3173:
2956:
3194:
3162:
3231:
3204:
3184:
1051:
in the sense that every point has a local base of compact neighborhoods. But the line through one origin does not contain a closed neighborhood of that origin, as any neighborhood of one origin contains the other origin in its closure. So the space is not a
1548:
889:
within the line through the second origin. But it is impossible to join the two origins with an arc, which is an injective path; intuitively, if one moves first to the left, one has to eventually backtrack and move back to the
1617:
1445:
477:
1697:, such as the sheaf of continuous real functions over a manifold, is a manifold that is often non-Hausdorff. (The etale space is Hausdorff if it is a sheaf of functions with some sort of
389:
135:
101:
1122:
1005:
951:
1495:
1313:
692:
542:
1318:
If there are infinitely many origins, the space illustrates that the closure of a compact set need not be compact in general. For example, the closure of the compact set
1284:
1234:
593:
278:
1679:
1154:
1186:
1212:
251:
161:
1040:
757:
355:
2813:
887:
814:
727:
667:
640:
571:
420:
325:
225:
193:
2004:
1394:
860:
837:
3234:
1639:
1465:
1254:
1094:
613:
517:
497:
298:
2808:
2095:
1056:, and even though every point has at least one closed compact neighborhood, the origin points do not admit a local base of closed compact neighborhoods.
2119:
2314:
1556:
2184:
1952:
1922:
1830:
1471:
in the sense that every point has a local base of compact neighborhoods. But the origin points do not have any closed compact neighborhood.
2410:
2463:
1991:
2868:
2747:
3222:
3217:
2512:
2104:
1076:
is similar to the line with two origins, but with an arbitrary number of origins. It is constructed by taking an arbitrary set
3212:
2495:
3114:
2707:
2692:
2415:
2189:
702:, in the sense that each point has a Hausdorff neighborhood. But the space is not Hausdorff, as every neighborhood of
2737:
1399:
425:
3122:
2742:
2712:
2420:
2376:
2357:
2124:
2068:
3255:
2279:
2144:
1489:
360:
66:
3193:
2921:
2664:
2529:
2221:
2063:
1718:
106:
72:
3207:
2361:
2331:
2255:
2245:
2201:
2031:
1984:
1748:
1726:
2129:
3142:
3137:
3063:
2940:
2928:
2901:
2861:
2702:
2321:
2216:
2036:
1940:
1099:
956:
902:
2984:
2911:
2351:
2346:
1698:
17:
3172:
3132:
3084:
3058:
2906:
2682:
2620:
2468:
2172:
2162:
2134:
2109:
2019:
1902:
1710:
1551:
37:
3183:
2979:
2820:
2793:
2502:
2380:
2365:
2294:
2053:
1694:
1289:
672:
522:
1259:
1217:
576:
261:
3260:
3177:
3127:
3048:
3038:
2916:
2896:
2762:
2717:
2614:
2485:
2289:
2114:
1977:
1944:
1892:
1863:
1742:
1644:
1127:
3147:
2299:
1159:
1822:
3265:
3165:
3031:
2989:
2854:
2697:
2677:
2672:
2579:
2490:
2304:
2284:
2139:
2078:
1958:
1948:
1918:
1826:
1191:
699:
695:
230:
140:
2945:
2891:
2835:
2629:
2584:
2507:
2478:
2336:
2269:
2264:
2259:
2249:
2041:
2024:
1873:
1814:
1753:
1722:
1010:
771:
732:
330:
29:
865:
792:
705:
645:
618:
549:
398:
303:
198:
166:
3004:
2999:
2778:
2687:
2517:
2473:
2239:
1730:
1714:
1543:{\displaystyle \mathbb {R} \times \{a\}\quad {\text{ and }}\quad \mathbb {R} \times \{b\}}
1468:
1048:
789:. In particular, to get a path from one origin to the other one can first move left from
41:
25:
1906:
1321:
842:
819:
3094:
3026:
2644:
2569:
2539:
2437:
2430:
2370:
2341:
2211:
2206:
2167:
1624:
1450:
1239:
1079:
782:
598:
502:
482:
283:
694:
Since every point has a neighborhood homeomorphic to the
Euclidean line, the space is
3249:
3104:
3014:
2994:
2830:
2654:
2649:
2634:
2624:
2574:
2551:
2425:
2385:
2326:
2274:
2073:
1932:
1815:
1053:
786:
3197:
3089:
3009:
2955:
2757:
2752:
2594:
2561:
2534:
2442:
2083:
1878:
1851:
1467:, and that closure is not compact. From being locally Euclidean, such a space is
3187:
3099:
2600:
2589:
2546:
2447:
2048:
1690:
896:
839:
within the line through the first origin, and then move back to the right from
3043:
2974:
2933:
2825:
2783:
2609:
2522:
2154:
2058:
1061:
392:
777:
The space exhibits several phenomena that do not happen in
Hausdorff spaces:
3068:
2639:
2604:
2309:
2196:
1962:
256:
1897:
1315:
The neighborhoods of each origin are described as in the two origin case.
3053:
3021:
2970:
2877:
2803:
2798:
2788:
2179:
2000:
760:
21:
1969:
1889:
A separable manifold failing to have the homotopy type of a CW-complex
2395:
1868:
2850:
1973:
1621:
This space has a single point for each negative real number
1096:
with the discrete topology and taking the quotient space of
2846:
1756: – Axioms in topology defining notions of "separation"
1612:{\displaystyle (x,a)\sim (x,b)\quad {\text{ if }}\;x<0.}
1745: – List of concrete topologies and topological spaces
1772:
1770:
1681:
for every non-negative number: it has a "fork" at zero.
1817:
255:
An equivalent description of the space is to take the
1647:
1627:
1559:
1498:
1453:
1402:
1324:
1292:
1262:
1242:
1220:
1194:
1162:
1130:
1102:
1082:
1013:
959:
905:
868:
845:
822:
795:
735:
708:
675:
648:
621:
601:
579:
552:
525:
505:
485:
428:
401:
363:
333:
306:
286:
264:
233:
201:
169:
143:
109:
75:
3113:
3077:
2963:
2884:
2771:
2730:
2663:
2560:
2456:
2403:
2394:
2230:
2153:
2092:
2012:
1673:
1633:
1611:
1542:
1459:
1439:
1388:
1307:
1278:
1248:
1228:
1206:
1180:
1148:
1116:
1088:
1034:
999:
945:
881:
854:
831:
808:
751:
721:
686:
661:
634:
607:
587:
565:
536:
511:
491:
471:
414:
383:
349:
319:
292:
272:
245:
219:
187:
155:
129:
95:
1856:Proceedings of the American Mathematical Society
1440:{\displaystyle A\cup \{0_{\beta }:\beta \in S\}}
472:{\displaystyle (U\setminus \{0\})\cup \{0_{i}\}}
57:The most familiar non-Hausdorff manifold is the
1060:The space does not have the homotopy type of a
2862:
1985:
1852:"Manifolds: Hausdorffness versus homogeneity"
391:retains its usual Euclidean topology. And a
8:
1850:Baillif, Mathieu; Gabard, Alexandre (2008).
1537:
1531:
1513:
1507:
1481:Similar to the line with two origins is the
1434:
1409:
1365:
1352:
994:
981:
940:
927:
899:need not be compact. For example, the sets
466:
453:
444:
438:
378:
372:
124:
118:
90:
84:
384:{\displaystyle \mathbb {R} \setminus \{0\}}
3230:
3203:
2869:
2855:
2847:
2400:
1992:
1978:
1970:
1599:
1896:
1877:
1867:
1665:
1652:
1646:
1626:
1594:
1558:
1524:
1523:
1517:
1500:
1499:
1497:
1452:
1416:
1401:
1359:
1323:
1291:
1267:
1261:
1241:
1222:
1221:
1219:
1193:
1161:
1129:
1104:
1103:
1101:
1081:
1012:
988:
958:
934:
904:
873:
867:
844:
821:
800:
794:
740:
734:
713:
707:
677:
676:
674:
653:
647:
626:
620:
600:
581:
580:
578:
557:
551:
527:
526:
524:
504:
484:
460:
427:
406:
400:
365:
364:
362:
338:
332:
311:
305:
285:
266:
265:
263:
232:
200:
168:
142:
111:
110:
108:
77:
76:
74:
32:, this axiom is relaxed, and one studies
130:{\displaystyle \mathbb {R} \times \{b\}}
96:{\displaystyle \mathbb {R} \times \{a\}}
1776:
1766:
435:
369:
1788:
1447:obtained by adding all the origins to
1214:Equivalently, it can be obtained from
1915:Introduction to topological manifolds
1821:. New York: Springer-Verlag. p.
395:of open neighborhoods at each origin
7:
1709:Because non-Hausdorff manifolds are
1117:{\displaystyle \mathbb {R} \times S}
1007:are compact, but their intersection
1000:{\displaystyle [-1,0)\cup \{0_{b}\}}
946:{\displaystyle [-1,0)\cup \{0_{a}\}}
1800:
729:intersects every neighbourhood of
163:), obtained by identifying points
14:
44:, but not necessarily Hausdorff.
3229:
3202:
3192:
3182:
3171:
3161:
3160:
2954:
69:of two copies of the real line,
1593:
1522:
1516:
1492:of two copies of the real line
2032:Differentiable/Smooth manifold
1590:
1578:
1572:
1560:
1383:
1371:
1346:
1331:
1175:
1163:
1143:
1131:
1029:
1014:
975:
960:
921:
906:
447:
429:
214:
202:
182:
170:
1:
1917:(Second ed.). Springer.
1879:10.1090/S0002-9939-07-09100-9
1308:{\displaystyle \alpha \in S.}
1064:, or of any Hausdorff space.
687:{\displaystyle \mathbb {R} .}
537:{\displaystyle \mathbb {R} .}
1279:{\displaystyle 0_{\alpha },}
1229:{\displaystyle \mathbb {R} }
588:{\displaystyle \mathbb {R} }
273:{\displaystyle \mathbb {R} }
2738:Classification of manifolds
1674:{\displaystyle x_{a},x_{b}}
1149:{\displaystyle (x,\alpha )}
642:is an open neighborhood of
573:the subspace obtained from
20:, it is a usual axiom of a
3282:
3123:Banach fixed-point theorem
1887:Gabard, Alexandre (2006),
1181:{\displaystyle (x,\beta )}
3156:
2952:
2814:over commutative algebras
1813:Warner, Frank W. (1983).
2530:Riemann curvature tensor
1236:by replacing the origin
1207:{\displaystyle x\neq 0.}
895:The intersection of two
698:. In particular, it is
499:an open neighborhood of
246:{\displaystyle x\neq 0.}
1803:, Problem 4-22, p. 125.
1749:Locally Hausdorff space
1124:that identifies points
280:and replace the origin
156:{\displaystyle a\neq b}
34:non-Hausdorff manifolds
3178:Mathematics portal
3078:Metrics and properties
3064:Second-countable space
2322:Manifold with boundary
2037:Differential structure
1941:Upper Saddle River, NJ
1675:
1635:
1613:
1544:
1461:
1441:
1390:
1309:
1280:
1250:
1230:
1208:
1182:
1150:
1118:
1090:
1074:line with many origins
1068:Line with many origins
1036:
1035:{\displaystyle [-1,0)}
1001:
947:
883:
856:
833:
810:
753:
752:{\displaystyle 0_{b}.}
723:
688:
663:
636:
609:
589:
567:
538:
513:
493:
473:
422:is formed by the sets
416:
385:
351:
350:{\displaystyle 0_{b}.}
321:
294:
274:
247:
221:
189:
157:
131:
97:
1913:Lee, John M. (2011).
1699:analytic continuation
1676:
1636:
1614:
1545:
1462:
1442:
1391:
1310:
1281:
1251:
1231:
1209:
1183:
1151:
1119:
1091:
1037:
1002:
948:
884:
882:{\displaystyle 0_{b}}
857:
834:
811:
809:{\displaystyle 0_{a}}
754:
724:
722:{\displaystyle 0_{a}}
689:
664:
662:{\displaystyle 0_{i}}
637:
635:{\displaystyle 0_{i}}
610:
590:
568:
566:{\displaystyle 0_{i}}
539:
514:
494:
474:
417:
415:{\displaystyle 0_{i}}
386:
352:
322:
320:{\displaystyle 0_{a}}
295:
275:
248:
222:
220:{\displaystyle (x,b)}
190:
188:{\displaystyle (x,a)}
158:
132:
98:
59:line with two origins
53:Line with two origins
18:geometry and topology
3133:Invariance of domain
3085:Euler characteristic
3059:Bundle (mathematics)
2469:Covariant derivative
2020:Topological manifold
1711:locally homeomorphic
1645:
1625:
1557:
1552:equivalence relation
1496:
1451:
1400:
1322:
1290:
1260:
1240:
1218:
1192:
1160:
1128:
1100:
1080:
1011:
957:
903:
866:
843:
820:
793:
733:
706:
673:
646:
619:
599:
577:
550:
523:
503:
483:
426:
399:
361:
331:
304:
284:
262:
231:
199:
167:
141:
107:
73:
38:locally homeomorphic
3143:Tychonoff's theorem
3138:Poincaré conjecture
2892:General (point-set)
2503:Exterior derivative
2105:Atiyah–Singer index
2054:Riemannian manifold
1939:(Second ed.).
1907:2006math......9665G
3128:De Rham cohomology
3049:Polyhedral complex
3039:Simplicial complex
2809:Secondary calculus
2763:Singularity theory
2718:Parallel transport
2486:De Rham cohomology
2125:Generalized Stokes
1945:Prentice Hall, Inc
1791:, Proposition 5.1.
1743:List of topologies
1719:locally metrizable
1671:
1631:
1609:
1540:
1457:
1437:
1389:{\displaystyle A=}
1386:
1305:
1276:
1256:with many origins
1246:
1226:
1204:
1178:
1146:
1114:
1086:
1032:
997:
943:
879:
855:{\displaystyle -1}
852:
832:{\displaystyle -1}
829:
806:
749:
719:
684:
659:
632:
605:
585:
563:
534:
509:
489:
469:
412:
381:
347:
317:
290:
270:
243:
217:
185:
153:
127:
93:
3243:
3242:
3032:fundamental group
2844:
2843:
2726:
2725:
2491:Differential form
2145:Whitney embedding
2079:Differential form
1954:978-0-13-181629-9
1933:Munkres, James R.
1924:978-1-4419-7939-1
1898:math.GT/0609665v1
1832:978-0-387-90894-6
1727:locally Hausdorff
1634:{\displaystyle r}
1597:
1520:
1460:{\displaystyle A}
1249:{\displaystyle 0}
1089:{\displaystyle S}
700:locally Hausdorff
696:locally Euclidean
608:{\displaystyle 0}
512:{\displaystyle 0}
492:{\displaystyle U}
300:with two origins
293:{\displaystyle 0}
3273:
3256:General topology
3233:
3232:
3206:
3205:
3196:
3186:
3176:
3175:
3164:
3163:
2958:
2871:
2864:
2857:
2848:
2836:Stratified space
2794:Fréchet manifold
2508:Interior product
2401:
2098:
1994:
1987:
1980:
1971:
1966:
1928:
1909:
1900:
1883:
1881:
1871:
1862:(3): 1105–1111.
1837:
1836:
1820:
1810:
1804:
1798:
1792:
1786:
1780:
1774:
1754:Separation axiom
1725:in general) and
1680:
1678:
1677:
1672:
1670:
1669:
1657:
1656:
1640:
1638:
1637:
1632:
1618:
1616:
1615:
1610:
1598:
1595:
1549:
1547:
1546:
1541:
1527:
1521:
1518:
1503:
1466:
1464:
1463:
1458:
1446:
1444:
1443:
1438:
1421:
1420:
1395:
1393:
1392:
1387:
1364:
1363:
1314:
1312:
1311:
1306:
1285:
1283:
1282:
1277:
1272:
1271:
1255:
1253:
1252:
1247:
1235:
1233:
1232:
1227:
1225:
1213:
1211:
1210:
1205:
1187:
1185:
1184:
1179:
1155:
1153:
1152:
1147:
1123:
1121:
1120:
1115:
1107:
1095:
1093:
1092:
1087:
1041:
1039:
1038:
1033:
1006:
1004:
1003:
998:
993:
992:
952:
950:
949:
944:
939:
938:
888:
886:
885:
880:
878:
877:
861:
859:
858:
853:
838:
836:
835:
830:
815:
813:
812:
807:
805:
804:
772:second countable
759:It is however a
758:
756:
755:
750:
745:
744:
728:
726:
725:
720:
718:
717:
693:
691:
690:
685:
680:
669:homeomorphic to
668:
666:
665:
660:
658:
657:
641:
639:
638:
633:
631:
630:
614:
612:
611:
606:
594:
592:
591:
586:
584:
572:
570:
569:
564:
562:
561:
546:For each origin
543:
541:
540:
535:
530:
518:
516:
515:
510:
498:
496:
495:
490:
478:
476:
475:
470:
465:
464:
421:
419:
418:
413:
411:
410:
390:
388:
387:
382:
368:
356:
354:
353:
348:
343:
342:
326:
324:
323:
318:
316:
315:
299:
297:
296:
291:
279:
277:
276:
271:
269:
252:
250:
249:
244:
226:
224:
223:
218:
194:
192:
191:
186:
162:
160:
159:
154:
136:
134:
133:
128:
114:
102:
100:
99:
94:
80:
30:general topology
3281:
3280:
3276:
3275:
3274:
3272:
3271:
3270:
3246:
3245:
3244:
3239:
3170:
3152:
3148:Urysohn's lemma
3109:
3073:
2959:
2950:
2922:low-dimensional
2880:
2875:
2845:
2840:
2779:Banach manifold
2772:Generalizations
2767:
2722:
2659:
2556:
2518:Ricci curvature
2474:Cotangent space
2452:
2390:
2232:
2226:
2185:Exponential map
2149:
2094:
2088:
2008:
1998:
1955:
1931:
1925:
1912:
1886:
1849:
1846:
1841:
1840:
1833:
1812:
1811:
1807:
1799:
1795:
1787:
1783:
1775:
1768:
1763:
1739:
1715:Euclidean space
1707:
1687:
1661:
1648:
1643:
1642:
1641:and two points
1623:
1622:
1555:
1554:
1519: and
1494:
1493:
1479:
1469:locally compact
1449:
1448:
1412:
1398:
1397:
1355:
1320:
1319:
1288:
1287:
1263:
1258:
1257:
1238:
1237:
1216:
1215:
1190:
1189:
1158:
1157:
1126:
1125:
1098:
1097:
1078:
1077:
1070:
1049:locally compact
1009:
1008:
984:
955:
954:
930:
901:
900:
869:
864:
863:
841:
840:
818:
817:
796:
791:
790:
764:
736:
731:
730:
709:
704:
703:
671:
670:
649:
644:
643:
622:
617:
616:
597:
596:
575:
574:
553:
548:
547:
521:
520:
501:
500:
481:
480:
456:
424:
423:
402:
397:
396:
359:
358:
334:
329:
328:
307:
302:
301:
282:
281:
260:
259:
229:
228:
197:
196:
165:
164:
139:
138:
105:
104:
71:
70:
65:. This is the
55:
50:
42:Euclidean space
26:Hausdorff space
12:
11:
5:
3279:
3277:
3269:
3268:
3263:
3258:
3248:
3247:
3241:
3240:
3238:
3237:
3227:
3226:
3225:
3220:
3215:
3200:
3190:
3180:
3168:
3157:
3154:
3153:
3151:
3150:
3145:
3140:
3135:
3130:
3125:
3119:
3117:
3111:
3110:
3108:
3107:
3102:
3097:
3095:Winding number
3092:
3087:
3081:
3079:
3075:
3074:
3072:
3071:
3066:
3061:
3056:
3051:
3046:
3041:
3036:
3035:
3034:
3029:
3027:homotopy group
3019:
3018:
3017:
3012:
3007:
3002:
2997:
2987:
2982:
2977:
2967:
2965:
2961:
2960:
2953:
2951:
2949:
2948:
2943:
2938:
2937:
2936:
2926:
2925:
2924:
2914:
2909:
2904:
2899:
2894:
2888:
2886:
2882:
2881:
2876:
2874:
2873:
2866:
2859:
2851:
2842:
2841:
2839:
2838:
2833:
2828:
2823:
2818:
2817:
2816:
2806:
2801:
2796:
2791:
2786:
2781:
2775:
2773:
2769:
2768:
2766:
2765:
2760:
2755:
2750:
2745:
2740:
2734:
2732:
2728:
2727:
2724:
2723:
2721:
2720:
2715:
2710:
2705:
2700:
2695:
2690:
2685:
2680:
2675:
2669:
2667:
2661:
2660:
2658:
2657:
2652:
2647:
2642:
2637:
2632:
2627:
2617:
2612:
2607:
2597:
2592:
2587:
2582:
2577:
2572:
2566:
2564:
2558:
2557:
2555:
2554:
2549:
2544:
2543:
2542:
2532:
2527:
2526:
2525:
2515:
2510:
2505:
2500:
2499:
2498:
2488:
2483:
2482:
2481:
2471:
2466:
2460:
2458:
2454:
2453:
2451:
2450:
2445:
2440:
2435:
2434:
2433:
2423:
2418:
2413:
2407:
2405:
2398:
2392:
2391:
2389:
2388:
2383:
2373:
2368:
2354:
2349:
2344:
2339:
2334:
2332:Parallelizable
2329:
2324:
2319:
2318:
2317:
2307:
2302:
2297:
2292:
2287:
2282:
2277:
2272:
2267:
2262:
2252:
2242:
2236:
2234:
2228:
2227:
2225:
2224:
2219:
2214:
2212:Lie derivative
2209:
2207:Integral curve
2204:
2199:
2194:
2193:
2192:
2182:
2177:
2176:
2175:
2168:Diffeomorphism
2165:
2159:
2157:
2151:
2150:
2148:
2147:
2142:
2137:
2132:
2127:
2122:
2117:
2112:
2107:
2101:
2099:
2090:
2089:
2087:
2086:
2081:
2076:
2071:
2066:
2061:
2056:
2051:
2046:
2045:
2044:
2039:
2029:
2028:
2027:
2016:
2014:
2013:Basic concepts
2010:
2009:
1999:
1997:
1996:
1989:
1982:
1974:
1968:
1967:
1953:
1929:
1923:
1910:
1884:
1845:
1842:
1839:
1838:
1831:
1805:
1793:
1781:
1779:, p. 227.
1765:
1764:
1762:
1759:
1758:
1757:
1751:
1746:
1738:
1735:
1706:
1703:
1686:
1683:
1668:
1664:
1660:
1655:
1651:
1630:
1608:
1605:
1602:
1596: if
1592:
1589:
1586:
1583:
1580:
1577:
1574:
1571:
1568:
1565:
1562:
1539:
1536:
1533:
1530:
1526:
1515:
1512:
1509:
1506:
1502:
1490:quotient space
1483:branching line
1478:
1477:Branching line
1475:
1456:
1436:
1433:
1430:
1427:
1424:
1419:
1415:
1411:
1408:
1405:
1385:
1382:
1379:
1376:
1373:
1370:
1367:
1362:
1358:
1354:
1351:
1348:
1345:
1342:
1339:
1336:
1333:
1330:
1327:
1304:
1301:
1298:
1295:
1275:
1270:
1266:
1245:
1224:
1203:
1200:
1197:
1177:
1174:
1171:
1168:
1165:
1145:
1142:
1139:
1136:
1133:
1113:
1110:
1106:
1085:
1069:
1066:
1058:
1057:
1044:
1043:
1031:
1028:
1025:
1022:
1019:
1016:
996:
991:
987:
983:
980:
977:
974:
971:
968:
965:
962:
942:
937:
933:
929:
926:
923:
920:
917:
914:
911:
908:
892:
891:
876:
872:
851:
848:
828:
825:
803:
799:
783:path connected
762:
748:
743:
739:
716:
712:
683:
679:
656:
652:
629:
625:
604:
583:
560:
556:
533:
529:
508:
488:
468:
463:
459:
455:
452:
449:
446:
443:
440:
437:
434:
431:
409:
405:
380:
377:
374:
371:
367:
346:
341:
337:
314:
310:
289:
268:
242:
239:
236:
216:
213:
210:
207:
204:
184:
181:
178:
175:
172:
152:
149:
146:
126:
123:
120:
117:
113:
92:
89:
86:
83:
79:
67:quotient space
54:
51:
49:
46:
13:
10:
9:
6:
4:
3:
2:
3278:
3267:
3264:
3262:
3259:
3257:
3254:
3253:
3251:
3236:
3228:
3224:
3221:
3219:
3216:
3214:
3211:
3210:
3209:
3201:
3199:
3195:
3191:
3189:
3185:
3181:
3179:
3174:
3169:
3167:
3159:
3158:
3155:
3149:
3146:
3144:
3141:
3139:
3136:
3134:
3131:
3129:
3126:
3124:
3121:
3120:
3118:
3116:
3112:
3106:
3105:Orientability
3103:
3101:
3098:
3096:
3093:
3091:
3088:
3086:
3083:
3082:
3080:
3076:
3070:
3067:
3065:
3062:
3060:
3057:
3055:
3052:
3050:
3047:
3045:
3042:
3040:
3037:
3033:
3030:
3028:
3025:
3024:
3023:
3020:
3016:
3013:
3011:
3008:
3006:
3003:
3001:
2998:
2996:
2993:
2992:
2991:
2988:
2986:
2983:
2981:
2978:
2976:
2972:
2969:
2968:
2966:
2962:
2957:
2947:
2944:
2942:
2941:Set-theoretic
2939:
2935:
2932:
2931:
2930:
2927:
2923:
2920:
2919:
2918:
2915:
2913:
2910:
2908:
2905:
2903:
2902:Combinatorial
2900:
2898:
2895:
2893:
2890:
2889:
2887:
2883:
2879:
2872:
2867:
2865:
2860:
2858:
2853:
2852:
2849:
2837:
2834:
2832:
2831:Supermanifold
2829:
2827:
2824:
2822:
2819:
2815:
2812:
2811:
2810:
2807:
2805:
2802:
2800:
2797:
2795:
2792:
2790:
2787:
2785:
2782:
2780:
2777:
2776:
2774:
2770:
2764:
2761:
2759:
2756:
2754:
2751:
2749:
2746:
2744:
2741:
2739:
2736:
2735:
2733:
2729:
2719:
2716:
2714:
2711:
2709:
2706:
2704:
2701:
2699:
2696:
2694:
2691:
2689:
2686:
2684:
2681:
2679:
2676:
2674:
2671:
2670:
2668:
2666:
2662:
2656:
2653:
2651:
2648:
2646:
2643:
2641:
2638:
2636:
2633:
2631:
2628:
2626:
2622:
2618:
2616:
2613:
2611:
2608:
2606:
2602:
2598:
2596:
2593:
2591:
2588:
2586:
2583:
2581:
2578:
2576:
2573:
2571:
2568:
2567:
2565:
2563:
2559:
2553:
2552:Wedge product
2550:
2548:
2545:
2541:
2538:
2537:
2536:
2533:
2531:
2528:
2524:
2521:
2520:
2519:
2516:
2514:
2511:
2509:
2506:
2504:
2501:
2497:
2496:Vector-valued
2494:
2493:
2492:
2489:
2487:
2484:
2480:
2477:
2476:
2475:
2472:
2470:
2467:
2465:
2462:
2461:
2459:
2455:
2449:
2446:
2444:
2441:
2439:
2436:
2432:
2429:
2428:
2427:
2426:Tangent space
2424:
2422:
2419:
2417:
2414:
2412:
2409:
2408:
2406:
2402:
2399:
2397:
2393:
2387:
2384:
2382:
2378:
2374:
2372:
2369:
2367:
2363:
2359:
2355:
2353:
2350:
2348:
2345:
2343:
2340:
2338:
2335:
2333:
2330:
2328:
2325:
2323:
2320:
2316:
2313:
2312:
2311:
2308:
2306:
2303:
2301:
2298:
2296:
2293:
2291:
2288:
2286:
2283:
2281:
2278:
2276:
2273:
2271:
2268:
2266:
2263:
2261:
2257:
2253:
2251:
2247:
2243:
2241:
2238:
2237:
2235:
2229:
2223:
2220:
2218:
2215:
2213:
2210:
2208:
2205:
2203:
2200:
2198:
2195:
2191:
2190:in Lie theory
2188:
2187:
2186:
2183:
2181:
2178:
2174:
2171:
2170:
2169:
2166:
2164:
2161:
2160:
2158:
2156:
2152:
2146:
2143:
2141:
2138:
2136:
2133:
2131:
2128:
2126:
2123:
2121:
2118:
2116:
2113:
2111:
2108:
2106:
2103:
2102:
2100:
2097:
2093:Main results
2091:
2085:
2082:
2080:
2077:
2075:
2074:Tangent space
2072:
2070:
2067:
2065:
2062:
2060:
2057:
2055:
2052:
2050:
2047:
2043:
2040:
2038:
2035:
2034:
2033:
2030:
2026:
2023:
2022:
2021:
2018:
2017:
2015:
2011:
2006:
2002:
1995:
1990:
1988:
1983:
1981:
1976:
1975:
1972:
1964:
1960:
1956:
1950:
1946:
1942:
1938:
1934:
1930:
1926:
1920:
1916:
1911:
1908:
1904:
1899:
1894:
1890:
1885:
1880:
1875:
1870:
1865:
1861:
1857:
1853:
1848:
1847:
1843:
1834:
1828:
1824:
1819:
1818:
1809:
1806:
1802:
1797:
1794:
1790:
1785:
1782:
1778:
1773:
1771:
1767:
1760:
1755:
1752:
1750:
1747:
1744:
1741:
1740:
1736:
1734:
1733:in general).
1732:
1728:
1724:
1720:
1716:
1712:
1704:
1702:
1700:
1696:
1692:
1684:
1682:
1666:
1662:
1658:
1653:
1649:
1628:
1619:
1606:
1603:
1600:
1587:
1584:
1581:
1575:
1569:
1566:
1563:
1553:
1534:
1528:
1510:
1504:
1491:
1486:
1484:
1476:
1474:
1472:
1470:
1454:
1431:
1428:
1425:
1422:
1417:
1413:
1406:
1403:
1380:
1377:
1374:
1368:
1360:
1356:
1349:
1343:
1340:
1337:
1334:
1328:
1325:
1316:
1302:
1299:
1296:
1293:
1286:one for each
1273:
1268:
1264:
1243:
1201:
1198:
1195:
1172:
1169:
1166:
1140:
1137:
1134:
1111:
1108:
1083:
1075:
1067:
1065:
1063:
1055:
1054:regular space
1050:
1047:The space is
1046:
1045:
1026:
1023:
1020:
1017:
989:
985:
978:
972:
969:
966:
963:
935:
931:
924:
918:
915:
912:
909:
898:
894:
893:
874:
870:
849:
846:
826:
823:
801:
797:
788:
787:arc connected
784:
781:The space is
780:
779:
778:
775:
773:
770:The space is
768:
766:
746:
741:
737:
714:
710:
701:
697:
681:
654:
650:
627:
623:
602:
595:by replacing
558:
554:
544:
531:
506:
486:
461:
457:
450:
441:
432:
407:
403:
394:
375:
357:The subspace
344:
339:
335:
312:
308:
287:
258:
253:
240:
237:
234:
211:
208:
205:
179:
176:
173:
150:
147:
144:
121:
115:
87:
81:
68:
64:
63:bug-eyed line
60:
52:
47:
45:
43:
39:
35:
31:
27:
23:
19:
3235:Publications
3100:Chern number
3090:Betti number
2973: /
2964:Key concepts
2912:Differential
2758:Moving frame
2753:Morse theory
2743:Gauge theory
2535:Tensor field
2464:Closed/Exact
2443:Vector field
2411:Distribution
2352:Hypercomplex
2347:Quaternionic
2084:Vector field
2042:Smooth atlas
1936:
1914:
1888:
1869:math/0609098
1859:
1855:
1816:
1808:
1796:
1784:
1777:Munkres 2000
1708:
1688:
1620:
1488:This is the
1487:
1482:
1480:
1473:
1317:
1073:
1071:
1059:
897:compact sets
776:
769:
545:
254:
62:
58:
56:
33:
15:
3198:Wikiversity
3115:Key results
2703:Levi-Civita
2693:Generalized
2665:Connections
2615:Lie algebra
2547:Volume form
2448:Vector flow
2421:Pushforward
2416:Lie bracket
2315:Lie algebra
2280:G-structure
2069:Pushforward
2049:Submanifold
1789:Gabard 2006
1717:, they are
1701:property.)
1691:etale space
1685:Etale space
1396:is the set
3250:Categories
3044:CW complex
2985:Continuity
2975:Closed set
2934:cohomology
2826:Stratifold
2784:Diffeology
2580:Associated
2381:Symplectic
2366:Riemannian
2295:Hyperbolic
2222:Submersion
2130:Hopf–Rinow
2064:Submersion
2059:Smooth map
1844:References
1723:metrizable
1705:Properties
1062:CW-complex
393:local base
3261:Manifolds
3223:geometric
3218:algebraic
3069:Cobordism
3005:Hausdorff
3000:connected
2917:Geometric
2907:Continuum
2897:Algebraic
2708:Principal
2683:Ehresmann
2640:Subbundle
2630:Principal
2605:Fibration
2585:Cotangent
2457:Covectors
2310:Lie group
2290:Hermitian
2233:manifolds
2202:Immersion
2197:Foliation
2135:Noether's
2120:Frobenius
2115:De Rham's
2110:Darboux's
2001:Manifolds
1731:Hausdorff
1729:(but not
1721:(but not
1576:∼
1550:with the
1529:×
1505:×
1429:∈
1426:β
1418:β
1407:∪
1369:∪
1361:α
1350:∪
1335:−
1297:∈
1294:α
1269:α
1199:≠
1188:whenever
1173:β
1141:α
1109:×
1018:−
979:∪
964:−
925:∪
910:−
847:−
824:−
451:∪
436:∖
370:∖
257:real line
238:≠
227:whenever
148:≠
116:×
82:×
36:: spaces
3266:Topology
3188:Wikibook
3166:Category
3054:Manifold
3022:Homotopy
2980:Interior
2971:Open set
2929:Homology
2878:Topology
2804:Orbifold
2799:K-theory
2789:Diffiety
2513:Pullback
2327:Oriented
2305:Kenmotsu
2285:Hadamard
2231:Types of
2180:Geodesic
2005:Glossary
1963:42683260
1937:Topology
1935:(2000).
1801:Lee 2011
1737:See also
785:but not
48:Examples
24:to be a
22:manifold
3213:general
3015:uniform
2995:compact
2946:Digital
2748:History
2731:Related
2645:Tangent
2623:)
2603:)
2570:Adjoint
2562:Bundles
2540:density
2438:Torsion
2404:Vectors
2396:Tensors
2379:)
2364:)
2360:,
2358:Pseudo−
2337:Poisson
2270:Finsler
2265:Fibered
2260:Contact
2258:)
2250:Complex
2248:)
2217:Section
1903:Bibcode
1042:is not.
3208:Topics
3010:metric
2885:Fields
2713:Vector
2698:Koszul
2678:Cartan
2673:Affine
2655:Vector
2650:Tensor
2635:Spinor
2625:Normal
2621:Stable
2575:Affine
2479:bundle
2431:bundle
2377:Almost
2300:Kähler
2256:Almost
2246:Almost
2240:Closed
2140:Sard's
2096:(list)
1961:
1951:
1921:
1829:
890:right.
137:(with
2990:Space
2821:Sheaf
2595:Fiber
2371:Rizza
2342:Prime
2173:Local
2163:Curve
2025:Atlas
1893:arXiv
1864:arXiv
1761:Notes
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