1949:
259:
877:
The distinction between a set and a topological space is often blurred notationally, for convenience, which can be a source of confusion when one first encounters these definitions. Thus, whenever
487:
2723:
2559:
640:
1197:
1086:
1780:
1744:
1722:
1652:
1616:
1579:
1554:
1532:
1459:
1437:
1415:
1393:
1371:
1339:
1305:
2149:
2424:
1255:
1229:
1118:
947:
402:
346:
129:
2108:
2053:
1943:
1913:
1861:
863:
795:
744:
601:
517:
2743:
2666:
2642:
2619:
2599:
2579:
2527:
2507:
2484:
2464:
2444:
2398:
2378:
2355:
2335:
2315:
2295:
2275:
2255:
2235:
2212:
2192:
2172:
2073:
2017:
1997:
1974:
1881:
1829:
1809:
1275:
1158:
1138:
1047:
1027:
1007:
987:
967:
915:
895:
839:
819:
771:
724:
700:
680:
660:
581:
561:
541:
445:
425:
366:
310:
282:
196:
172:
152:
204:
2997:
2901:
17:
1948:
404:. Subsets of topological spaces are usually assumed to be equipped with the subspace topology unless otherwise stated.
3022:
2941:
460:
2671:
2532:
2980:
368:
is equipped with the subspace topology then it is a topological space in its own right, and is called a
3042:
2847:
289:
1956:
This property is characteristic in the sense that it can be used to define the subspace topology on
606:
1163:
1052:
1763:
1727:
1705:
1635:
1599:
1562:
1537:
1515:
1442:
1420:
1398:
1376:
1354:
1322:
1288:
2113:
1466:
68:
2403:
1234:
2975:
1202:
1091:
920:
375:
319:
102:
2081:
2026:
1922:
1886:
1834:
2782:
747:
493:
2883:
848:
780:
729:
586:
502:
30:"Induced topology" redirects here. For the topology generated by a family of functions, see
3007:
2911:
2823:
2766:
2753:
8:
2857:
2833:
2888:, EMS Textbooks in Mathematics, European Mathematical Society (EMS), Zürich, p. 5,
2985:
2829:
2728:
2651:
2627:
2604:
2584:
2564:
2512:
2492:
2469:
2449:
2429:
2383:
2363:
2340:
2320:
2300:
2280:
2260:
2240:
2220:
2197:
2177:
2157:
2058:
2002:
1982:
1959:
1866:
1814:
1794:
1586:
1260:
1143:
1123:
1032:
1012:
992:
972:
952:
900:
880:
824:
804:
756:
709:
685:
665:
645:
566:
546:
526:
520:
430:
410:
351:
295:
267:
181:
157:
137:
3037:
3018:
2993:
2971:
2937:
2897:
2816:
1342:
448:
50:
2960:
2929:
2889:
2852:
2645:
31:
3003:
2989:
2907:
2803:
1349:
603:
is continuous. The open sets in this topology are precisely the ones of the form
2933:
1979:
We list some further properties of the subspace topology. In the following let
1916:
1316:
285:
3031:
2924:
Pinoli, Jean-Charles (June 2014), "The
Geometric and Topological Framework",
2796:
452:
2810:
2749:
1120:
as the topological spaces, related as discussed above. So phrases such as "
703:
2789:
1395:
do not have the discrete topology ({0} for example is not an open set in
1308:
42:
866:
2769:
implies its subspaces have that property, then we say the property is
1791:
The subspace topology has the following characteristic property. Let
1760:
is both open and closed as a subset of itself but not as a subset of
798:
313:
38:
2297:
with the same topology. In other words the subspace topology that
254:{\displaystyle \tau _{S}=\lbrace S\cap U\mid U\in \tau \rbrace .}
2893:
2756:
to this subset coincides with subspace topology for this subset.
407:
Alternatively we can define the subspace topology for a subset
132:
57:
2773:. If only closed subspaces must share the property we call it
1585:
subsets (which happen also to be closed), and is therefore a
2926:
Mathematical
Foundations of Image Processing and Analysis 2
2760:
1277:
is considered to be endowed with the subspace topology.
2731:
2674:
2654:
2630:
2607:
2587:
2567:
2535:
2515:
2495:
2472:
2452:
2432:
2406:
2386:
2366:
2343:
2323:
2303:
2283:
2263:
2243:
2223:
2200:
2180:
2160:
2116:
2084:
2061:
2029:
2005:
1985:
1962:
1925:
1889:
1869:
1863:
be the inclusion map. Then for any topological space
1837:
1817:
1797:
1766:
1730:
1708:
1638:
1602:
1565:
1540:
1518:
1445:
1423:
1401:
1379:
1357:
1325:
1291:
1263:
1237:
1205:
1166:
1146:
1126:
1094:
1055:
1035:
1015:
995:
975:
955:
923:
903:
883:
851:
827:
807:
783:
759:
732:
712:
688:
668:
648:
609:
589:
569:
549:
529:
505:
463:
433:
413:
378:
354:
322:
298:
270:
207:
184:
160:
140:
105:
949:
is a topological space, then the unadorned symbols "
2737:
2717:
2660:
2636:
2613:
2593:
2573:
2553:
2521:
2501:
2478:
2458:
2438:
2418:
2392:
2372:
2349:
2329:
2309:
2289:
2269:
2249:
2229:
2206:
2186:
2166:
2143:
2102:
2067:
2047:
2011:
1991:
1968:
1937:
1907:
1875:
1855:
1823:
1803:
1774:
1738:
1716:
1646:
1610:
1573:
1548:
1526:
1453:
1431:
1409:
1387:
1365:
1333:
1299:
1269:
1249:
1223:
1191:
1152:
1132:
1112:
1080:
1041:
1021:
1001:
981:
961:
941:
909:
889:
857:
833:
813:
789:
765:
738:
718:
694:
674:
654:
634:
595:
575:
555:
535:
511:
481:
439:
419:
396:
360:
340:
304:
276:
253:
190:
166:
146:
123:
3029:
1952:Characteristic property of the subspace topology
1534:is both open and closed, whereas as a subset of
1485:) and are respectively open and closed, but if
801:, i.e., if the forward image of an open set of
583:is defined as the coarsest topology for which
2988:reprint of 1978 ed.), Berlin, New York:
1746:that can intersect with [0, 1) to result in [
1493:are irrational, then the set of all rational
2970:
2712:
2688:
245:
221:
1654:(as for example the intersection between (-
2781:Every open and every closed subspace of a
2965:Elements of Mathematics: General Topology
2947:; see Section 26.2.4. Submanifolds, p. 59
2881:
1768:
1732:
1710:
1640:
1604:
1567:
1542:
1520:
1447:
1425:
1403:
1381:
1359:
1327:
1293:
482:{\displaystyle \iota :S\hookrightarrow X}
2718:{\displaystyle B_{S}=\{U\cap S:U\in B\}}
2337:is the same as the one it inherits from
1596:= [0, 1) be a subspace of the real line
1581:, ∪ is composed of two disjoint
1231:, in the sense used above; that is: (i)
14:
3030:
2923:
2761:Preservation of topological properties
2748:The topology induced on a subset of a
2055:is continuous then the restriction to
726:(also with the subspace topology) and
2877:
2875:
2873:
2554:{\displaystyle X\setminus S\in \tau }
989:" can often be used to refer both to
2765:If a topological space having some
2174:are precisely the intersections of
1417:because there is no open subset of
24:
2870:
1947:
1477:are rational, then the intervals (
25:
3054:
2539:
284:is open in the subspace topology
563:. Then the subspace topology on
2785:space is completely metrizable.
2601:if and only if it is closed in
1724:(as there is no open subset of
2917:
2138:
2132:
2126:
2094:
2039:
1899:
1847:
1218:
1206:
1186:
1167:
1107:
1095:
1075:
1056:
936:
924:
872:
635:{\displaystyle \iota ^{-1}(U)}
629:
623:
473:
391:
379:
335:
323:
118:
106:
13:
1:
2954:
2466:if and only if it is open in
1786:
1315:The subspace topology of the
1192:{\displaystyle (S,\tau _{S})}
1081:{\displaystyle (S,\tau _{S})}
1029:considered as two subsets of
94:
3017:, Dover Publications (2004)
1775:{\displaystyle \mathbb {R} }
1739:{\displaystyle \mathbb {R} }
1717:{\displaystyle \mathbb {R} }
1647:{\displaystyle \mathbb {R} }
1611:{\displaystyle \mathbb {R} }
1574:{\displaystyle \mathbb {R} }
1549:{\displaystyle \mathbb {R} }
1527:{\displaystyle \mathbb {R} }
1454:{\displaystyle \mathbb {Q} }
1432:{\displaystyle \mathbb {R} }
1410:{\displaystyle \mathbb {Q} }
1388:{\displaystyle \mathbb {R} }
1373:considered as a subspace of
1366:{\displaystyle \mathbb {Q} }
1334:{\displaystyle \mathbb {R} }
1300:{\displaystyle \mathbb {R} }
7:
2981:Counterexamples in Topology
2840:
2795:Every closed subspace of a
2144:{\displaystyle f:X\to f(X)}
1311:with their usual topology.
1280:
10:
3059:
2934:10.1002/9781118984574.ch26
2419:{\displaystyle S\in \tau }
1512:The set as a subspace of
1250:{\displaystyle S\in \tau }
841:. Likewise it is called a
99:Given a topological space
29:
2928:, Wiley, pp. 57–69,
2882:tom Dieck, Tammo (2008),
2788:Every open subspace of a
1224:{\displaystyle (X,\tau )}
1113:{\displaystyle (X,\tau )}
942:{\displaystyle (X,\tau )}
499:More generally, suppose
397:{\displaystyle (X,\tau )}
341:{\displaystyle (X,\tau )}
124:{\displaystyle (X,\tau )}
67:which is equipped with a
2863:
2509:is a closed subspace of
2103:{\displaystyle f:X\to Y}
2048:{\displaystyle f:X\to Y}
1938:{\displaystyle i\circ f}
1908:{\displaystyle f:Z\to Y}
1856:{\displaystyle i:Y\to X}
1509:is both open and closed.
1439:whose intersection with
1160:" are used to mean that
2967:, Addison-Wesley (1966)
2380:is an open subspace of
1199:is an open subspace of
543:to a topological space
2976:Seebach, J. Arthur Jr.
2739:
2719:
2662:
2638:
2615:
2595:
2575:
2555:
2523:
2503:
2480:
2460:
2440:
2420:
2394:
2374:
2351:
2331:
2311:
2291:
2277:is also a subspace of
2271:
2251:
2231:
2208:
2188:
2168:
2145:
2104:
2069:
2049:
2013:
1993:
1970:
1953:
1939:
1909:
1877:
1857:
1825:
1805:
1776:
1740:
1718:
1648:
1612:
1575:
1550:
1528:
1455:
1433:
1411:
1389:
1367:
1335:
1301:
1271:
1251:
1225:
1193:
1154:
1134:
1114:
1082:
1043:
1023:
1003:
983:
963:
943:
911:
891:
859:
858:{\displaystyle \iota }
835:
815:
791:
790:{\displaystyle \iota }
767:
740:
739:{\displaystyle \iota }
720:
696:
676:
656:
636:
597:
596:{\displaystyle \iota }
577:
557:
537:
513:
512:{\displaystyle \iota }
483:
441:
421:
398:
362:
342:
306:
278:
255:
192:
168:
148:
125:
2813:is weakly hereditary.
2783:completely metrizable
2740:
2720:
2663:
2639:
2616:
2596:
2576:
2556:
2524:
2504:
2481:
2461:
2441:
2421:
2395:
2375:
2352:
2332:
2312:
2292:
2272:
2252:
2232:
2209:
2189:
2169:
2146:
2105:
2070:
2050:
2014:
1994:
1971:
1951:
1940:
1910:
1878:
1858:
1826:
1806:
1777:
1741:
1719:
1649:
1613:
1576:
1551:
1529:
1456:
1434:
1412:
1390:
1368:
1336:
1302:
1272:
1252:
1226:
1194:
1155:
1135:
1115:
1083:
1044:
1024:
1004:
984:
964:
944:
912:
892:
860:
836:
816:
792:
768:
748:topological embedding
741:
721:
697:
677:
657:
637:
598:
578:
558:
538:
514:
484:
442:
422:
399:
363:
343:
307:
279:
264:That is, a subset of
256:
193:
169:
149:
126:
71:induced from that of
41:and related areas of
2824:totally disconnected
2767:topological property
2729:
2672:
2652:
2628:
2605:
2585:
2565:
2561:). Then a subset of
2533:
2513:
2493:
2470:
2450:
2430:
2426:). Then a subset of
2404:
2384:
2364:
2341:
2321:
2301:
2281:
2261:
2241:
2221:
2198:
2194:with closed sets in
2178:
2158:
2114:
2082:
2059:
2027:
2003:
1983:
1960:
1923:
1887:
1867:
1835:
1815:
1795:
1764:
1728:
1706:
1636:
1600:
1563:
1538:
1516:
1443:
1421:
1399:
1377:
1355:
1323:
1289:
1261:
1235:
1203:
1164:
1144:
1140:an open subspace of
1124:
1092:
1053:
1033:
1013:
993:
973:
953:
921:
901:
881:
849:
825:
805:
781:
757:
730:
710:
686:
666:
646:
607:
587:
567:
547:
527:
503:
461:
431:
411:
376:
352:
320:
296:
268:
205:
182:
158:
138:
103:
2858:direct sum topology
2834:second countability
2752:by restricting the
2154:The closed sets in
2110:is continuous then
1319:, as a subspace of
18:Subspace (topology)
3013:Willard, Stephen.
2972:Steen, Lynn Arthur
2885:Algebraic topology
2830:First countability
2735:
2715:
2658:
2634:
2611:
2591:
2571:
2551:
2519:
2499:
2476:
2456:
2436:
2416:
2390:
2370:
2347:
2327:
2307:
2287:
2267:
2247:
2227:
2204:
2184:
2164:
2141:
2100:
2065:
2045:
2009:
1989:
1966:
1954:
1935:
1919:the composite map
1905:
1873:
1853:
1821:
1801:
1772:
1736:
1714:
1698:, 1) is closed in
1644:
1608:
1587:disconnected space
1571:
1556:it is only closed.
1546:
1524:
1451:
1429:
1407:
1385:
1363:
1331:
1297:
1285:In the following,
1267:
1247:
1221:
1189:
1150:
1130:
1110:
1078:
1039:
1019:
999:
979:
959:
939:
907:
887:
855:
831:
811:
787:
763:
736:
716:
692:
672:
652:
632:
593:
573:
553:
533:
509:
479:
437:
417:
394:
358:
338:
302:
274:
251:
188:
164:
144:
121:
27:Inherited topology
2999:978-0-486-68735-3
2961:Bourbaki, Nicolas
2903:978-3-03719-048-7
2817:Total boundedness
2792:is a Baire space.
2775:weakly hereditary
2738:{\displaystyle S}
2661:{\displaystyle X}
2637:{\displaystyle B}
2614:{\displaystyle X}
2594:{\displaystyle S}
2574:{\displaystyle S}
2522:{\displaystyle X}
2502:{\displaystyle S}
2479:{\displaystyle X}
2459:{\displaystyle S}
2439:{\displaystyle S}
2393:{\displaystyle X}
2373:{\displaystyle S}
2350:{\displaystyle X}
2330:{\displaystyle S}
2310:{\displaystyle A}
2290:{\displaystyle X}
2270:{\displaystyle A}
2250:{\displaystyle S}
2237:is a subspace of
2230:{\displaystyle A}
2207:{\displaystyle X}
2187:{\displaystyle S}
2167:{\displaystyle S}
2068:{\displaystyle S}
2012:{\displaystyle X}
1999:be a subspace of
1992:{\displaystyle S}
1969:{\displaystyle Y}
1876:{\displaystyle Z}
1824:{\displaystyle X}
1811:be a subspace of
1804:{\displaystyle Y}
1559:As a subspace of
1343:discrete topology
1270:{\displaystyle S}
1153:{\displaystyle X}
1133:{\displaystyle S}
1042:{\displaystyle X}
1022:{\displaystyle X}
1002:{\displaystyle S}
982:{\displaystyle X}
962:{\displaystyle S}
910:{\displaystyle X}
890:{\displaystyle S}
845:if the injection
834:{\displaystyle X}
814:{\displaystyle S}
777:if the injection
766:{\displaystyle S}
719:{\displaystyle X}
695:{\displaystyle S}
675:{\displaystyle X}
655:{\displaystyle U}
576:{\displaystyle S}
556:{\displaystyle X}
536:{\displaystyle S}
449:coarsest topology
440:{\displaystyle X}
420:{\displaystyle S}
361:{\displaystyle S}
305:{\displaystyle S}
277:{\displaystyle S}
191:{\displaystyle S}
176:subspace topology
167:{\displaystyle X}
147:{\displaystyle S}
81:relative topology
77:subspace topology
51:topological space
16:(Redirected from
3050:
3043:General topology
3015:General Topology
3010:
2948:
2946:
2921:
2915:
2914:
2879:
2853:product topology
2846:the dual notion
2744:
2742:
2741:
2736:
2724:
2722:
2721:
2716:
2684:
2683:
2667:
2665:
2664:
2659:
2643:
2641:
2640:
2635:
2620:
2618:
2617:
2612:
2600:
2598:
2597:
2592:
2580:
2578:
2577:
2572:
2560:
2558:
2557:
2552:
2528:
2526:
2525:
2520:
2508:
2506:
2505:
2500:
2485:
2483:
2482:
2477:
2465:
2463:
2462:
2457:
2445:
2443:
2442:
2437:
2425:
2423:
2422:
2417:
2399:
2397:
2396:
2391:
2379:
2377:
2376:
2371:
2356:
2354:
2353:
2348:
2336:
2334:
2333:
2328:
2316:
2314:
2313:
2308:
2296:
2294:
2293:
2288:
2276:
2274:
2273:
2268:
2256:
2254:
2253:
2248:
2236:
2234:
2233:
2228:
2213:
2211:
2210:
2205:
2193:
2191:
2190:
2185:
2173:
2171:
2170:
2165:
2150:
2148:
2147:
2142:
2109:
2107:
2106:
2101:
2074:
2072:
2071:
2066:
2054:
2052:
2051:
2046:
2018:
2016:
2015:
2010:
1998:
1996:
1995:
1990:
1975:
1973:
1972:
1967:
1945:is continuous.
1944:
1942:
1941:
1936:
1914:
1912:
1911:
1906:
1882:
1880:
1879:
1874:
1862:
1860:
1859:
1854:
1830:
1828:
1827:
1822:
1810:
1808:
1807:
1802:
1781:
1779:
1778:
1773:
1771:
1755:
1754:
1750:
1745:
1743:
1742:
1737:
1735:
1723:
1721:
1720:
1715:
1713:
1697:
1696:
1692:
1687:
1686:
1682:
1673:
1672:
1668:
1663:
1662:
1658:
1653:
1651:
1650:
1645:
1643:
1627:
1626:
1622:
1617:
1615:
1614:
1609:
1607:
1580:
1578:
1577:
1572:
1570:
1555:
1553:
1552:
1547:
1545:
1533:
1531:
1530:
1525:
1523:
1460:
1458:
1457:
1452:
1450:
1438:
1436:
1435:
1430:
1428:
1416:
1414:
1413:
1408:
1406:
1394:
1392:
1391:
1386:
1384:
1372:
1370:
1369:
1364:
1362:
1350:rational numbers
1340:
1338:
1337:
1332:
1330:
1306:
1304:
1303:
1298:
1296:
1276:
1274:
1273:
1268:
1256:
1254:
1253:
1248:
1230:
1228:
1227:
1222:
1198:
1196:
1195:
1190:
1185:
1184:
1159:
1157:
1156:
1151:
1139:
1137:
1136:
1131:
1119:
1117:
1116:
1111:
1087:
1085:
1084:
1079:
1074:
1073:
1048:
1046:
1045:
1040:
1028:
1026:
1025:
1020:
1008:
1006:
1005:
1000:
988:
986:
985:
980:
968:
966:
965:
960:
948:
946:
945:
940:
916:
914:
913:
908:
896:
894:
893:
888:
864:
862:
861:
856:
840:
838:
837:
832:
820:
818:
817:
812:
796:
794:
793:
788:
772:
770:
769:
764:
745:
743:
742:
737:
725:
723:
722:
717:
706:to its image in
701:
699:
698:
693:
681:
679:
678:
673:
661:
659:
658:
653:
641:
639:
638:
633:
622:
621:
602:
600:
599:
594:
582:
580:
579:
574:
562:
560:
559:
554:
542:
540:
539:
534:
518:
516:
515:
510:
488:
486:
485:
480:
446:
444:
443:
438:
426:
424:
423:
418:
403:
401:
400:
395:
367:
365:
364:
359:
347:
345:
344:
339:
311:
309:
308:
303:
283:
281:
280:
275:
260:
258:
257:
252:
217:
216:
197:
195:
194:
189:
173:
171:
170:
165:
153:
151:
150:
145:
130:
128:
127:
122:
85:induced topology
32:Initial topology
21:
3058:
3057:
3053:
3052:
3051:
3049:
3048:
3047:
3028:
3027:
3000:
2990:Springer-Verlag
2957:
2952:
2951:
2944:
2922:
2918:
2904:
2880:
2871:
2866:
2843:
2836:are hereditary.
2804:Hausdorff space
2763:
2730:
2727:
2726:
2725:is a basis for
2679:
2675:
2673:
2670:
2669:
2653:
2650:
2649:
2629:
2626:
2625:
2606:
2603:
2602:
2586:
2583:
2582:
2566:
2563:
2562:
2534:
2531:
2530:
2514:
2511:
2510:
2494:
2491:
2490:
2471:
2468:
2467:
2451:
2448:
2447:
2431:
2428:
2427:
2405:
2402:
2401:
2385:
2382:
2381:
2365:
2362:
2361:
2342:
2339:
2338:
2322:
2319:
2318:
2302:
2299:
2298:
2282:
2279:
2278:
2262:
2259:
2258:
2242:
2239:
2238:
2222:
2219:
2218:
2199:
2196:
2195:
2179:
2176:
2175:
2159:
2156:
2155:
2115:
2112:
2111:
2083:
2080:
2079:
2060:
2057:
2056:
2028:
2025:
2024:
2004:
2001:
2000:
1984:
1981:
1980:
1961:
1958:
1957:
1924:
1921:
1920:
1888:
1885:
1884:
1868:
1865:
1864:
1836:
1833:
1832:
1816:
1813:
1812:
1796:
1793:
1792:
1789:
1767:
1765:
1762:
1761:
1752:
1748:
1747:
1731:
1729:
1726:
1725:
1709:
1707:
1704:
1703:
1694:
1690:
1689:
1684:
1680:
1679:
1678:results in [0,
1670:
1666:
1665:
1660:
1656:
1655:
1639:
1637:
1634:
1633:
1624:
1620:
1619:
1603:
1601:
1598:
1597:
1566:
1564:
1561:
1560:
1541:
1539:
1536:
1535:
1519:
1517:
1514:
1513:
1446:
1444:
1441:
1440:
1424:
1422:
1419:
1418:
1402:
1400:
1397:
1396:
1380:
1378:
1375:
1374:
1358:
1356:
1353:
1352:
1326:
1324:
1321:
1320:
1317:natural numbers
1307:represents the
1292:
1290:
1287:
1286:
1283:
1262:
1259:
1258:
1236:
1233:
1232:
1204:
1201:
1200:
1180:
1176:
1165:
1162:
1161:
1145:
1142:
1141:
1125:
1122:
1121:
1093:
1090:
1089:
1069:
1065:
1054:
1051:
1050:
1034:
1031:
1030:
1014:
1011:
1010:
994:
991:
990:
974:
971:
970:
954:
951:
950:
922:
919:
918:
902:
899:
898:
897:is a subset of
882:
879:
878:
875:
850:
847:
846:
843:closed subspace
826:
823:
822:
806:
803:
802:
782:
779:
778:
758:
755:
754:
731:
728:
727:
711:
708:
707:
687:
684:
683:
667:
664:
663:
647:
644:
643:
614:
610:
608:
605:
604:
588:
585:
584:
568:
565:
564:
548:
545:
544:
528:
525:
524:
504:
501:
500:
462:
459:
458:
432:
429:
428:
412:
409:
408:
377:
374:
373:
353:
350:
349:
321:
318:
317:
297:
294:
293:
269:
266:
265:
212:
208:
206:
203:
202:
183:
180:
179:
159:
156:
155:
139:
136:
135:
104:
101:
100:
97:
35:
28:
23:
22:
15:
12:
11:
5:
3056:
3046:
3045:
3040:
3026:
3025:
3011:
2998:
2968:
2956:
2953:
2950:
2949:
2942:
2916:
2902:
2868:
2867:
2865:
2862:
2861:
2860:
2855:
2850:
2848:quotient space
2842:
2839:
2838:
2837:
2827:
2826:is hereditary.
2820:
2819:is hereditary.
2814:
2807:
2806:is hereditary.
2800:
2793:
2786:
2762:
2759:
2758:
2757:
2746:
2734:
2714:
2711:
2708:
2705:
2702:
2699:
2696:
2693:
2690:
2687:
2682:
2678:
2657:
2633:
2622:
2610:
2590:
2570:
2550:
2547:
2544:
2541:
2538:
2518:
2498:
2487:
2475:
2455:
2435:
2415:
2412:
2409:
2389:
2369:
2358:
2346:
2326:
2317:inherits from
2306:
2286:
2266:
2246:
2226:
2215:
2203:
2183:
2163:
2152:
2151:is continuous.
2140:
2137:
2134:
2131:
2128:
2125:
2122:
2119:
2099:
2096:
2093:
2090:
2087:
2076:
2075:is continuous.
2064:
2044:
2041:
2038:
2035:
2032:
2008:
1988:
1965:
1934:
1931:
1928:
1917:if and only if
1915:is continuous
1904:
1901:
1898:
1895:
1892:
1872:
1852:
1849:
1846:
1843:
1840:
1820:
1800:
1788:
1785:
1784:
1783:
1770:
1734:
1712:
1688:)). Likewise [
1642:
1606:
1590:
1569:
1557:
1544:
1522:
1510:
1461:can result in
1449:
1427:
1405:
1383:
1361:
1346:
1329:
1295:
1282:
1279:
1266:
1246:
1243:
1240:
1220:
1217:
1214:
1211:
1208:
1188:
1183:
1179:
1175:
1172:
1169:
1149:
1129:
1109:
1106:
1103:
1100:
1097:
1077:
1072:
1068:
1064:
1061:
1058:
1049:, and also to
1038:
1018:
998:
978:
958:
938:
935:
932:
929:
926:
906:
886:
874:
871:
854:
830:
810:
786:
762:
735:
715:
691:
671:
651:
631:
628:
625:
620:
617:
613:
592:
572:
552:
532:
508:
490:
489:
478:
475:
472:
469:
466:
451:for which the
436:
416:
393:
390:
387:
384:
381:
357:
337:
334:
331:
328:
325:
301:
286:if and only if
273:
262:
261:
250:
247:
244:
241:
238:
235:
232:
229:
226:
223:
220:
215:
211:
198:is defined by
187:
163:
143:
120:
117:
114:
111:
108:
96:
93:
89:trace topology
26:
9:
6:
4:
3:
2:
3055:
3044:
3041:
3039:
3036:
3035:
3033:
3024:
3023:0-486-43479-6
3020:
3016:
3012:
3009:
3005:
3001:
2995:
2991:
2987:
2983:
2982:
2977:
2973:
2969:
2966:
2962:
2959:
2958:
2945:
2943:9781118984574
2939:
2935:
2931:
2927:
2920:
2913:
2909:
2905:
2899:
2895:
2891:
2887:
2886:
2878:
2876:
2874:
2869:
2859:
2856:
2854:
2851:
2849:
2845:
2844:
2835:
2831:
2828:
2825:
2821:
2818:
2815:
2812:
2808:
2805:
2801:
2798:
2797:compact space
2794:
2791:
2787:
2784:
2780:
2779:
2778:
2776:
2772:
2768:
2755:
2751:
2747:
2732:
2709:
2706:
2703:
2700:
2697:
2694:
2691:
2685:
2680:
2676:
2655:
2647:
2631:
2623:
2608:
2588:
2581:is closed in
2568:
2548:
2545:
2542:
2536:
2516:
2496:
2488:
2473:
2453:
2433:
2413:
2410:
2407:
2387:
2367:
2359:
2344:
2324:
2304:
2284:
2264:
2244:
2224:
2216:
2201:
2181:
2161:
2153:
2135:
2129:
2123:
2120:
2117:
2097:
2091:
2088:
2085:
2077:
2062:
2042:
2036:
2033:
2030:
2022:
2021:
2020:
2006:
1986:
1977:
1963:
1950:
1946:
1932:
1929:
1926:
1918:
1902:
1896:
1893:
1890:
1870:
1850:
1844:
1841:
1838:
1818:
1798:
1759:
1701:
1677:
1631:
1628:) is open in
1595:
1591:
1588:
1584:
1558:
1511:
1508:
1504:
1500:
1496:
1492:
1488:
1484:
1480:
1476:
1472:
1468:
1464:
1351:
1347:
1344:
1318:
1314:
1313:
1312:
1310:
1278:
1264:
1244:
1241:
1238:
1215:
1212:
1209:
1181:
1177:
1173:
1170:
1147:
1127:
1104:
1101:
1098:
1070:
1066:
1062:
1059:
1036:
1016:
996:
976:
956:
933:
930:
927:
904:
884:
870:
868:
852:
844:
828:
808:
800:
784:
776:
775:open subspace
773:is called an
760:
751:
749:
733:
713:
705:
689:
669:
649:
626:
618:
615:
611:
590:
570:
550:
530:
522:
506:
497:
495:
476:
470:
467:
464:
457:
456:
455:
454:
453:inclusion map
450:
434:
414:
405:
388:
385:
382:
371:
355:
332:
329:
326:
315:
299:
291:
287:
271:
248:
242:
239:
236:
233:
230:
227:
224:
218:
213:
209:
201:
200:
199:
185:
177:
161:
141:
134:
115:
112:
109:
92:
90:
86:
82:
78:
74:
70:
66:
62:
59:
55:
52:
48:
44:
40:
33:
19:
3014:
2979:
2964:
2925:
2919:
2884:
2811:normal space
2774:
2770:
2764:
2750:metric space
1978:
1955:
1790:
1757:
1699:
1675:
1629:
1593:
1582:
1506:
1502:
1498:
1494:
1490:
1486:
1482:
1478:
1474:
1470:
1462:
1309:real numbers
1284:
876:
842:
774:
752:
746:is called a
704:homeomorphic
498:
491:
406:
369:
290:intersection
263:
175:
98:
88:
84:
80:
76:
72:
64:
60:
53:
46:
36:
2894:10.4171/048
2799:is compact.
2790:Baire space
2446:is open in
1702:but not in
1632:but not in
1618:. Then [0,
1257:; and (ii)
873:Terminology
821:is open in
753:A subspace
523:from a set
75:called the
43:mathematics
3032:Categories
2955:References
2771:hereditary
1787:Properties
867:closed map
494:continuous
288:it is the
95:Definition
2978:(1995) ,
2707:∈
2695:∩
2549:τ
2546:∈
2540:∖
2414:τ
2411:∈
2127:→
2095:→
2040:→
1930:∘
1900:→
1848:→
1469:{0}). If
1467:singleton
1341:, is the
1245:τ
1242:∈
1216:τ
1178:τ
1105:τ
1067:τ
934:τ
853:ι
785:ι
734:ι
616:−
612:ι
591:ι
521:injection
507:ι
474:↪
465:ι
389:τ
333:τ
243:τ
240:∈
234:∣
228:∩
210:τ
116:τ
87:, or the
83:, or the
3038:Topology
2841:See also
2809:Being a
2802:Being a
2489:Suppose
2360:Suppose
1831:and let
1281:Examples
799:open map
702:is then
662:open in
370:subspace
314:open set
312:with an
79:(or the
69:topology
47:subspace
39:topology
3008:0507446
2912:2456045
1756:, 1)).
1751:⁄
1693:⁄
1683:⁄
1669:⁄
1659:⁄
1623:⁄
969:" and "
447:as the
3021:
3006:
2996:
2940:
2910:
2900:
2822:Being
2754:metric
1883:a map
1674:) and
917:, and
797:is an
519:is an
174:, the
133:subset
131:and a
58:subset
2986:Dover
2864:Notes
2668:then
2646:basis
2644:is a
2257:then
1505:<
1501:<
1497:with
865:is a
348:. If
56:is a
49:of a
3019:ISBN
2994:ISBN
2938:ISBN
2898:ISBN
2832:and
2648:for
2529:(so
2400:(so
1592:Let
1583:open
1489:and
1473:and
1465:the
1463:only
1348:The
1088:and
1009:and
642:for
45:, a
2930:doi
2890:doi
2624:If
2217:If
2078:If
2023:If
492:is
427:of
372:of
316:in
292:of
178:on
154:of
91:).
63:of
37:In
3034::
3004:MR
3002:,
2992:,
2974:;
2963:,
2936:,
2908:MR
2906:,
2896:,
2872:^
2777:.
2019:.
1976:.
1664:,
1481:,
869:.
750:.
682:.
496:.
2984:(
2932::
2892::
2745:.
2733:S
2713:}
2710:B
2704:U
2701::
2698:S
2692:U
2689:{
2686:=
2681:S
2677:B
2656:X
2632:B
2621:.
2609:X
2589:S
2569:S
2543:S
2537:X
2517:X
2497:S
2486:.
2474:X
2454:S
2434:S
2408:S
2388:X
2368:S
2357:.
2345:X
2325:S
2305:A
2285:X
2265:A
2245:S
2225:A
2214:.
2202:X
2182:S
2162:S
2139:)
2136:X
2133:(
2130:f
2124:X
2121::
2118:f
2098:Y
2092:X
2089::
2086:f
2063:S
2043:Y
2037:X
2034::
2031:f
2007:X
1987:S
1964:Y
1933:f
1927:i
1903:Y
1897:Z
1894::
1891:f
1871:Z
1851:X
1845:Y
1842::
1839:i
1819:X
1799:Y
1782:.
1769:R
1758:S
1753:2
1749:1
1733:R
1711:R
1700:S
1695:2
1691:1
1685:2
1681:1
1676:S
1671:2
1667:1
1661:2
1657:1
1641:R
1630:S
1625:2
1621:1
1605:R
1594:S
1589:.
1568:R
1543:R
1521:R
1507:b
1503:x
1499:a
1495:x
1491:b
1487:a
1483:b
1479:a
1475:b
1471:a
1448:Q
1426:R
1404:Q
1382:R
1360:Q
1345:.
1328:R
1294:R
1265:S
1239:S
1219:)
1213:,
1210:X
1207:(
1187:)
1182:S
1174:,
1171:S
1168:(
1148:X
1128:S
1108:)
1102:,
1099:X
1096:(
1076:)
1071:S
1063:,
1060:S
1057:(
1037:X
1017:X
997:S
977:X
957:S
937:)
931:,
928:X
925:(
905:X
885:S
829:X
809:S
761:S
714:X
690:S
670:X
650:U
630:)
627:U
624:(
619:1
571:S
551:X
531:S
477:X
471:S
468::
435:X
415:S
392:)
386:,
383:X
380:(
356:S
336:)
330:,
327:X
324:(
300:S
272:S
249:.
246:}
237:U
231:U
225:S
222:{
219:=
214:S
186:S
162:X
142:S
119:)
113:,
110:X
107:(
73:X
65:X
61:S
54:X
34:.
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
