2642:
2425:
2663:
2631:
2700:
2673:
2653:
702:
1383:
1273:
839:
1164:
161:
1087:
759:
927:
1031:
676:
485:
1583:
1554:
1496:
1467:
584:
453:
232:
120:
63:
1430:
1223:
999:
977:
1671:
1525:
203:
515:
420:
1772:
1896:
1832:
1812:
1792:
1739:
1719:
1695:
1635:
1615:
1403:
1184:
870:
696:
644:
624:
604:
555:
535:
387:
367:
340:
296:
276:
252:
87:
2272:
1278:
2703:
698:
in the subspace topology is automatic.) Both of these statements are not at all obvious and are not generally true if one leaves
Euclidean space.
2729:
2114:
2057:
1232:
764:
2337:
2299:
1092:
2691:
2686:
2220:
2084:
1948:
125:
2180:
1036:
708:
2681:
879:
2138:
2739:
2583:
2309:
2724:
2304:
311:
2591:
2243:
2212:
2163:
2130:
2052:. Texts and Readings in Mathematics. Vol. 27 (Second ed.). New Delhi: Hindustan Book Agency.
1674:
1004:
649:
458:
1559:
1530:
1472:
1443:
560:
429:
208:
96:
39:
1408:
1201:
701:
2390:
982:
2734:
2676:
2662:
1698:
932:
1189:
The theorem is also not generally true in infinitely many dimensions. Consider for instance the
2611:
2532:
2409:
2397:
2370:
2330:
2606:
1644:
1501:
2453:
2380:
2121:(see p. 72–73 for Hirsch's proof utilizing non-existence of a differentiable retraction)
1900:
173:
2079:. Cahiers Scientifiques (in French). Vol. IX. Paris: Gauthier-Villars. pp. 44–47.
490:
395:
2553:
2527:
2375:
2251:
2230:
2199:
2171:
2148:
2094:
2067:
2032:
2011:
1989:
1958:
1854:
845:
1748:
8:
2448:
2006:
1968:
1835:
849:
2652:
2646:
2616:
2596:
2517:
2507:
2385:
2365:
2157:
2036:
1993:
1881:
1817:
1797:
1777:
1724:
1704:
1680:
1620:
1600:
1388:
1169:
855:
681:
629:
609:
589:
540:
520:
372:
352:
325:
307:
281:
261:
237:
72:
2273:"Brouwer's fixed point and invariance of domain theorems, and Hilbert's fifth problem"
2641:
2634:
2500:
2458:
2323:
2216:
2134:
2110:
2080:
2053:
2040:
1997:
1944:
1378:{\displaystyle f\left(x_{1},x_{2},\ldots \right)=\left(0,x_{1},x_{2},\ldots \right).}
2666:
1186:
is injective and continuous but does not even yield a homeomorphism onto its image.
2414:
2360:
2020:
1977:
390:
303:
1981:
2473:
2468:
2247:
2226:
2195:
2167:
2144:
2090:
2063:
2028:
1985:
1954:
1940:
34:
2656:
2563:
2495:
1874:
167:
1966:
Cao Labora, Daniel (2020). "When is a continuous bijection a homeomorphism?".
2718:
2573:
2483:
2463:
2102:
255:
2558:
2478:
2424:
1842:
1190:
27:
2209:
From calculus to cohomology: de Rham cohomology and characteristic classes
1841:
There are also generalizations to certain types of continuous maps from a
2568:
2268:
2512:
2443:
2402:
2024:
1914:
979:
This map is injective and continuous, the domain is an open subset of
2537:
1857:
for other conditions that ensure that a given continuous map is open.
164:
2522:
2490:
2439:
2346:
1742:
1594:
1226:
1193:
343:
90:
23:
2242:. Annals of Mathematics Studies. Vol. 54 (Revised ed.).
1440:
An important consequence of the domain invariance theorem is that
322:
The conclusion of the theorem can equivalently be formulated as: "
517:
is automatic. Furthermore, the theorem says that if two subsets
16:
Theorem in topology about homeomorphic subsets of
Euclidean space
30:
2315:
422:
are continuous; the theorem says that if the domain is an
1904:
71 (1912), pages 305–315; see also 72 (1912), pages 55–56
2075:
Dieudonné, Jean (1982). "8. Les théorèmes de
Brouwer".
369:
is a homeomorphism, one would have to verify that both
2127:
Homology theory: An introduction to algebraic topology
1884:
1820:
1800:
1780:
1751:
1727:
1707:
1683:
1647:
1623:
1603:
1562:
1533:
1504:
1475:
1446:
1411:
1391:
1281:
1235:
1204:
1172:
1095:
1039:
1007:
985:
935:
882:
858:
767:
711:
684:
652:
632:
612:
592:
563:
543:
523:
493:
461:
432:
398:
375:
355:
328:
284:
264:
240:
211:
176:
128:
99:
75:
42:
1721:
restricted to this neighborhood is injective), then
1593:
The domain invariance theorem may be generalized to
1268:{\displaystyle f:\ell ^{\infty }\to \ell ^{\infty }}
678:
and not just in the subspace topology. Openness of
1405:is injective and continuous, the domain is open in
705:A map which is not a homeomorphism onto its image:
1890:
1826:
1806:
1786:
1766:
1733:
1713:
1689:
1665:
1629:
1609:
1577:
1548:
1519:
1490:
1461:
1424:
1397:
1377:
1267:
1217:
1178:
1158:
1081:
1025:
993:
971:
921:
864:
834:{\displaystyle g(t)=\left(t^{2}-1,t^{3}-t\right).}
833:
753:
690:
670:
638:
618:
598:
578:
549:
529:
509:
479:
447:
414:
381:
361:
334:
290:
270:
246:
226:
197:
155:
114:
81:
57:
1159:{\displaystyle g(t)=\left(t^{2}-1,t^{3}-t\right)}
2716:
1939:. Graduate Texts in Mathematics. Vol. 139.
2206:
2155:
156:{\displaystyle f:U\rightarrow \mathbb {R} ^{n}}
2162:. Princeton Mathematical Series. Vol. 4.
1918:Topologie des espaces abstraits de M. Banach.
1082:{\displaystyle g:(-1.1,1)\to \mathbb {R} ^{2}}
754:{\displaystyle g:(-1.1,1)\to \mathbb {R} ^{2}}
306:, published in 1912. The proof uses tools of
2331:
2124:
2007:"Méthodes modernes en topologie algébrique"
922:{\displaystyle f:(0,1)\to \mathbb {R} ^{2}}
2699:
2672:
2338:
2324:
1965:
1556:can be homeomorphic to any open subset of
2156:Hurewicz, Witold; Wallman, Henry (1941).
2074:
1565:
1536:
1478:
1449:
1069:
1010:
987:
909:
741:
655:
566:
464:
435:
214:
143:
102:
45:
2181:"Poincaré and domain invariance theorem"
700:
2263:. New York-Toronto-London: McGraw-Hill.
2258:
2237:
2125:Hilton, Peter J.; Wylie, Shaun (1960).
2717:
2207:Madsen, Ib; Tornehave, Jørgen (1997).
2101:
2004:
1934:
844:It is of crucial importance that both
2319:
2178:
302:The theorem and its proof are due to
2297:
1527:Indeed, no non-empty open subset of
2267:
2047:
13:
1588:
1417:
1260:
1247:
1210:
1033:A more extreme example is the map
626:must be open as well. (Note that
14:
2751:
2291:
1026:{\displaystyle \mathbb {R} ^{2}.}
876:. Consider for instance the map
872:are contained in Euclidean space
671:{\displaystyle \mathbb {R} ^{n},}
480:{\displaystyle \mathbb {R} ^{n},}
2698:
2671:
2661:
2651:
2640:
2630:
2629:
2423:
2240:Elementary differential topology
1641:-manifolds without boundary and
1578:{\displaystyle \mathbb {R} ^{m}}
1549:{\displaystyle \mathbb {R} ^{n}}
1491:{\displaystyle \mathbb {R} ^{m}}
1462:{\displaystyle \mathbb {R} ^{n}}
579:{\displaystyle \mathbb {R} ^{n}}
448:{\displaystyle \mathbb {R} ^{n}}
227:{\displaystyle \mathbb {R} ^{n}}
115:{\displaystyle \mathbb {R} ^{n}}
58:{\displaystyle \mathbb {R} ^{n}}
1435:
1425:{\displaystyle \ell ^{\infty }}
1218:{\displaystyle \ell ^{\infty }}
1001:, but the image is not open in
2730:Theory of continuous functions
2188:Acta Univ. Carolin. Math. Phys
1907:
1867:
1761:
1755:
1657:
1252:
1105:
1099:
1064:
1061:
1046:
963:
951:
945:
939:
904:
901:
889:
777:
771:
736:
733:
718:
192:
186:
138:
1:
1982:10.1080/00029890.2020.1738826
1928:
1922:, 200 (1935) pages 1083–1093
1677:(meaning that every point in
1673:is a continuous map which is
2345:
2050:Algebraic topology: A primer
994:{\displaystyle \mathbb {R} }
7:
2305:Encyclopedia of Mathematics
1848:
972:{\displaystyle f(t)=(t,0).}
312:Brouwer fixed point theorem
10:
2756:
2592:Banach fixed-point theorem
2259:Spanier, Edwin H. (1966).
2244:Princeton University Press
2238:Munkres, James R. (1966).
2213:Cambridge University Press
2164:Princeton University Press
2131:Cambridge University Press
1861:
1469:cannot be homeomorphic to
2625:
2582:
2546:
2432:
2421:
2353:
2179:Kulpa, Władysław (1998).
1878:Beweis der Invarianz des
455:and the image is also in
1935:Bredon, Glen E. (1993).
1898:-dimensionalen Gebiets,
1666:{\displaystyle f:M\to N}
1520:{\displaystyle m\neq n.}
1432:, but the image is not.
349:Normally, to check that
317:
646:is open as a subset of
198:{\displaystyle V:=f(U)}
2647:Mathematics portal
2547:Metrics and properties
2533:Second-countable space
2298:Mill, J. van (2001) ,
2277:terrytao.wordpress.com
2109:. New York: Springer.
2005:Cartan, Henri (1945).
1920:C. R. Acad. Sci. Paris
1892:
1828:
1808:
1788:
1768:
1735:
1715:
1691:
1667:
1631:
1611:
1579:
1550:
1521:
1492:
1463:
1426:
1399:
1379:
1269:
1219:
1180:
1160:
1083:
1027:
995:
973:
923:
866:
841:
835:
755:
692:
672:
640:
620:
600:
586:are homeomorphic, and
580:
551:
531:
511:
510:{\displaystyle f^{-1}}
481:
449:
416:
415:{\displaystyle f^{-1}}
383:
363:
336:
292:
272:
248:
228:
199:
157:
116:
83:
59:
2107:Differential Topology
1937:Topology and geometry
1901:Mathematische Annalen
1893:
1829:
1814:is an open subset of
1809:
1789:
1769:
1736:
1716:
1692:
1668:
1632:
1612:
1580:
1551:
1522:
1493:
1464:
1427:
1400:
1380:
1270:
1220:
1181:
1161:
1084:
1028:
996:
974:
924:
874:of the same dimension
867:
836:
756:
704:
693:
673:
641:
621:
601:
581:
552:
532:
512:
482:
450:
417:
384:
364:
337:
293:
273:
249:
229:
200:
158:
117:
84:
60:
2740:Theorems in topology
2602:Invariance of domain
2554:Euler characteristic
2528:Bundle (mathematics)
2012:Comment. Math. Helv.
1882:
1855:Open mapping theorem
1818:
1798:
1778:
1767:{\displaystyle f(U)}
1749:
1725:
1705:
1681:
1645:
1621:
1601:
1560:
1531:
1502:
1473:
1444:
1409:
1389:
1279:
1233:
1225:of all bounded real
1202:
1170:
1093:
1037:
1005:
983:
933:
880:
856:
765:
709:
682:
650:
630:
610:
590:
561:
541:
521:
491:
459:
430:
396:
373:
353:
326:
282:
262:
238:
209:
174:
126:
97:
73:
40:
20:Invariance of domain
2612:Tychonoff's theorem
2607:Poincaré conjecture
2361:General (point-set)
2300:"Domain invariance"
2048:Deo, Satya (2018).
1969:Amer. Math. Monthly
1836:local homeomorphism
487:then continuity of
2725:Algebraic topology
2597:De Rham cohomology
2518:Polyhedral complex
2508:Simplicial complex
2261:Algebraic topology
2077:Éléments d'analyse
2025:10.1007/BF02568096
1888:
1824:
1804:
1784:
1764:
1731:
1711:
1687:
1675:locally one-to-one
1663:
1627:
1607:
1575:
1546:
1517:
1488:
1459:
1422:
1395:
1375:
1265:
1215:
1176:
1156:
1079:
1023:
991:
969:
919:
862:
842:
831:
751:
688:
668:
636:
616:
596:
576:
547:
527:
507:
477:
445:
412:
379:
359:
332:
308:algebraic topology
288:
268:
244:
224:
195:
153:
112:
79:
55:
2712:
2711:
2501:fundamental group
2116:978-0-387-90148-0
2103:Hirsch, Morris W.
2059:978-93-86279-67-5
1891:{\displaystyle n}
1827:{\displaystyle M}
1807:{\displaystyle U}
1787:{\displaystyle N}
1734:{\displaystyle f}
1714:{\displaystyle f}
1690:{\displaystyle M}
1630:{\displaystyle N}
1610:{\displaystyle M}
1398:{\displaystyle f}
1179:{\displaystyle g}
865:{\displaystyle f}
691:{\displaystyle V}
639:{\displaystyle V}
619:{\displaystyle V}
599:{\displaystyle U}
550:{\displaystyle V}
530:{\displaystyle U}
382:{\displaystyle f}
362:{\displaystyle f}
335:{\displaystyle f}
291:{\displaystyle V}
271:{\displaystyle U}
247:{\displaystyle f}
82:{\displaystyle U}
2747:
2702:
2701:
2675:
2674:
2665:
2655:
2645:
2644:
2633:
2632:
2427:
2340:
2333:
2326:
2317:
2316:
2312:
2287:
2285:
2283:
2264:
2255:
2234:
2203:
2185:
2175:
2159:Dimension Theory
2152:
2120:
2098:
2071:
2044:
2001:
1962:
1923:
1917:
1911:
1905:
1897:
1895:
1894:
1889:
1877:
1871:
1833:
1831:
1830:
1825:
1813:
1811:
1810:
1805:
1793:
1791:
1790:
1785:
1773:
1771:
1770:
1765:
1740:
1738:
1737:
1732:
1720:
1718:
1717:
1712:
1696:
1694:
1693:
1688:
1672:
1670:
1669:
1664:
1640:
1637:are topological
1636:
1634:
1633:
1628:
1616:
1614:
1613:
1608:
1584:
1582:
1581:
1576:
1574:
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1555:
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1125:
1124:
1088:
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1080:
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1032:
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1029:
1024:
1019:
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1000:
998:
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992:
990:
978:
976:
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928:
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920:
918:
917:
912:
871:
869:
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863:
840:
838:
837:
832:
827:
823:
816:
815:
797:
796:
760:
758:
757:
752:
750:
749:
744:
697:
695:
694:
689:
677:
675:
674:
669:
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658:
645:
643:
642:
637:
625:
623:
622:
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605:
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602:
597:
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583:
582:
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569:
556:
554:
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536:
534:
533:
528:
516:
514:
513:
508:
506:
505:
486:
484:
483:
478:
473:
472:
467:
454:
452:
451:
446:
444:
443:
438:
421:
419:
418:
413:
411:
410:
391:inverse function
388:
386:
385:
380:
368:
366:
365:
360:
341:
339:
338:
333:
304:L. E. J. Brouwer
297:
295:
294:
289:
277:
275:
274:
269:
253:
251:
250:
245:
233:
231:
230:
225:
223:
222:
217:
204:
202:
201:
196:
162:
160:
159:
154:
152:
151:
146:
121:
119:
118:
113:
111:
110:
105:
88:
86:
85:
80:
64:
62:
61:
56:
54:
53:
48:
22:is a theorem in
2755:
2754:
2750:
2749:
2748:
2746:
2745:
2744:
2715:
2714:
2713:
2708:
2639:
2621:
2617:Urysohn's lemma
2578:
2542:
2428:
2419:
2391:low-dimensional
2349:
2344:
2294:
2281:
2279:
2223:
2183:
2141:
2117:
2087:
2060:
1951:
1941:Springer-Verlag
1931:
1926:
1913:
1912:
1908:
1883:
1880:
1879:
1873:
1872:
1868:
1864:
1851:
1819:
1816:
1815:
1799:
1796:
1795:
1779:
1776:
1775:
1750:
1747:
1746:
1726:
1723:
1722:
1706:
1703:
1702:
1682:
1679:
1678:
1646:
1643:
1642:
1638:
1622:
1619:
1618:
1602:
1599:
1598:
1591:
1589:Generalizations
1569:
1564:
1563:
1561:
1558:
1557:
1540:
1535:
1534:
1532:
1529:
1528:
1503:
1500:
1499:
1482:
1477:
1476:
1474:
1471:
1470:
1453:
1448:
1447:
1445:
1442:
1441:
1438:
1416:
1412:
1410:
1407:
1406:
1390:
1387:
1386:
1355:
1351:
1342:
1338:
1331:
1327:
1307:
1303:
1294:
1290:
1289:
1285:
1280:
1277:
1276:
1259:
1255:
1246:
1242:
1234:
1231:
1230:
1209:
1205:
1203:
1200:
1199:
1194:
1171:
1168:
1167:
1139:
1135:
1120:
1116:
1115:
1111:
1094:
1091:
1090:
1073:
1068:
1067:
1038:
1035:
1034:
1014:
1009:
1008:
1006:
1003:
1002:
986:
984:
981:
980:
934:
931:
930:
913:
908:
907:
881:
878:
877:
857:
854:
853:
811:
807:
792:
788:
787:
783:
766:
763:
762:
745:
740:
739:
710:
707:
706:
683:
680:
679:
659:
654:
653:
651:
648:
647:
631:
628:
627:
611:
608:
607:
591:
588:
587:
570:
565:
564:
562:
559:
558:
542:
539:
538:
522:
519:
518:
498:
494:
492:
489:
488:
468:
463:
462:
460:
457:
456:
439:
434:
433:
431:
428:
427:
403:
399:
397:
394:
393:
374:
371:
370:
354:
351:
350:
327:
324:
323:
320:
283:
280:
279:
263:
260:
259:
239:
236:
235:
218:
213:
212:
210:
207:
206:
175:
172:
171:
147:
142:
141:
127:
124:
123:
106:
101:
100:
98:
95:
94:
74:
71:
70:
65:. It states:
49:
44:
43:
41:
38:
37:
35:Euclidean space
17:
12:
11:
5:
2753:
2743:
2742:
2737:
2735:Homeomorphisms
2732:
2727:
2710:
2709:
2707:
2706:
2696:
2695:
2694:
2689:
2684:
2669:
2659:
2649:
2637:
2626:
2623:
2622:
2620:
2619:
2614:
2609:
2604:
2599:
2594:
2588:
2586:
2580:
2579:
2577:
2576:
2571:
2566:
2564:Winding number
2561:
2556:
2550:
2548:
2544:
2543:
2541:
2540:
2535:
2530:
2525:
2520:
2515:
2510:
2505:
2504:
2503:
2498:
2496:homotopy group
2488:
2487:
2486:
2481:
2476:
2471:
2466:
2456:
2451:
2446:
2436:
2434:
2430:
2429:
2422:
2420:
2418:
2417:
2412:
2407:
2406:
2405:
2395:
2394:
2393:
2383:
2378:
2373:
2368:
2363:
2357:
2355:
2351:
2350:
2343:
2342:
2335:
2328:
2320:
2314:
2313:
2293:
2292:External links
2290:
2289:
2288:
2265:
2256:
2235:
2221:
2204:
2194:(1): 129–136.
2176:
2153:
2139:
2122:
2115:
2099:
2085:
2072:
2058:
2045:
2002:
1976:(6): 547–553.
1963:
1949:
1930:
1927:
1925:
1924:
1906:
1887:
1875:Brouwer L.E.J.
1865:
1863:
1860:
1859:
1858:
1850:
1847:
1823:
1803:
1783:
1763:
1760:
1757:
1754:
1745:(meaning that
1730:
1710:
1686:
1662:
1659:
1656:
1653:
1650:
1626:
1606:
1590:
1587:
1585:in this case.
1572:
1567:
1543:
1538:
1516:
1513:
1510:
1507:
1485:
1480:
1456:
1451:
1437:
1434:
1419:
1415:
1394:
1374:
1370:
1366:
1363:
1358:
1354:
1350:
1345:
1341:
1337:
1334:
1330:
1326:
1322:
1318:
1315:
1310:
1306:
1302:
1297:
1293:
1288:
1284:
1262:
1258:
1254:
1249:
1245:
1241:
1238:
1212:
1208:
1175:
1154:
1150:
1147:
1142:
1138:
1134:
1131:
1128:
1123:
1119:
1114:
1110:
1107:
1104:
1101:
1098:
1076:
1071:
1066:
1063:
1060:
1057:
1054:
1051:
1048:
1045:
1042:
1022:
1017:
1012:
989:
968:
965:
962:
959:
956:
953:
950:
947:
944:
941:
938:
916:
911:
906:
903:
900:
897:
894:
891:
888:
885:
875:
861:
830:
826:
822:
819:
814:
810:
806:
803:
800:
795:
791:
786:
782:
779:
776:
773:
770:
748:
743:
738:
735:
732:
729:
726:
723:
720:
717:
714:
687:
667:
662:
657:
635:
615:
606:is open, then
595:
573:
568:
546:
526:
504:
501:
497:
476:
471:
466:
442:
437:
425:
409:
406:
402:
378:
358:
331:
319:
316:
310:, notably the
300:
299:
287:
267:
243:
221:
216:
194:
191:
188:
185:
182:
179:
168:continuous map
150:
145:
140:
137:
134:
131:
109:
104:
78:
52:
47:
15:
9:
6:
4:
3:
2:
2752:
2741:
2738:
2736:
2733:
2731:
2728:
2726:
2723:
2722:
2720:
2705:
2697:
2693:
2690:
2688:
2685:
2683:
2680:
2679:
2678:
2670:
2668:
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2660:
2658:
2654:
2650:
2648:
2643:
2638:
2636:
2628:
2627:
2624:
2618:
2615:
2613:
2610:
2608:
2605:
2603:
2600:
2598:
2595:
2593:
2590:
2589:
2587:
2585:
2581:
2575:
2574:Orientability
2572:
2570:
2567:
2565:
2562:
2560:
2557:
2555:
2552:
2551:
2549:
2545:
2539:
2536:
2534:
2531:
2529:
2526:
2524:
2521:
2519:
2516:
2514:
2511:
2509:
2506:
2502:
2499:
2497:
2494:
2493:
2492:
2489:
2485:
2482:
2480:
2477:
2475:
2472:
2470:
2467:
2465:
2462:
2461:
2460:
2457:
2455:
2452:
2450:
2447:
2445:
2441:
2438:
2437:
2435:
2431:
2426:
2416:
2413:
2411:
2410:Set-theoretic
2408:
2404:
2401:
2400:
2399:
2396:
2392:
2389:
2388:
2387:
2384:
2382:
2379:
2377:
2374:
2372:
2371:Combinatorial
2369:
2367:
2364:
2362:
2359:
2358:
2356:
2352:
2348:
2341:
2336:
2334:
2329:
2327:
2322:
2321:
2318:
2311:
2307:
2306:
2301:
2296:
2295:
2278:
2274:
2270:
2266:
2262:
2257:
2253:
2249:
2245:
2241:
2236:
2232:
2228:
2224:
2222:0-521-58059-5
2218:
2214:
2210:
2205:
2201:
2197:
2193:
2189:
2182:
2177:
2173:
2169:
2165:
2161:
2160:
2154:
2150:
2146:
2142:
2136:
2132:
2128:
2123:
2118:
2112:
2108:
2104:
2100:
2096:
2092:
2088:
2086:2-04-011499-8
2082:
2078:
2073:
2069:
2065:
2061:
2055:
2051:
2046:
2042:
2038:
2034:
2030:
2026:
2022:
2018:
2015:(in French).
2014:
2013:
2008:
2003:
1999:
1995:
1991:
1987:
1983:
1979:
1975:
1971:
1970:
1964:
1960:
1956:
1952:
1950:0-387-97926-3
1946:
1942:
1938:
1933:
1932:
1921:
1916:
1910:
1903:
1902:
1885:
1876:
1870:
1866:
1856:
1853:
1852:
1846:
1844:
1839:
1837:
1821:
1801:
1781:
1758:
1752:
1744:
1728:
1708:
1700:
1684:
1676:
1660:
1654:
1651:
1648:
1624:
1604:
1596:
1586:
1570:
1541:
1514:
1511:
1508:
1505:
1483:
1454:
1433:
1413:
1392:
1372:
1368:
1364:
1361:
1356:
1352:
1348:
1343:
1339:
1335:
1332:
1328:
1324:
1320:
1316:
1313:
1308:
1304:
1300:
1295:
1291:
1286:
1282:
1275:as the shift
1256:
1243:
1239:
1236:
1228:
1206:
1198:
1192:
1187:
1173:
1166:because here
1152:
1148:
1145:
1140:
1136:
1132:
1129:
1126:
1121:
1117:
1112:
1108:
1102:
1096:
1074:
1058:
1055:
1052:
1049:
1043:
1040:
1020:
1015:
966:
960:
957:
954:
948:
942:
936:
914:
898:
895:
892:
886:
883:
873:
859:
851:
847:
828:
824:
820:
817:
812:
808:
804:
801:
798:
793:
789:
784:
780:
774:
768:
746:
730:
727:
724:
721:
715:
712:
703:
699:
685:
665:
660:
633:
613:
593:
571:
544:
524:
502:
499:
495:
474:
469:
440:
423:
407:
404:
400:
392:
376:
356:
347:
345:
329:
315:
313:
309:
305:
285:
265:
257:
256:homeomorphism
241:
219:
189:
183:
180:
177:
169:
166:
148:
135:
132:
129:
107:
92:
76:
68:
67:
66:
50:
36:
32:
29:
25:
21:
2704:Publications
2601:
2569:Chern number
2559:Betti number
2442: /
2433:Key concepts
2381:Differential
2303:
2280:. Retrieved
2276:
2269:Tao, Terence
2260:
2239:
2208:
2191:
2187:
2158:
2129:. New York:
2126:
2106:
2076:
2049:
2016:
2010:
1973:
1967:
1936:
1919:
1909:
1899:
1869:
1843:Banach space
1840:
1699:neighborhood
1592:
1439:
1436:Consequences
1188:
843:
348:
321:
301:
28:homeomorphic
19:
18:
2667:Wikiversity
2584:Key results
1845:to itself.
1774:is open in
1089:defined by
929:defined by
205:is open in
91:open subset
2719:Categories
2513:CW complex
2454:Continuity
2444:Closed set
2403:cohomology
2282:2 February
2140:0521094224
1929:References
1701:such that
1229:. Define
426:subset of
2692:geometric
2687:algebraic
2538:Cobordism
2474:Hausdorff
2469:connected
2386:Geometric
2376:Continuum
2366:Algebraic
2310:EMS Press
2041:124671921
1998:221066737
1794:whenever
1658:→
1595:manifolds
1509:≠
1418:∞
1414:ℓ
1365:…
1317:…
1261:∞
1257:ℓ
1253:→
1248:∞
1244:ℓ
1227:sequences
1211:∞
1207:ℓ
1146:−
1127:−
1065:→
1050:−
905:→
818:−
799:−
737:→
722:−
500:−
405:−
165:injective
139:→
2657:Wikibook
2635:Category
2523:Manifold
2491:Homotopy
2449:Interior
2440:Open set
2398:Homology
2347:Topology
2271:(2011).
2105:(1988).
2019:: 1–15.
1915:Leray J.
1849:See also
1834:) and a
1743:open map
389:and its
344:open map
258:between
24:topology
2682:general
2484:uniform
2464:compact
2415:Digital
2252:0198479
2231:1454127
2200:1696596
2172:0006493
2149:0115161
2095:0658305
2068:3887626
2033:0013313
1990:4101407
1959:1224675
170:, then
31:subsets
2677:Topics
2479:metric
2354:Fields
2250:
2229:
2219:
2198:
2170:
2147:
2137:
2113:
2093:
2083:
2066:
2056:
2039:
2031:
1996:
1988:
1957:
1947:
1741:is an
1697:has a
1191:Banach
846:domain
342:is an
163:is an
89:is an
26:about
2459:Space
2184:(PDF)
2037:S2CID
1994:S2CID
1862:Notes
1597:: if
1385:Then
1197:space
850:image
761:with
318:Notes
254:is a
2284:2022
2217:ISBN
2135:ISBN
2111:ISBN
2081:ISBN
2054:ISBN
1945:ISBN
1617:and
848:and
537:and
424:open
278:and
234:and
122:and
2021:doi
1978:doi
1974:127
1498:if
1053:1.1
852:of
725:1.1
557:of
346:".
93:of
69:If
33:of
2721::
2308:,
2302:,
2275:.
2248:MR
2246:.
2227:MR
2225:.
2215:.
2211:.
2196:MR
2192:39
2190:.
2186:.
2168:MR
2166:.
2145:MR
2143:.
2133:.
2091:MR
2089:.
2064:MR
2062:.
2035:.
2029:MR
2027:.
2017:18
2009:.
1992:.
1986:MR
1984:.
1972:.
1955:MR
1953:.
1943:.
1838:.
314:.
181::=
2339:e
2332:t
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2286:.
2254:.
2233:.
2202:.
2174:.
2151:.
2119:.
2097:.
2070:.
2043:.
2023::
2000:.
1980::
1961:.
1886:n
1822:M
1802:U
1782:N
1762:)
1759:U
1756:(
1753:f
1729:f
1709:f
1685:M
1661:N
1655:M
1652::
1649:f
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1625:N
1605:M
1571:m
1566:R
1542:n
1537:R
1515:.
1512:n
1506:m
1484:m
1479:R
1455:n
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1393:f
1373:.
1369:)
1362:,
1357:2
1353:x
1349:,
1344:1
1340:x
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1333:0
1329:(
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1321:)
1314:,
1309:2
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1301:,
1296:1
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1283:f
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1113:(
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964:)
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952:(
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937:f
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896:,
893:0
890:(
887::
884:f
860:f
829:.
825:)
821:t
813:3
809:t
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802:1
794:2
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775:t
772:(
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731:1
728:,
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666:,
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614:V
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298:.
286:V
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187:(
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77:U
51:n
46:R
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