085:
3081:
3073:
3069:
3065:
3061:
3059:
2980:
2976:
2972:
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1972:
1729:
1687:
1425:
1254:
1212:
842:metric space
747:
741:
675:
642:
600:
597:
585:intersection
578:
516:
500:
479:
469:
454:
436:
430:January 2018
427:
417:
410:
403:
396:
384:
372:Please help
367:verification
364:
334:
328:January 2018
325:
314:Please help
306:
270:
263:
257:
256:Please help
253:
115:completely T
55:(Kolmogorov)
3612:Wikiversity
3529:Key results
3157:Pervin 1964
1061:derived set
513:Definitions
486:of a given
476:mathematics
320:introducing
162:(Tychonoff)
91:(Hausdorff)
3664:Categories
3458:CW complex
3399:Continuity
3389:Closed set
3348:cohomology
2937:complement
2650:such that
2538:less than
2100:such that
1820:such that
1344:such that
846:open balls
593:set theory
400:newspapers
259:improve it
35:Kolmogorov
3637:geometric
3632:algebraic
3483:Cobordism
3419:Hausdorff
3414:connected
3331:Geometric
3321:Continuum
3311:Algebraic
3144:Citations
3034:∅
3014:∅
2994:∅
2985:empty set
2953:connected
2874:separated
2714:−
2669:−
2633:→
2568:The sets
2481:−
2467:−
2424:−
2413:−
2164:−
2156:⊆
2119:−
2111:⊆
2041:→
1973:The sets
1688:The sets
1521:−
1213:The sets
1142:∩
1128:∅
1119:∩
1035:≥
964:∈
883:∈
814:real line
752:intervals
722:∩
716:¯
704:∅
695:¯
686:∩
645:separated
601:The sets
589:empty set
265:talk page
109:(Urysohn)
73:(Fréchet)
3675:Topology
3602:Wikibook
3580:Category
3468:Manifold
3436:Homotopy
3394:Interior
3385:Open set
3343:Homology
3292:Topology
3207:Topology
3205:(2000).
3174:Topology
3172:(2000).
3120:See also
3078:open set
2878:distinct
2772:. Since
2372:preimage
1197:′
1172:′
1138:′
1115:′
580:disjoint
472:topology
3627:general
3429:uniform
3409:compact
3360:Digital
3195:Sources
3159:, p. 51
3104:} and {
3096:} and {
2892:} and {
2880:points
2534:is any
671:closure
587:is the
484:subsets
414:scholar
316:improve
229:History
155:3½
3622:Topics
3424:metric
3299:Fields
3240:
3217:
3180:
2971:is an
2961:subset
2911:spaces
2514:where
1738:closed
844:, two
416:
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102:½
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3404:Space
421:JSTOR
407:books
301:, or
3238:ISBN
3215:ISBN
3178:ISBN
3112:and
3084:and
3072:are
3068:and
2884:and
2864:The
2798:and
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2608:are
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2145:and
2013:are
1993:and
1922:are
1840:and
1728:are
1708:and
1652:and
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1427:open
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1364:and
1304:and
1253:are
1233:and
1185:and
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991:<
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910:<
641:are
621:and
537:and
393:news
2975:of
2963:of
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2394:as
2374:of
1946:or
1924:not
1800:of
1760:of
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649:in
470:In
376:by
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