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1971:
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366:
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192:
149:
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2548:
1466:{\displaystyle f^{-1}(K)\subseteq f^{-1}\left(\cup _{i=1}^{s}V_{k_{i}}\right)\subseteq \cup _{a\in \Gamma }U_{a}}
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2934:
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673:
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A topological space is compact if and only if the map from that space to a single point is proper.
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Since the latter is assumed to be compact, it has a finite subcover. In other words, for every
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Map between topological spaces with the property that the preimage of every compact is compact
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8:
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3003:
2418: – A function that sends open (resp. closed) subsets to open (resp. closed) subsets
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998:{\displaystyle V_{k}=Y\setminus f\left(X\setminus \cup _{a\in \gamma _{k}}U_{a}\right)}
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2424: – Continuous closed surjective map, each of whose fibers are also compact sets
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It is possible to generalize the notion of proper maps of topological spaces to
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67:
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2412: – Map that satisfies a condition similar to that of being an open map.
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2605:(1973). "Sequentially proper maps and a sequential compactification".
1639:{\displaystyle f\times \operatorname {id} _{Z}:X\times Z\to Y\times Z}
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2583:
63:
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2841:
2790:
2697:
124:
20:
2598:, esp. p. 90 "Proper maps" and the Exercises to Section 3.6.
2444:
774:{\displaystyle f^{-1}(k)\subseteq \cup _{a\in \gamma _{k}}U_{a}.}
1666:
is
Hausdorff, this is equivalent to requiring that for any map
2393:
2666:
1569:. A map is universally closed if for any topological space
1563:
is locally compact
Hausdorff then proper is equivalent to
1147:
is assumed to be compact, there are finitely many points
830:{\displaystyle X\setminus \cup _{a\in \gamma _{k}}U_{a}}
2432:
Pages displaying short descriptions of redirect targets
1795:
An equivalent, possibly more intuitive definition when
2005:
is proper if and only if for every sequence of points
1311:{\displaystyle \Gamma =\cup _{i=1}^{s}\gamma _{k_{i}}}
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is as follows: we say an infinite sequence of points
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There are several competing definitions of a "proper
2639:. Vol. 218 (Second ed.). New York London:
1250:{\displaystyle K\subseteq \cup _{i=1}^{s}V_{k_{i}}.}
2553:Sketches of an elephant: a topos theory compendium
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2339:
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2104:{\displaystyle \left\{f\left(p_{i}\right)\right\}}
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773:
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662:
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591:
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523:
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432:
409:
389:
360:
340:
301:
246:
222:
186:
166:
143:
106:
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2473:Proceedings of the American Mathematical Society
2351:(this includes Hausdorff spaces that are either
2142:Every continuous map from a compact space to a
1318:is a finite union of finite sets, which makes
1120:{\displaystyle K\subseteq \cup _{k\in K}V_{k}}
2682:
2521:. Elements of Mathematics. Berlin, New York:
1859:
1846:
524:{\displaystyle \left\{U_{a}:a\in A\right\}}
3050:
3023:
2689:
2675:
2608:Journal of the London Mathematical Society
2547:
2485:
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1731:be closed, as follows from the fact that
2513:
234:. The two definitions are equivalent if
1473:and we have found a finite subcover of
3068:
2462:
2157:proper map is a compact covering map.
696:{\displaystyle \gamma _{k}\subseteq A}
2670:
2601:
2574:
2456:
19:This article is about the concept in
2033:{\displaystyle \left\{p_{i}\right\}}
2627:
2450:
2349:compactly generated Hausdorff space
210:and the preimage of every point in
13:
1724:{\displaystyle X\times _{Y}Z\to Z}
1448:
1325:
1264:
1186:{\displaystyle k_{1},\dots ,k_{s}}
14:
3087:
2571:, esp. section C3.2 "Proper maps"
2487:10.1090/s0002-9939-1970-0254818-x
2383:
2224:there exists some compact subset
954:
940:
791:
82:". Some authors call a function
3049:
3022:
3012:
3002:
2991:
2981:
2980:
2774:
2633:Introduction to Smooth Manifolds
2453:, p. 610, above Prop. A.53.
2519:General topology. Chapters 5–10
2327:is a proper continuous map and
917:is a closed map. Hence the set
3076:Theory of continuous functions
2311:
2266:
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2176:
1989:
1715:
1676:
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1499:
1493:
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1361:
729:
723:
625:
619:
599:this is also an open cover of
557:
551:
466:
460:
335:
329:
293:
98:
1:
2637:Graduate Texts in Mathematics
2506:
2468:"When proper maps are closed"
2136:
1757:{\displaystyle X\times _{Y}Z}
670:there exists a finite subset
73:
2696:
2437:
2430: – Mathematics glossary
2243:{\displaystyle C\subseteq X}
2217:{\displaystyle K\subseteq Y}
2198:if for every compact subset
2040:that escapes to infinity in
1916:{\displaystyle S\subseteq X}
1646:is closed. In the case that
270:Partial proof of equivalence
7:
2403:
1515:which completes the proof.
309:be a closed map, such that
10:
3092:
2943:Banach fixed-point theorem
1923:only finitely many points
1897:if, for every compact set
1786:{\displaystyle X\times Z.}
1508:{\displaystyle f^{-1}(K),}
634:{\displaystyle f^{-1}(k).}
566:{\displaystyle f^{-1}(K).}
200:closed with compact fibers
18:
2976:
2933:
2897:
2783:
2772:
2704:
1865:{\displaystyle \{p_{i}\}}
472:{\displaystyle f^{-1}(K)}
341:{\displaystyle f^{-1}(y)}
174:Other authors call a map
2320:{\displaystyle f:X\to Y}
2185:{\displaystyle f:X\to Y}
1998:{\displaystyle f:X\to Y}
1764:is a closed subspace of
440:It remains to show that
302:{\displaystyle f:X\to Y}
197:if it is continuous and
107:{\displaystyle f:X\to Y}
2621:10.1112/jlms/s2-7.3.515
2557:Oxford University Press
2281:{\displaystyle f(C)=K.}
2111:escapes to infinity in
1872:in a topological space
1331:{\displaystyle \Gamma }
663:{\displaystyle k\in K,}
417:be a compact subset of
390:{\displaystyle y\in Y.}
2998:Mathematics portal
2898:Metrics and properties
2884:Second-countable space
2580:Topology and groupoids
2373:
2341:
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2218:
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2105:
2057:
2034:
1999:
1973:Then a continuous map
1967:
1944:
1917:
1886:
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1829:
1809:
1787:
1758:
1725:
1686:
1685:{\displaystyle Z\to Y}
1660:
1640:
1583:
1557:
1537:
1509:
1467:
1332:
1312:
1251:
1187:
1141:
1121:
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1049:
1022:
999:
911:
891:
871:
851:
831:
775:
697:
664:
635:
593:
592:{\displaystyle k\in K}
567:
525:
473:
434:
411:
391:
362:
342:
303:
248:
224:
188:
168:
145:
108:
29:proper convex function
2374:
2342:
2322:
2283:
2245:
2219:
2187:
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2035:
2000:
1968:
1945:
1943:{\displaystyle p_{i}}
1918:
1887:
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1810:
1788:
1759:
1726:
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1661:
1641:
1584:
1558:
1538:
1510:
1468:
1333:
1313:
1257:Furthermore, the set
1252:
1188:
1142:
1122:
1073:
1050:
1048:{\displaystyle V_{k}}
1023:
1000:
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832:
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363:
343:
304:
249:
225:
203:; that is if it is a
189:
169:
146:
109:
23:. For the concept in
2953:Invariance of domain
2905:Euler characteristic
2879:Bundle (mathematics)
2416:Open and closed maps
2363:
2331:
2299:
2254:
2228:
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2115:
2067:
2044:
2009:
1977:
1954:
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1901:
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1843:
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1799:
1768:
1735:
1696:
1670:
1650:
1593:
1573:
1547:
1527:
1477:
1345:
1341:Now it follows that
1322:
1261:
1197:
1151:
1131:
1082:
1059:
1032:
1009:
921:
901:
881:
861:
857:and its image under
841:
785:
707:
674:
645:
603:
577:
535:
531:be an open cover of
486:
444:
421:
401:
372:
352:
313:
281:
238:
214:
178:
155:
135:
86:
66:concept is called a
2963:Tychonoff's theorem
2958:Poincaré conjecture
2712:General (point-set)
2146:is both proper and
1894:escapes to infinity
1411:
1290:
1226:
1055:contains the point
2948:De Rham cohomology
2869:Polyhedral complex
2859:Simplicial complex
2582:. North Carolina:
2464:Palais, Richard S.
2369:
2337:
2317:
2278:
2240:
2214:
2182:
2127:{\displaystyle Y.}
2124:
2101:
2056:{\displaystyle X,}
2053:
2030:
1995:
1966:{\displaystyle S.}
1963:
1940:
1913:
1882:
1862:
1825:
1805:
1783:
1754:
1721:
1682:
1656:
1636:
1579:
1566:universally closed
1553:
1533:
1505:
1463:
1391:
1328:
1308:
1270:
1247:
1206:
1183:
1137:
1117:
1071:{\displaystyle k.}
1068:
1045:
1021:{\displaystyle Y.}
1018:
995:
907:
887:
867:
847:
827:
771:
693:
660:
631:
589:
563:
521:
469:
433:{\displaystyle Y.}
430:
407:
387:
358:
338:
299:
244:
220:
184:
167:{\displaystyle X.}
164:
141:
116:topological spaces
104:
60:algebraic geometry
44:topological spaces
3063:
3062:
2852:fundamental group
2650:978-1-4419-9981-8
2611:. Second series.
2532:978-3-540-64563-4
2515:Bourbaki, Nicolas
2428:Topology glossary
2372:{\displaystyle f}
2340:{\displaystyle Y}
1885:{\displaystyle X}
1828:{\displaystyle Y}
1808:{\displaystyle X}
1659:{\displaystyle Y}
1582:{\displaystyle Z}
1556:{\displaystyle Y}
1543:is Hausdorff and
1536:{\displaystyle X}
1520:
1519:
1140:{\displaystyle K}
910:{\displaystyle f}
890:{\displaystyle Y}
870:{\displaystyle f}
850:{\displaystyle X}
410:{\displaystyle K}
361:{\displaystyle X}
247:{\displaystyle Y}
223:{\displaystyle Y}
187:{\displaystyle f}
144:{\displaystyle Y}
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3053:
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3025:
3016:
3006:
2996:
2995:
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2983:
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2624:
2597:
2570:
2549:Johnstone, Peter
2544:
2500:
2499:
2489:
2460:
2454:
2448:
2433:
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2370:
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2343:
2338:
2326:
2324:
2323:
2318:
2287:
2285:
2284:
2279:
2249:
2247:
2246:
2241:
2223:
2221:
2220:
2215:
2195:compact covering
2191:
2189:
2188:
2183:
2133:
2131:
2130:
2125:
2110:
2108:
2107:
2102:
2100:
2096:
2095:
2091:
2090:
2062:
2060:
2059:
2054:
2039:
2037:
2036:
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2029:
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2004:
2002:
2001:
1996:
1972:
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1964:
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1947:
1946:
1941:
1939:
1938:
1922:
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1914:
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1889:
1888:
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1863:
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1834:
1832:
1831:
1826:
1814:
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1811:
1806:
1792:
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1789:
1784:
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1755:
1750:
1749:
1730:
1728:
1727:
1722:
1711:
1710:
1691:
1689:
1688:
1683:
1665:
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1662:
1657:
1645:
1643:
1642:
1637:
1611:
1610:
1588:
1586:
1585:
1580:
1562:
1560:
1559:
1554:
1542:
1540:
1539:
1534:
1514:
1512:
1511:
1506:
1492:
1491:
1472:
1470:
1469:
1464:
1462:
1461:
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1451:
1433:
1429:
1428:
1427:
1426:
1425:
1410:
1405:
1385:
1384:
1360:
1359:
1337:
1335:
1334:
1329:
1317:
1315:
1314:
1309:
1307:
1306:
1305:
1304:
1289:
1284:
1256:
1254:
1253:
1248:
1243:
1242:
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1240:
1225:
1220:
1192:
1190:
1189:
1184:
1182:
1181:
1163:
1162:
1146:
1144:
1143:
1138:
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1124:
1123:
1118:
1116:
1115:
1106:
1105:
1077:
1075:
1074:
1069:
1054:
1052:
1051:
1046:
1044:
1043:
1028:It follows that
1027:
1025:
1024:
1019:
1004:
1002:
1001:
996:
994:
990:
989:
988:
979:
978:
977:
976:
933:
932:
916:
914:
913:
908:
896:
894:
893:
888:
876:
874:
873:
868:
856:
854:
853:
848:
836:
834:
833:
828:
826:
825:
816:
815:
814:
813:
780:
778:
777:
772:
767:
766:
757:
756:
755:
754:
722:
721:
702:
700:
699:
694:
686:
685:
669:
667:
666:
661:
640:
638:
637:
632:
618:
617:
598:
596:
595:
590:
572:
570:
569:
564:
550:
549:
530:
528:
527:
522:
520:
516:
503:
502:
478:
476:
475:
470:
459:
458:
439:
437:
436:
431:
416:
414:
413:
408:
396:
394:
393:
388:
367:
365:
364:
359:
347:
345:
344:
339:
328:
327:
308:
306:
305:
300:
266:
265:
253:
251:
250:
245:
229:
227:
226:
221:
193:
191:
190:
185:
173:
171:
170:
165:
150:
148:
147:
142:
113:
111:
110:
105:
58:are compact. In
3091:
3090:
3086:
3085:
3084:
3082:
3081:
3080:
3066:
3065:
3064:
3059:
2990:
2972:
2968:Urysohn's lemma
2929:
2893:
2779:
2770:
2742:low-dimensional
2700:
2695:
2665:
2651:
2641:Springer-Verlag
2594:
2567:
2533:
2523:Springer-Verlag
2509:
2504:
2503:
2461:
2457:
2449:
2445:
2440:
2431:
2410:Almost open map
2406:
2386:
2364:
2361:
2360:
2357:locally compact
2353:first-countable
2332:
2329:
2328:
2300:
2297:
2296:
2255:
2252:
2251:
2229:
2226:
2225:
2203:
2200:
2199:
2165:
2162:
2161:
2144:Hausdorff space
2139:
2116:
2113:
2112:
2086:
2082:
2078:
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2064:
2045:
2042:
2041:
2020:
2016:
2012:
2010:
2007:
2006:
1978:
1975:
1974:
1955:
1952:
1951:
1934:
1930:
1928:
1925:
1924:
1902:
1899:
1898:
1877:
1874:
1873:
1853:
1849:
1844:
1841:
1840:
1820:
1817:
1816:
1800:
1797:
1796:
1769:
1766:
1765:
1745:
1741:
1736:
1733:
1732:
1706:
1702:
1697:
1694:
1693:
1671:
1668:
1667:
1651:
1648:
1647:
1606:
1602:
1594:
1591:
1590:
1574:
1571:
1570:
1548:
1545:
1544:
1528:
1525:
1524:
1521:
1484:
1480:
1478:
1475:
1474:
1457:
1453:
1441:
1437:
1421:
1417:
1416:
1412:
1406:
1395:
1390:
1386:
1377:
1373:
1352:
1348:
1346:
1343:
1342:
1323:
1320:
1319:
1300:
1296:
1295:
1291:
1285:
1274:
1262:
1259:
1258:
1236:
1232:
1231:
1227:
1221:
1210:
1198:
1195:
1194:
1177:
1173:
1158:
1154:
1152:
1149:
1148:
1132:
1129:
1128:
1111:
1107:
1095:
1091:
1083:
1080:
1079:
1060:
1057:
1056:
1039:
1035:
1033:
1030:
1029:
1010:
1007:
1006:
984:
980:
972:
968:
961:
957:
950:
946:
928:
924:
922:
919:
918:
902:
899:
898:
882:
879:
878:
862:
859:
858:
842:
839:
838:
821:
817:
809:
805:
798:
794:
786:
783:
782:
762:
758:
750:
746:
739:
735:
714:
710:
708:
705:
704:
681:
677:
675:
672:
671:
646:
643:
642:
610:
606:
604:
601:
600:
578:
575:
574:
542:
538:
536:
533:
532:
498:
494:
493:
489:
487:
484:
483:
451:
447:
445:
442:
441:
422:
419:
418:
402:
399:
398:
373:
370:
369:
353:
350:
349:
348:is compact (in
320:
316:
314:
311:
310:
282:
279:
278:
271:
256:locally compact
239:
236:
235:
215:
212:
211:
179:
176:
175:
156:
153:
152:
136:
133:
132:
87:
84:
83:
76:
68:proper morphism
56:compact subsets
32:
25:convex analysis
17:
12:
11:
5:
3089:
3079:
3078:
3061:
3060:
3058:
3057:
3047:
3046:
3045:
3040:
3035:
3020:
3010:
3000:
2988:
2977:
2974:
2973:
2971:
2970:
2965:
2960:
2955:
2950:
2945:
2939:
2937:
2931:
2930:
2928:
2927:
2922:
2917:
2915:Winding number
2912:
2907:
2901:
2899:
2895:
2894:
2892:
2891:
2886:
2881:
2876:
2871:
2866:
2861:
2856:
2855:
2854:
2849:
2847:homotopy group
2839:
2838:
2837:
2832:
2827:
2822:
2817:
2807:
2802:
2797:
2787:
2785:
2781:
2780:
2773:
2771:
2769:
2768:
2763:
2758:
2757:
2756:
2746:
2745:
2744:
2734:
2729:
2724:
2719:
2714:
2708:
2706:
2702:
2701:
2694:
2693:
2686:
2679:
2671:
2664:
2663:
2649:
2625:
2615:(3): 515–522.
2599:
2592:
2572:
2565:
2545:
2531:
2510:
2508:
2505:
2502:
2501:
2480:(4): 835–836.
2455:
2442:
2441:
2439:
2436:
2435:
2434:
2425:
2419:
2413:
2405:
2402:
2398:Johnstone 2002
2385:
2384:Generalization
2382:
2381:
2380:
2368:
2336:
2316:
2313:
2310:
2307:
2304:
2293:
2290:
2289:
2288:
2277:
2274:
2271:
2268:
2265:
2262:
2259:
2239:
2236:
2233:
2213:
2210:
2207:
2196:
2181:
2178:
2175:
2172:
2169:
2151:
2138:
2135:
2123:
2120:
2099:
2094:
2089:
2085:
2081:
2077:
2073:
2052:
2049:
2028:
2023:
2019:
2015:
1994:
1991:
1988:
1985:
1982:
1962:
1959:
1937:
1933:
1912:
1909:
1906:
1895:
1881:
1861:
1856:
1852:
1848:
1824:
1804:
1782:
1779:
1776:
1773:
1753:
1748:
1744:
1740:
1720:
1717:
1714:
1709:
1705:
1701:
1681:
1678:
1675:
1655:
1635:
1632:
1629:
1626:
1623:
1620:
1617:
1614:
1609:
1605:
1601:
1598:
1578:
1567:
1552:
1532:
1518:
1517:
1504:
1501:
1498:
1495:
1490:
1487:
1483:
1460:
1456:
1450:
1447:
1444:
1440:
1436:
1432:
1424:
1420:
1415:
1409:
1404:
1401:
1398:
1394:
1389:
1383:
1380:
1376:
1372:
1369:
1366:
1363:
1358:
1355:
1351:
1338:a finite set.
1327:
1303:
1299:
1294:
1288:
1283:
1280:
1277:
1273:
1269:
1266:
1246:
1239:
1235:
1230:
1224:
1219:
1216:
1213:
1209:
1205:
1202:
1180:
1176:
1172:
1169:
1166:
1161:
1157:
1136:
1114:
1110:
1104:
1101:
1098:
1094:
1090:
1087:
1067:
1064:
1042:
1038:
1017:
1014:
993:
987:
983:
975:
971:
967:
964:
960:
956:
953:
949:
945:
942:
939:
936:
931:
927:
906:
886:
866:
846:
824:
820:
812:
808:
804:
801:
797:
793:
790:
770:
765:
761:
753:
749:
745:
742:
738:
734:
731:
728:
725:
720:
717:
713:
692:
689:
684:
680:
659:
656:
653:
650:
630:
627:
624:
621:
616:
613:
609:
588:
585:
582:
562:
559:
556:
553:
548:
545:
541:
519:
515:
512:
509:
506:
501:
497:
492:
468:
465:
462:
457:
454:
450:
429:
426:
406:
386:
383:
380:
377:
357:
337:
334:
331:
326:
323:
319:
298:
295:
292:
289:
286:
273:
272:
269:
264:
243:
219:
201:
196:
183:
163:
160:
151:is compact in
140:
121:
103:
100:
97:
94:
91:
75:
72:
52:inverse images
15:
9:
6:
4:
3:
2:
3088:
3077:
3074:
3073:
3071:
3056:
3048:
3044:
3041:
3039:
3036:
3034:
3031:
3030:
3029:
3021:
3019:
3015:
3011:
3009:
3005:
3001:
2999:
2994:
2989:
2987:
2979:
2978:
2975:
2969:
2966:
2964:
2961:
2959:
2956:
2954:
2951:
2949:
2946:
2944:
2941:
2940:
2938:
2936:
2932:
2926:
2925:Orientability
2923:
2921:
2918:
2916:
2913:
2911:
2908:
2906:
2903:
2902:
2900:
2896:
2890:
2887:
2885:
2882:
2880:
2877:
2875:
2872:
2870:
2867:
2865:
2862:
2860:
2857:
2853:
2850:
2848:
2845:
2844:
2843:
2840:
2836:
2833:
2831:
2828:
2826:
2823:
2821:
2818:
2816:
2813:
2812:
2811:
2808:
2806:
2803:
2801:
2798:
2796:
2792:
2789:
2788:
2786:
2782:
2777:
2767:
2764:
2762:
2761:Set-theoretic
2759:
2755:
2752:
2751:
2750:
2747:
2743:
2740:
2739:
2738:
2735:
2733:
2730:
2728:
2725:
2723:
2722:Combinatorial
2720:
2718:
2715:
2713:
2710:
2709:
2707:
2703:
2699:
2692:
2687:
2685:
2680:
2678:
2673:
2672:
2669:
2660:
2656:
2652:
2646:
2642:
2638:
2634:
2630:
2626:
2622:
2618:
2614:
2610:
2609:
2604:
2603:Brown, Ronald
2600:
2595:
2593:1-4196-2722-8
2589:
2585:
2581:
2577:
2576:Brown, Ronald
2573:
2568:
2566:0-19-851598-7
2562:
2558:
2554:
2550:
2546:
2542:
2538:
2534:
2528:
2524:
2520:
2516:
2512:
2511:
2497:
2493:
2488:
2483:
2479:
2475:
2474:
2469:
2465:
2459:
2452:
2447:
2443:
2429:
2426:
2423:
2420:
2417:
2414:
2411:
2408:
2407:
2401:
2399:
2395:
2391:
2366:
2358:
2354:
2350:
2334:
2314:
2308:
2305:
2302:
2294:
2291:
2275:
2272:
2269:
2263:
2257:
2237:
2234:
2231:
2211:
2208:
2205:
2197:
2194:
2179:
2173:
2170:
2167:
2159:
2158:
2156:
2152:
2149:
2145:
2141:
2140:
2134:
2121:
2118:
2097:
2092:
2087:
2083:
2079:
2075:
2071:
2063:the sequence
2050:
2047:
2026:
2021:
2017:
2013:
1992:
1986:
1983:
1980:
1960:
1957:
1935:
1931:
1910:
1907:
1904:
1896:
1893:
1879:
1854:
1850:
1838:
1837:metric spaces
1822:
1802:
1793:
1780:
1777:
1774:
1771:
1751:
1746:
1742:
1738:
1718:
1712:
1707:
1703:
1699:
1692:the pullback
1679:
1673:
1653:
1633:
1630:
1627:
1621:
1618:
1615:
1612:
1607:
1603:
1599:
1596:
1576:
1568:
1565:
1550:
1530:
1516:
1502:
1496:
1488:
1485:
1481:
1458:
1454:
1445:
1442:
1438:
1434:
1430:
1422:
1418:
1413:
1407:
1402:
1399:
1396:
1392:
1387:
1381:
1378:
1374:
1370:
1364:
1356:
1353:
1349:
1339:
1301:
1297:
1292:
1286:
1281:
1278:
1275:
1271:
1267:
1244:
1237:
1233:
1228:
1222:
1217:
1214:
1211:
1207:
1203:
1200:
1178:
1174:
1170:
1167:
1164:
1159:
1155:
1134:
1112:
1108:
1102:
1099:
1096:
1092:
1088:
1085:
1065:
1062:
1040:
1036:
1015:
1012:
991:
985:
981:
973:
969:
965:
962:
958:
951:
947:
943:
937:
934:
929:
925:
904:
884:
877:is closed in
864:
844:
837:is closed in
822:
818:
810:
806:
802:
799:
795:
788:
768:
763:
759:
751:
747:
743:
740:
736:
732:
726:
718:
715:
711:
690:
687:
682:
678:
657:
654:
651:
648:
628:
622:
614:
611:
607:
586:
583:
580:
573:Then for all
560:
554:
546:
543:
539:
517:
513:
510:
507:
504:
499:
495:
490:
480:
463:
455:
452:
448:
427:
424:
404:
384:
381:
378:
375:
355:
332:
324:
321:
317:
296:
290:
287:
284:
275:
274:
268:
267:
263:
261:
257:
241:
233:
217:
209:
206:
202:
199:
194:
181:
161:
158:
138:
130:
126:
122:
119:
117:
101:
95:
92:
89:
81:
71:
69:
65:
61:
57:
53:
49:
45:
41:
37:
30:
26:
22:
3055:Publications
2920:Chern number
2910:Betti number
2793: /
2784:Key concepts
2732:Differential
2632:
2629:Lee, John M.
2612:
2606:
2579:
2552:
2518:
2477:
2471:
2458:
2446:
2387:
2193:
2192:is called a
1892:
1794:
1564:
1522:
1340:
1127:and because
481:
479:is compact.
276:
198:
118:
114:between two
77:
47:
33:
3018:Wikiversity
2935:Key results
2422:Perfect map
1005:is open in
36:mathematics
2864:CW complex
2805:Continuity
2795:Closed set
2754:cohomology
2555:. Oxford:
2507:References
2379:is closed.
2250:such that
2155:surjective
2137:Properties
1193:such that
703:such that
368:) for all
208:closed map
205:continuous
74:Definition
46:is called
3043:geometric
3038:algebraic
2889:Cobordism
2825:Hausdorff
2820:connected
2737:Geometric
2727:Continuum
2717:Algebraic
2659:808682771
2584:Booksurge
2438:Citations
2312:→
2235:⊆
2209:⊆
2177:→
1990:→
1908:⊆
1775:×
1743:×
1716:→
1704:×
1677:→
1631:×
1625:→
1619:×
1600:×
1486:−
1449:Γ
1446:∈
1439:∪
1435:⊆
1393:∪
1379:−
1371:⊆
1354:−
1326:Γ
1293:γ
1272:∪
1265:Γ
1208:∪
1204:⊆
1168:…
1100:∈
1093:∪
1089:⊆
970:γ
966:∈
959:∪
955:∖
941:∖
807:γ
803:∈
796:∪
792:∖
748:γ
744:∈
737:∪
733:⊆
716:−
688:⊆
679:γ
652:∈
612:−
584:∈
544:−
511:∈
453:−
379:∈
322:−
294:→
260:Hausdorff
127:of every
99:→
64:analogous
3070:Category
3008:Wikibook
2986:Category
2874:Manifold
2842:Homotopy
2800:Interior
2791:Open set
2749:Homology
2698:Topology
2631:(2012).
2578:(2006).
2551:(2002).
2517:(1998).
2466:(1970).
2451:Lee 2012
2404:See also
2359:), then
1589:the map
897:because
781:The set
125:preimage
80:function
42:between
40:function
21:topology
3033:general
2835:uniform
2815:compact
2766:Digital
2541:1726872
2496:0254818
2396:, see (
2390:locales
1950:are in
232:compact
131:set in
129:compact
123:if the
3028:Topics
2830:metric
2705:Fields
2657:
2647:
2590:
2563:
2539:
2529:
2494:
2160:A map
2153:Every
2148:closed
195:proper
120:proper
62:, the
48:proper
27:, see
2810:Space
2394:topoi
2347:is a
2655:OCLC
2645:ISBN
2588:ISBN
2561:ISBN
2527:ISBN
2392:and
1835:are
1815:and
1078:Now
482:Let
397:Let
277:Let
258:and
38:, a
2617:doi
2482:doi
2400:).
2355:or
2295:If
1523:If
254:is
230:is
54:of
50:if
34:In
3072::
2653:.
2643:.
2635:.
2586:.
2559:.
2537:MR
2535:.
2525:.
2492:MR
2490:.
2478:24
2476:.
2470:.
1604:id
262:.
70:.
2690:e
2683:t
2676:v
2661:.
2623:.
2619::
2613:7
2596:.
2569:.
2543:.
2498:.
2484::
2367:f
2335:Y
2315:Y
2309:X
2306::
2303:f
2276:.
2273:K
2270:=
2267:)
2264:C
2261:(
2258:f
2238:X
2232:C
2212:Y
2206:K
2180:Y
2174:X
2171::
2168:f
2150:.
2122:.
2119:Y
2098:}
2093:)
2088:i
2084:p
2080:(
2076:f
2072:{
2051:,
2048:X
2027:}
2022:i
2018:p
2014:{
1993:Y
1987:X
1984::
1981:f
1961:.
1958:S
1936:i
1932:p
1911:X
1905:S
1880:X
1860:}
1855:i
1851:p
1847:{
1823:Y
1803:X
1781:.
1778:Z
1772:X
1752:Z
1747:Y
1739:X
1719:Z
1713:Z
1708:Y
1700:X
1680:Y
1674:Z
1654:Y
1634:Z
1628:Y
1622:Z
1616:X
1613::
1608:Z
1597:f
1577:Z
1551:Y
1531:X
1503:,
1500:)
1497:K
1494:(
1489:1
1482:f
1459:a
1455:U
1443:a
1431:)
1423:i
1419:k
1414:V
1408:s
1403:1
1400:=
1397:i
1388:(
1382:1
1375:f
1368:)
1365:K
1362:(
1357:1
1350:f
1302:i
1298:k
1287:s
1282:1
1279:=
1276:i
1268:=
1245:.
1238:i
1234:k
1229:V
1223:s
1218:1
1215:=
1212:i
1201:K
1179:s
1175:k
1171:,
1165:,
1160:1
1156:k
1135:K
1113:k
1109:V
1103:K
1097:k
1086:K
1066:.
1063:k
1041:k
1037:V
1016:.
1013:Y
992:)
986:a
982:U
974:k
963:a
952:X
948:(
944:f
938:Y
935:=
930:k
926:V
905:f
885:Y
865:f
845:X
823:a
819:U
811:k
800:a
789:X
769:.
764:a
760:U
752:k
741:a
730:)
727:k
724:(
719:1
712:f
691:A
683:k
658:,
655:K
649:k
629:.
626:)
623:k
620:(
615:1
608:f
587:K
581:k
561:.
558:)
555:K
552:(
547:1
540:f
518:}
514:A
508:a
505::
500:a
496:U
491:{
467:)
464:K
461:(
456:1
449:f
428:.
425:Y
405:K
385:.
382:Y
376:y
356:X
336:)
333:y
330:(
325:1
318:f
297:Y
291:X
288::
285:f
242:Y
218:Y
182:f
162:.
159:X
139:Y
102:Y
96:X
93::
90:f
31:.
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