3184:
2967:
3205:
3173:
3242:
3215:
3195:
1062:
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
1559:
900:
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
1628:
1456:
488:
1708:, 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
400:
146:
112:
1133:
1016:
962:
1506:
1324:
703:
553:
1329:
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
1295:
1245:
604:
289:
1690:
1165:
1197:
1223:
262:
172:
1051:
768:
366:
898:
825:
738:
678:
651:
582:
431:
336:
236:
204:
1405:
871:
848:
1650:
1476:
1265:
1105:
624:
528:
508:
309:
1067:, 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.
3245:
2824:
1567:
2015:
1482:
in the sense that every point has a local base of compact neighborhoods. But the origin points do not have any closed compact neighborhood.
2819:
2106:
2130:
2325:
1087:
is similar to the line with two origins, but with an arbitrary number of origins. It is constructed by taking an arbitrary set
2195:
1963:
1933:
1841:
17:
2421:
2474:
2002:
2879:
2758:
713:, in the sense that each point has a Hausdorff neighborhood. But the space is not Hausdorff, as every neighborhood of
1410:
436:
3233:
3228:
2523:
3223:
2506:
2115:
371:
117:
83:
3125:
2718:
2125:
2703:
2426:
2200:
1110:
967:
913:
2748:
3133:
2753:
2723:
2431:
2387:
2368:
2135:
2079:
3266:
2290:
2155:
1500:
77:
2932:
2675:
2540:
2232:
2074:
1729:
1300:
683:
533:
1270:
1228:
587:
272:
3218:
3204:
2372:
2342:
2266:
2256:
2212:
2042:
1995:
1759:
1737:
1655:
1138:
1170:
3153:
3074:
2951:
2939:
2912:
2872:
2713:
2332:
2227:
2140:
2047:
1951:
1833:
3148:
1202:
241:
2995:
2922:
2362:
2357:
1709:
151:
28:
1825:
1021:
743:
341:
3143:
3095:
3069:
2917:
2693:
2631:
2479:
2183:
2173:
2145:
2120:
2030:
1913:
1721:
1562:
876:
803:
716:
656:
629:
560:
409:
314:
209:
177:
48:
1554:{\displaystyle \mathbb {R} \times \{a\}\quad {\text{ and }}\quad \mathbb {R} \times \{b\}}
800:. In particular, to get a path from one origin to the other one can first move left from
8:
2990:
2831:
2513:
2391:
2376:
2305:
2064:
1705:
3194:
2804:
1917:
1332:
853:
830:
3271:
3188:
3158:
3138:
3059:
3049:
2927:
2907:
2773:
2728:
2625:
2496:
2300:
1988:
1955:
1903:
1874:
1753:
1635:
1461:
1250:
1090:
609:
513:
493:
294:
2310:
705:
Since every point has a neighborhood homeomorphic to the
Euclidean line, the space is
3276:
3183:
3176:
3042:
3000:
2865:
2708:
2688:
2683:
2590:
2501:
2315:
2295:
2150:
2089:
1969:
1959:
1929:
1837:
1826:
710:
706:
3208:
2956:
2902:
2846:
2640:
2595:
2518:
2489:
2347:
2280:
2275:
2270:
2260:
2052:
2035:
1884:
1764:
1733:
782:
40:
1889:
1862:
1478:, and that closure is not compact. From being locally Euclidean, such a space is
3015:
3010:
2789:
2698:
2528:
2484:
2250:
1741:
1725:
1479:
1059:
52:
36:
3198:
3105:
3037:
2655:
2580:
2550:
2448:
2441:
2381:
2352:
2222:
2217:
2178:
850:
within the line through the first origin, and then move back to the right from
793:
3260:
3115:
3025:
3005:
2841:
2665:
2660:
2645:
2635:
2585:
2562:
2436:
2396:
2337:
2285:
2084:
1943:
1064:
797:
788:
The space exhibits several phenomena that do not happen in
Hausdorff spaces:
1973:
3100:
3020:
2966:
2768:
2763:
2605:
2572:
2545:
2453:
2094:
1908:
1326:
The neighborhoods of each origin are described as in the two origin case.
3110:
2611:
2600:
2557:
2458:
2059:
1701:
907:
3054:
2985:
2944:
2836:
2794:
2620:
2533:
2165:
2069:
1980:
1072:
403:
1900:
A separable manifold failing to have the homotopy type of a CW-complex
3079:
2650:
2615:
2320:
2207:
1879:
267:
3064:
3032:
2981:
2888:
2814:
2809:
2799:
2190:
2011:
771:
32:
1805:
1793:
2406:
1632:
This space has a single point for each negative real number
1107:
with the discrete topology and taking the quotient space of
2857:
1767: – Axioms in topology defining notions of "separation"
1623:{\displaystyle (x,a)\sim (x,b)\quad {\text{ if }}\;x<0.}
1756: – List of concrete topologies and topological spaces
1783:
1781:
1692:
for every non-negative number: it has a "fork" at zero.
1828:
266:
An equivalent description of the space is to take the
1778:
1658:
1638:
1570:
1509:
1464:
1413:
1335:
1303:
1273:
1253:
1231:
1205:
1173:
1141:
1113:
1093:
1024:
970:
916:
879:
856:
833:
806:
746:
719:
686:
659:
632:
612:
590:
563:
536:
516:
496:
439:
412:
374:
344:
317:
297:
275:
244:
212:
180:
154:
120:
86:
1684:
1644:
1622:
1553:
1470:
1450:
1399:
1318:
1289:
1259:
1239:
1217:
1191:
1159:
1127:
1099:
1045:
1010:
956:
892:
865:
842:
819:
762:
732:
697:
672:
645:
618:
598:
576:
547:
522:
502:
482:
425:
394:
360:
330:
303:
283:
256:
230:
198:
166:
140:
106:
3258:
1867:Proceedings of the American Mathematical Society
1451:{\displaystyle A\cup \{0_{\beta }:\beta \in S\}}
483:{\displaystyle (U\setminus \{0\})\cup \{0_{i}\}}
68:The most familiar non-Hausdorff manifold is the
1071:The space does not have the homotopy type of a
2873:
1996:
1863:"Manifolds: Hausdorffness versus homogeneity"
1860:
402:retains its usual Euclidean topology. And a
1861:Baillif, Mathieu; Gabard, Alexandre (2008).
1548:
1542:
1524:
1518:
1492:Similar to the line with two origins is the
1445:
1420:
1376:
1363:
1005:
992:
951:
938:
910:need not be compact. For example, the sets
477:
464:
455:
449:
389:
383:
135:
129:
101:
95:
395:{\displaystyle \mathbb {R} \setminus \{0\}}
3241:
3214:
2880:
2866:
2003:
1989:
1610:
1907:
1888:
1878:
1535:
1511:
1233:
1115:
1078:
688:
592:
538:
376:
277:
122:
88:
43:, this axiom is relaxed, and one studies
2010:
141:{\displaystyle \mathbb {R} \times \{b\}}
107:{\displaystyle \mathbb {R} \times \{a\}}
63:
1942:
1787:
14:
3259:
1897:
1823:
1799:
1458:obtained by adding all the origins to
1225:Equivalently, it can be obtained from
2861:
1984:
1926:Introduction to topological manifolds
1832:. New York: Springer-Verlag. p.
1817:
406:of open neighborhoods at each origin
1720:Because non-Hausdorff manifolds are
1128:{\displaystyle \mathbb {R} \times S}
1018:are compact, but their intersection
1011:{\displaystyle [-1,0)\cup \{0_{b}\}}
957:{\displaystyle [-1,0)\cup \{0_{a}\}}
1923:
1811:
24:
740:intersects every neighbourhood of
174:), obtained by identifying points
25:
3288:
1487:
446:
380:
55:, but not necessarily Hausdorff.
3240:
3213:
3203:
3193:
3182:
3172:
3171:
2965:
80:of two copies of the real line,
1604:
1533:
1527:
1503:of two copies of the real line
2043:Differentiable/Smooth manifold
1695:
1601:
1589:
1583:
1571:
1394:
1382:
1357:
1342:
1186:
1174:
1154:
1142:
1040:
1025:
986:
971:
932:
917:
458:
440:
225:
213:
193:
181:
13:
1:
1928:(Second ed.). Springer.
1890:10.1090/S0002-9939-07-09100-9
1854:
1715:
1319:{\displaystyle \alpha \in S.}
1075:, or of any Hausdorff space.
698:{\displaystyle \mathbb {R} .}
548:{\displaystyle \mathbb {R} .}
2887:
1290:{\displaystyle 0_{\alpha },}
1240:{\displaystyle \mathbb {R} }
599:{\displaystyle \mathbb {R} }
284:{\displaystyle \mathbb {R} }
7:
2749:Classification of manifolds
1747:
1685:{\displaystyle x_{a},x_{b}}
1160:{\displaystyle (x,\alpha )}
653:is an open neighborhood of
584:the subspace obtained from
58:
31:, it is a usual axiom of a
10:
3293:
3134:Banach fixed-point theorem
1898:Gabard, Alexandre (2006),
1192:{\displaystyle (x,\beta )}
3167:
3124:
3088:
2974:
2963:
2895:
2825:over commutative algebras
2782:
2741:
2674:
2571:
2467:
2414:
2405:
2241:
2164:
2103:
2023:
1824:Warner, Frank W. (1983).
2541:Riemann curvature tensor
1771:
1247:by replacing the origin
1218:{\displaystyle x\neq 0.}
906:The intersection of two
709:. In particular, it is
510:an open neighborhood of
257:{\displaystyle x\neq 0.}
1814:, Problem 4-22, p. 125.
1760:Locally Hausdorff space
1135:that identifies points
291:and replace the origin
167:{\displaystyle a\neq b}
45:non-Hausdorff manifolds
3189:Mathematics portal
3089:Metrics and properties
3075:Second-countable space
2333:Manifold with boundary
2048:Differential structure
1952:Upper Saddle River, NJ
1686:
1646:
1624:
1555:
1472:
1452:
1401:
1320:
1291:
1261:
1241:
1219:
1193:
1161:
1129:
1101:
1085:line with many origins
1079:Line with many origins
1047:
1046:{\displaystyle [-1,0)}
1012:
958:
894:
867:
844:
821:
764:
763:{\displaystyle 0_{b}.}
734:
699:
674:
647:
620:
600:
578:
549:
524:
504:
484:
433:is formed by the sets
427:
396:
362:
361:{\displaystyle 0_{b}.}
332:
305:
285:
258:
232:
200:
168:
142:
108:
1924:Lee, John M. (2011).
1710:analytic continuation
1687:
1647:
1625:
1556:
1473:
1453:
1402:
1321:
1292:
1262:
1242:
1220:
1194:
1162:
1130:
1102:
1048:
1013:
959:
895:
893:{\displaystyle 0_{b}}
868:
845:
822:
820:{\displaystyle 0_{a}}
765:
735:
733:{\displaystyle 0_{a}}
700:
675:
673:{\displaystyle 0_{i}}
648:
646:{\displaystyle 0_{i}}
621:
601:
579:
577:{\displaystyle 0_{i}}
550:
525:
505:
485:
428:
426:{\displaystyle 0_{i}}
397:
363:
333:
331:{\displaystyle 0_{a}}
306:
286:
259:
233:
231:{\displaystyle (x,b)}
201:
199:{\displaystyle (x,a)}
169:
143:
109:
70:line with two origins
64:Line with two origins
29:geometry and topology
18:Line with two origins
3144:Invariance of domain
3096:Euler characteristic
3070:Bundle (mathematics)
2480:Covariant derivative
2031:Topological manifold
1722:locally homeomorphic
1656:
1636:
1568:
1563:equivalence relation
1507:
1462:
1411:
1333:
1301:
1271:
1251:
1229:
1203:
1171:
1139:
1111:
1091:
1022:
968:
914:
877:
854:
831:
804:
744:
717:
684:
657:
630:
610:
588:
561:
534:
514:
494:
437:
410:
372:
342:
315:
295:
273:
242:
210:
178:
152:
118:
84:
49:locally homeomorphic
3154:Tychonoff's theorem
3149:Poincaré conjecture
2903:General (point-set)
2514:Exterior derivative
2116:Atiyah–Singer index
2065:Riemannian manifold
1950:(Second ed.).
1918:2006math......9665G
3139:De Rham cohomology
3060:Polyhedral complex
3050:Simplicial complex
2820:Secondary calculus
2774:Singularity theory
2729:Parallel transport
2497:De Rham cohomology
2136:Generalized Stokes
1956:Prentice Hall, Inc
1802:, Proposition 5.1.
1754:List of topologies
1730:locally metrizable
1682:
1642:
1620:
1551:
1468:
1448:
1400:{\displaystyle A=}
1397:
1316:
1287:
1267:with many origins
1257:
1237:
1215:
1189:
1157:
1125:
1097:
1043:
1008:
954:
890:
866:{\displaystyle -1}
863:
843:{\displaystyle -1}
840:
817:
760:
730:
695:
670:
643:
616:
596:
574:
545:
520:
500:
480:
423:
392:
358:
328:
301:
281:
254:
228:
196:
164:
138:
104:
3254:
3253:
3043:fundamental group
2855:
2854:
2737:
2736:
2502:Differential form
2156:Whitney embedding
2090:Differential form
1965:978-0-13-181629-9
1944:Munkres, James R.
1935:978-1-4419-7939-1
1909:math.GT/0609665v1
1843:978-0-387-90894-6
1738:locally Hausdorff
1645:{\displaystyle r}
1608:
1531:
1471:{\displaystyle A}
1260:{\displaystyle 0}
1100:{\displaystyle S}
711:locally Hausdorff
707:locally Euclidean
619:{\displaystyle 0}
523:{\displaystyle 0}
503:{\displaystyle U}
311:with two origins
304:{\displaystyle 0}
16:(Redirected from
3284:
3267:General topology
3244:
3243:
3217:
3216:
3207:
3197:
3187:
3186:
3175:
3174:
2969:
2882:
2875:
2868:
2859:
2858:
2847:Stratified space
2805:Fréchet manifold
2519:Interior product
2412:
2411:
2109:
2005:
1998:
1991:
1982:
1981:
1977:
1939:
1920:
1911:
1894:
1892:
1882:
1873:(3): 1105–1111.
1848:
1847:
1831:
1821:
1815:
1809:
1803:
1797:
1791:
1785:
1765:Separation axiom
1736:in general) and
1691:
1689:
1688:
1683:
1681:
1680:
1668:
1667:
1651:
1649:
1648:
1643:
1629:
1627:
1626:
1621:
1609:
1606:
1560:
1558:
1557:
1552:
1538:
1532:
1529:
1514:
1477:
1475:
1474:
1469:
1457:
1455:
1454:
1449:
1432:
1431:
1406:
1404:
1403:
1398:
1375:
1374:
1325:
1323:
1322:
1317:
1296:
1294:
1293:
1288:
1283:
1282:
1266:
1264:
1263:
1258:
1246:
1244:
1243:
1238:
1236:
1224:
1222:
1221:
1216:
1198:
1196:
1195:
1190:
1166:
1164:
1163:
1158:
1134:
1132:
1131:
1126:
1118:
1106:
1104:
1103:
1098:
1052:
1050:
1049:
1044:
1017:
1015:
1014:
1009:
1004:
1003:
963:
961:
960:
955:
950:
949:
899:
897:
896:
891:
889:
888:
872:
870:
869:
864:
849:
847:
846:
841:
826:
824:
823:
818:
816:
815:
783:second countable
770:It is however a
769:
767:
766:
761:
756:
755:
739:
737:
736:
731:
729:
728:
704:
702:
701:
696:
691:
680:homeomorphic to
679:
677:
676:
671:
669:
668:
652:
650:
649:
644:
642:
641:
625:
623:
622:
617:
605:
603:
602:
597:
595:
583:
581:
580:
575:
573:
572:
557:For each origin
554:
552:
551:
546:
541:
529:
527:
526:
521:
509:
507:
506:
501:
489:
487:
486:
481:
476:
475:
432:
430:
429:
424:
422:
421:
401:
399:
398:
393:
379:
367:
365:
364:
359:
354:
353:
337:
335:
334:
329:
327:
326:
310:
308:
307:
302:
290:
288:
287:
282:
280:
263:
261:
260:
255:
237:
235:
234:
229:
205:
203:
202:
197:
173:
171:
170:
165:
147:
145:
144:
139:
125:
113:
111:
110:
105:
91:
41:general topology
21:
3292:
3291:
3287:
3286:
3285:
3283:
3282:
3281:
3257:
3256:
3255:
3250:
3181:
3163:
3159:Urysohn's lemma
3120:
3084:
2970:
2961:
2933:low-dimensional
2891:
2886:
2856:
2851:
2790:Banach manifold
2783:Generalizations
2778:
2733:
2670:
2567:
2529:Ricci curvature
2485:Cotangent space
2463:
2401:
2243:
2237:
2196:Exponential map
2160:
2105:
2099:
2019:
2009:
1966:
1936:
1857:
1852:
1851:
1844:
1822:
1818:
1810:
1806:
1798:
1794:
1786:
1779:
1774:
1750:
1726:Euclidean space
1718:
1698:
1676:
1672:
1663:
1659:
1657:
1654:
1653:
1652:and two points
1637:
1634:
1633:
1605:
1569:
1566:
1565:
1534:
1530: and
1528:
1510:
1508:
1505:
1504:
1490:
1480:locally compact
1463:
1460:
1459:
1427:
1423:
1412:
1409:
1408:
1370:
1366:
1334:
1331:
1330:
1302:
1299:
1298:
1278:
1274:
1272:
1269:
1268:
1252:
1249:
1248:
1232:
1230:
1227:
1226:
1204:
1201:
1200:
1172:
1169:
1168:
1140:
1137:
1136:
1114:
1112:
1109:
1108:
1092:
1089:
1088:
1081:
1060:locally compact
1023:
1020:
1019:
999:
995:
969:
966:
965:
945:
941:
915:
912:
911:
884:
880:
878:
875:
874:
855:
852:
851:
832:
829:
828:
811:
807:
805:
802:
801:
775:
751:
747:
745:
742:
741:
724:
720:
718:
715:
714:
687:
685:
682:
681:
664:
660:
658:
655:
654:
637:
633:
631:
628:
627:
611:
608:
607:
591:
589:
586:
585:
568:
564:
562:
559:
558:
537:
535:
532:
531:
515:
512:
511:
495:
492:
491:
471:
467:
438:
435:
434:
417:
413:
411:
408:
407:
375:
373:
370:
369:
349:
345:
343:
340:
339:
322:
318:
316:
313:
312:
296:
293:
292:
276:
274:
271:
270:
243:
240:
239:
211:
208:
207:
179:
176:
175:
153:
150:
149:
121:
119:
116:
115:
87:
85:
82:
81:
76:. This is the
66:
61:
53:Euclidean space
37:Hausdorff space
23:
22:
15:
12:
11:
5:
3290:
3280:
3279:
3274:
3269:
3252:
3251:
3249:
3248:
3238:
3237:
3236:
3231:
3226:
3211:
3201:
3191:
3179:
3168:
3165:
3164:
3162:
3161:
3156:
3151:
3146:
3141:
3136:
3130:
3128:
3122:
3121:
3119:
3118:
3113:
3108:
3106:Winding number
3103:
3098:
3092:
3090:
3086:
3085:
3083:
3082:
3077:
3072:
3067:
3062:
3057:
3052:
3047:
3046:
3045:
3040:
3038:homotopy group
3030:
3029:
3028:
3023:
3018:
3013:
3008:
2998:
2993:
2988:
2978:
2976:
2972:
2971:
2964:
2962:
2960:
2959:
2954:
2949:
2948:
2947:
2937:
2936:
2935:
2925:
2920:
2915:
2910:
2905:
2899:
2897:
2893:
2892:
2885:
2884:
2877:
2870:
2862:
2853:
2852:
2850:
2849:
2844:
2839:
2834:
2829:
2828:
2827:
2817:
2812:
2807:
2802:
2797:
2792:
2786:
2784:
2780:
2779:
2777:
2776:
2771:
2766:
2761:
2756:
2751:
2745:
2743:
2739:
2738:
2735:
2734:
2732:
2731:
2726:
2721:
2716:
2711:
2706:
2701:
2696:
2691:
2686:
2680:
2678:
2672:
2671:
2669:
2668:
2663:
2658:
2653:
2648:
2643:
2638:
2628:
2623:
2618:
2608:
2603:
2598:
2593:
2588:
2583:
2577:
2575:
2569:
2568:
2566:
2565:
2560:
2555:
2554:
2553:
2543:
2538:
2537:
2536:
2526:
2521:
2516:
2511:
2510:
2509:
2499:
2494:
2493:
2492:
2482:
2477:
2471:
2469:
2465:
2464:
2462:
2461:
2456:
2451:
2446:
2445:
2444:
2434:
2429:
2424:
2418:
2416:
2409:
2403:
2402:
2400:
2399:
2394:
2384:
2379:
2365:
2360:
2355:
2350:
2345:
2343:Parallelizable
2340:
2335:
2330:
2329:
2328:
2318:
2313:
2308:
2303:
2298:
2293:
2288:
2283:
2278:
2273:
2263:
2253:
2247:
2245:
2239:
2238:
2236:
2235:
2230:
2225:
2223:Lie derivative
2220:
2218:Integral curve
2215:
2210:
2205:
2204:
2203:
2193:
2188:
2187:
2186:
2179:Diffeomorphism
2176:
2170:
2168:
2162:
2161:
2159:
2158:
2153:
2148:
2143:
2138:
2133:
2128:
2123:
2118:
2112:
2110:
2101:
2100:
2098:
2097:
2092:
2087:
2082:
2077:
2072:
2067:
2062:
2057:
2056:
2055:
2050:
2040:
2039:
2038:
2027:
2025:
2024:Basic concepts
2021:
2020:
2008:
2007:
2000:
1993:
1985:
1979:
1978:
1964:
1940:
1934:
1921:
1895:
1856:
1853:
1850:
1849:
1842:
1816:
1804:
1792:
1790:, p. 227.
1776:
1775:
1773:
1770:
1769:
1768:
1762:
1757:
1749:
1746:
1717:
1714:
1697:
1694:
1679:
1675:
1671:
1666:
1662:
1641:
1619:
1616:
1613:
1607: if
1603:
1600:
1597:
1594:
1591:
1588:
1585:
1582:
1579:
1576:
1573:
1550:
1547:
1544:
1541:
1537:
1526:
1523:
1520:
1517:
1513:
1501:quotient space
1494:branching line
1489:
1488:Branching line
1486:
1467:
1447:
1444:
1441:
1438:
1435:
1430:
1426:
1422:
1419:
1416:
1396:
1393:
1390:
1387:
1384:
1381:
1378:
1373:
1369:
1365:
1362:
1359:
1356:
1353:
1350:
1347:
1344:
1341:
1338:
1315:
1312:
1309:
1306:
1286:
1281:
1277:
1256:
1235:
1214:
1211:
1208:
1188:
1185:
1182:
1179:
1176:
1156:
1153:
1150:
1147:
1144:
1124:
1121:
1117:
1096:
1080:
1077:
1069:
1068:
1055:
1054:
1042:
1039:
1036:
1033:
1030:
1027:
1007:
1002:
998:
994:
991:
988:
985:
982:
979:
976:
973:
953:
948:
944:
940:
937:
934:
931:
928:
925:
922:
919:
903:
902:
887:
883:
862:
859:
839:
836:
814:
810:
794:path connected
773:
759:
754:
750:
727:
723:
694:
690:
667:
663:
640:
636:
615:
594:
571:
567:
544:
540:
519:
499:
479:
474:
470:
466:
463:
460:
457:
454:
451:
448:
445:
442:
420:
416:
391:
388:
385:
382:
378:
357:
352:
348:
325:
321:
300:
279:
253:
250:
247:
227:
224:
221:
218:
215:
195:
192:
189:
186:
183:
163:
160:
157:
137:
134:
131:
128:
124:
103:
100:
97:
94:
90:
78:quotient space
65:
62:
60:
57:
9:
6:
4:
3:
2:
3289:
3278:
3275:
3273:
3270:
3268:
3265:
3264:
3262:
3247:
3239:
3235:
3232:
3230:
3227:
3225:
3222:
3221:
3220:
3212:
3210:
3206:
3202:
3200:
3196:
3192:
3190:
3185:
3180:
3178:
3170:
3169:
3166:
3160:
3157:
3155:
3152:
3150:
3147:
3145:
3142:
3140:
3137:
3135:
3132:
3131:
3129:
3127:
3123:
3117:
3116:Orientability
3114:
3112:
3109:
3107:
3104:
3102:
3099:
3097:
3094:
3093:
3091:
3087:
3081:
3078:
3076:
3073:
3071:
3068:
3066:
3063:
3061:
3058:
3056:
3053:
3051:
3048:
3044:
3041:
3039:
3036:
3035:
3034:
3031:
3027:
3024:
3022:
3019:
3017:
3014:
3012:
3009:
3007:
3004:
3003:
3002:
2999:
2997:
2994:
2992:
2989:
2987:
2983:
2980:
2979:
2977:
2973:
2968:
2958:
2955:
2953:
2952:Set-theoretic
2950:
2946:
2943:
2942:
2941:
2938:
2934:
2931:
2930:
2929:
2926:
2924:
2921:
2919:
2916:
2914:
2913:Combinatorial
2911:
2909:
2906:
2904:
2901:
2900:
2898:
2894:
2890:
2883:
2878:
2876:
2871:
2869:
2864:
2863:
2860:
2848:
2845:
2843:
2842:Supermanifold
2840:
2838:
2835:
2833:
2830:
2826:
2823:
2822:
2821:
2818:
2816:
2813:
2811:
2808:
2806:
2803:
2801:
2798:
2796:
2793:
2791:
2788:
2787:
2785:
2781:
2775:
2772:
2770:
2767:
2765:
2762:
2760:
2757:
2755:
2752:
2750:
2747:
2746:
2744:
2740:
2730:
2727:
2725:
2722:
2720:
2717:
2715:
2712:
2710:
2707:
2705:
2702:
2700:
2697:
2695:
2692:
2690:
2687:
2685:
2682:
2681:
2679:
2677:
2673:
2667:
2664:
2662:
2659:
2657:
2654:
2652:
2649:
2647:
2644:
2642:
2639:
2637:
2633:
2629:
2627:
2624:
2622:
2619:
2617:
2613:
2609:
2607:
2604:
2602:
2599:
2597:
2594:
2592:
2589:
2587:
2584:
2582:
2579:
2578:
2576:
2574:
2570:
2564:
2563:Wedge product
2561:
2559:
2556:
2552:
2549:
2548:
2547:
2544:
2542:
2539:
2535:
2532:
2531:
2530:
2527:
2525:
2522:
2520:
2517:
2515:
2512:
2508:
2507:Vector-valued
2505:
2504:
2503:
2500:
2498:
2495:
2491:
2488:
2487:
2486:
2483:
2481:
2478:
2476:
2473:
2472:
2470:
2466:
2460:
2457:
2455:
2452:
2450:
2447:
2443:
2440:
2439:
2438:
2437:Tangent space
2435:
2433:
2430:
2428:
2425:
2423:
2420:
2419:
2417:
2413:
2410:
2408:
2404:
2398:
2395:
2393:
2389:
2385:
2383:
2380:
2378:
2374:
2370:
2366:
2364:
2361:
2359:
2356:
2354:
2351:
2349:
2346:
2344:
2341:
2339:
2336:
2334:
2331:
2327:
2324:
2323:
2322:
2319:
2317:
2314:
2312:
2309:
2307:
2304:
2302:
2299:
2297:
2294:
2292:
2289:
2287:
2284:
2282:
2279:
2277:
2274:
2272:
2268:
2264:
2262:
2258:
2254:
2252:
2249:
2248:
2246:
2240:
2234:
2231:
2229:
2226:
2224:
2221:
2219:
2216:
2214:
2211:
2209:
2206:
2202:
2201:in Lie theory
2199:
2198:
2197:
2194:
2192:
2189:
2185:
2182:
2181:
2180:
2177:
2175:
2172:
2171:
2169:
2167:
2163:
2157:
2154:
2152:
2149:
2147:
2144:
2142:
2139:
2137:
2134:
2132:
2129:
2127:
2124:
2122:
2119:
2117:
2114:
2113:
2111:
2108:
2104:Main results
2102:
2096:
2093:
2091:
2088:
2086:
2085:Tangent space
2083:
2081:
2078:
2076:
2073:
2071:
2068:
2066:
2063:
2061:
2058:
2054:
2051:
2049:
2046:
2045:
2044:
2041:
2037:
2034:
2033:
2032:
2029:
2028:
2026:
2022:
2017:
2013:
2006:
2001:
1999:
1994:
1992:
1987:
1986:
1983:
1975:
1971:
1967:
1961:
1957:
1953:
1949:
1945:
1941:
1937:
1931:
1927:
1922:
1919:
1915:
1910:
1905:
1901:
1896:
1891:
1886:
1881:
1876:
1872:
1868:
1864:
1859:
1858:
1845:
1839:
1835:
1830:
1829:
1820:
1813:
1808:
1801:
1796:
1789:
1784:
1782:
1777:
1766:
1763:
1761:
1758:
1755:
1752:
1751:
1745:
1744:in general).
1743:
1739:
1735:
1731:
1727:
1723:
1713:
1711:
1707:
1703:
1693:
1677:
1673:
1669:
1664:
1660:
1639:
1630:
1617:
1614:
1611:
1598:
1595:
1592:
1586:
1580:
1577:
1574:
1564:
1545:
1539:
1521:
1515:
1502:
1497:
1495:
1485:
1483:
1481:
1465:
1442:
1439:
1436:
1433:
1428:
1424:
1417:
1414:
1391:
1388:
1385:
1379:
1371:
1367:
1360:
1354:
1351:
1348:
1345:
1339:
1336:
1327:
1313:
1310:
1307:
1304:
1297:one for each
1284:
1279:
1275:
1254:
1212:
1209:
1206:
1183:
1180:
1177:
1151:
1148:
1145:
1122:
1119:
1094:
1086:
1076:
1074:
1066:
1065:regular space
1061:
1058:The space is
1057:
1056:
1037:
1034:
1031:
1028:
1000:
996:
989:
983:
980:
977:
974:
946:
942:
935:
929:
926:
923:
920:
909:
905:
904:
885:
881:
860:
857:
837:
834:
812:
808:
799:
798:arc connected
795:
792:The space is
791:
790:
789:
786:
784:
781:The space is
779:
777:
757:
752:
748:
725:
721:
712:
708:
692:
665:
661:
638:
634:
613:
606:by replacing
569:
565:
555:
542:
517:
497:
472:
468:
461:
452:
443:
418:
414:
405:
386:
368:The subspace
355:
350:
346:
323:
319:
298:
269:
264:
251:
248:
245:
222:
219:
216:
190:
187:
184:
161:
158:
155:
132:
126:
98:
92:
79:
75:
74:bug-eyed line
71:
56:
54:
50:
46:
42:
38:
34:
30:
19:
3246:Publications
3111:Chern number
3101:Betti number
2984: /
2975:Key concepts
2923:Differential
2769:Moving frame
2764:Morse theory
2754:Gauge theory
2546:Tensor field
2475:Closed/Exact
2454:Vector field
2422:Distribution
2363:Hypercomplex
2358:Quaternionic
2095:Vector field
2053:Smooth atlas
1947:
1925:
1899:
1880:math/0609098
1870:
1866:
1827:
1819:
1807:
1795:
1788:Munkres 2000
1719:
1699:
1631:
1499:This is the
1498:
1493:
1491:
1484:
1328:
1084:
1082:
1070:
908:compact sets
787:
780:
556:
265:
73:
69:
67:
44:
26:
3209:Wikiversity
3126:Key results
2714:Levi-Civita
2704:Generalized
2676:Connections
2626:Lie algebra
2558:Volume form
2459:Vector flow
2432:Pushforward
2427:Lie bracket
2326:Lie algebra
2291:G-structure
2080:Pushforward
2060:Submanifold
1800:Gabard 2006
1728:, they are
1712:property.)
1702:etale space
1696:Etale space
1407:is the set
3261:Categories
3055:CW complex
2996:Continuity
2986:Closed set
2945:cohomology
2837:Stratifold
2795:Diffeology
2591:Associated
2392:Symplectic
2377:Riemannian
2306:Hyperbolic
2233:Submersion
2141:Hopf–Rinow
2075:Submersion
2070:Smooth map
1855:References
1734:metrizable
1716:Properties
1073:CW-complex
404:local base
3272:Manifolds
3234:geometric
3229:algebraic
3080:Cobordism
3016:Hausdorff
3011:connected
2928:Geometric
2918:Continuum
2908:Algebraic
2719:Principal
2694:Ehresmann
2651:Subbundle
2641:Principal
2616:Fibration
2596:Cotangent
2468:Covectors
2321:Lie group
2301:Hermitian
2244:manifolds
2213:Immersion
2208:Foliation
2146:Noether's
2131:Frobenius
2126:De Rham's
2121:Darboux's
2012:Manifolds
1742:Hausdorff
1740:(but not
1732:(but not
1587:∼
1561:with the
1540:×
1516:×
1440:∈
1437:β
1429:β
1418:∪
1380:∪
1372:α
1361:∪
1346:−
1308:∈
1305:α
1280:α
1210:≠
1199:whenever
1184:β
1152:α
1120:×
1029:−
990:∪
975:−
936:∪
921:−
858:−
835:−
462:∪
447:∖
381:∖
268:real line
249:≠
238:whenever
159:≠
127:×
93:×
47:: spaces
3277:Topology
3199:Wikibook
3177:Category
3065:Manifold
3033:Homotopy
2991:Interior
2982:Open set
2940:Homology
2889:Topology
2815:Orbifold
2810:K-theory
2800:Diffiety
2524:Pullback
2338:Oriented
2316:Kenmotsu
2296:Hadamard
2242:Types of
2191:Geodesic
2016:Glossary
1974:42683260
1948:Topology
1946:(2000).
1812:Lee 2011
1748:See also
796:but not
59:Examples
35:to be a
33:manifold
3224:general
3026:uniform
3006:compact
2957:Digital
2759:History
2742:Related
2656:Tangent
2634:)
2614:)
2581:Adjoint
2573:Bundles
2551:density
2449:Torsion
2415:Vectors
2407:Tensors
2390:)
2375:)
2371:,
2369:Pseudo−
2348:Poisson
2281:Finsler
2276:Fibered
2271:Contact
2269:)
2261:Complex
2259:)
2228:Section
1914:Bibcode
1053:is not.
3219:Topics
3021:metric
2896:Fields
2724:Vector
2709:Koszul
2689:Cartan
2684:Affine
2666:Vector
2661:Tensor
2646:Spinor
2636:Normal
2632:Stable
2586:Affine
2490:bundle
2442:bundle
2388:Almost
2311:Kähler
2267:Almost
2257:Almost
2251:Closed
2151:Sard's
2107:(list)
1972:
1962:
1932:
1840:
901:right.
148:(with
3001:Space
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