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Informally, an object in our space is simply connected if it consists of one piece and does not have any "holes" that pass all the way through it. For example, neither a doughnut nor a coffee cup (with a handle) is simply connected, but a hollow rubber ball is simply connected. In two dimensions, a
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between two points can be continuously transformed into any other such path while preserving the two endpoints in question. Intuitively, this corresponds to a space that has no disjoint parts and no holes that go completely through it, because two paths going around different sides of such a hole
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are connected. The set of complex numbers with imaginary part strictly greater than zero and less than one furnishes an example of an unbounded, connected, open subset of the plane whose complement is not connected. It is nevertheless simply connected. A relaxation of the requirement that
1083:-shaped holes. A sphere (or, equivalently, a rubber ball with a hollow center) is simply connected, because any loop on the surface of a sphere can contract to a point even though it has a "hole" in the hollow center. The stronger condition, that the object has no holes of
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be connected leads to an exploration of open subsets of the plane with connected extended complement. For example, a (not necessarily connected) open set has a connected extended complement exactly when each of its connected components is simply connected.
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of a topological space is an indicator of the failure for the space to be simply connected: a path-connected topological space is simply connected if and only if its fundamental group is trivial.
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The image of a simply connected set under a continuous function need not be simply connected. Take for example the complex plane under the exponential map: the image is
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A torus is not a simply connected surface. Neither of the two colored loops shown here can be contracted to a point without leaving the surface. A
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The notion of simple connectedness is also a crucial condition in the
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The integral thus does not depend on the particular path connecting
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while keeping both endpoints fixed. Explicitly, there exists a
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can be contracted to a point: there exists a continuous map
2317:. Academic Search Complete. North Charleston: CreateSpace.
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states that any non-empty open simply connected subset of
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cannot be continuously transformed into each other. The
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at each point is trivial, i.e. consists only of the
1837:The notion of simple connectedness is important in
1495:{\displaystyle \operatorname {SO} (n,\mathbb {R} )}
930:{\displaystyle \operatorname {Hom} _{\Pi (X)}(x,y)}
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1827:{\displaystyle \mathbb {C} \setminus \{0\},}
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1646:A surface (two-dimensional topological
989:{\displaystyle X\subseteq \mathbb {C} }
1530:{\displaystyle \operatorname {SU} (n)}
1292:minus the origin are simply connected.
67:Definition and equivalent formulations
1562:is not simply connected (even though
7:
2085:of the path, and can be computed as
1055:but not simply connected are called
75:This shape represents a set that is
2366:Functions of One Complex Variable I
1924:{\displaystyle f:U\to \mathbb {C} }
1419:is simply connected; this includes
1328:is simply connected if and only if
777:is simply connected if and only if
351:An equivalent formulation is this:
27:Space which has no holes through it
2402:Gamelin, Theodore (January 2001).
1540:The one-point compactification of
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567:can be continuously deformed into
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1841:because of the following facts:
2455:Properties of topological spaces
2423:Introduction to General Topology
2345:Spanier, Edwin (December 1994).
1772:is simply connected, then so is
1502:is not simply connected and the
1382:{\displaystyle \mathbb {R} ^{n}}
1285:{\displaystyle \mathbb {R} ^{n}}
1256:{\displaystyle \mathbb {R} ^{n}}
1166:{\displaystyle \mathbb {R} ^{2}}
1137:{\displaystyle \mathbb {R} ^{2}}
110:if it is path-connected and any
2045:depends only on the end points
1834:which is not simply connected.
2260:Locally simply connected space
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645:{\displaystyle F:\times \to X}
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1890:{\displaystyle \mathbb {C} ,}
2421:Joshi, Kapli (August 1983).
2218:{\displaystyle \mathbb {C} }
2196:{\displaystyle \mathbb {C} }
1577:{\displaystyle \mathbb {R} }
1555:{\displaystyle \mathbb {R} }
748:{\displaystyle F(x,1)=q(x).}
205:{\displaystyle F:D^{2}\to X}
166:{\displaystyle f:S^{1}\to X}
2385:Lie Groups and Lie Algebras
2313:Ronald, Brown (June 2006).
2290:"n-connected space in nLab"
695:{\displaystyle F(x,0)=p(x)}
2471:
2383:Bourbaki, Nicolas (2005).
2125:{\displaystyle F(v)-F(u).}
1016:and its complement in the
797:is path-connected and the
1847:Cauchy's integral theorem
1412:are not simply connected.
1144:is simply connected, but
873:{\displaystyle x,y\in X,}
540:{\displaystyle p(1)=q(1)}
496:{\displaystyle p(0)=q(0)}
1461:special orthogonal group
1452:{\displaystyle n\geq 2,}
1417:topological vector space
1347:{\displaystyle n\geq 2.}
2179:Riemann mapping theorem
2001:and the value of every
1227:{\displaystyle n>2,}
452:{\displaystyle q:\to X}
408:{\displaystyle p:\to X}
2425:. New Age Publishers.
2315:Topology and Groupoids
2227:conformally equivalent
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1630:{\displaystyle L^{*}}
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1584:is simply connected).
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1504:special unitary group
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1321:{\displaystyle S^{n}}
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1198:{\displaystyle (0,0)}
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329:{\displaystyle D^{2}}
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2251:Deformation retract
2239:Poincaré conjecture
1750:homotopy equivalent
1300:-dimensional sphere
1046:Informal discussion
2450:Algebraic topology
2347:Algebraic Topology
2215:
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2168:{\displaystyle v,}
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1994:{\displaystyle U,}
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1788:{\displaystyle Y.}
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275:{\displaystyle f.}
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48:1-simply connected
2270:Unicoherent space
2265:n-connected space
2145:{\displaystyle u}
2078:{\displaystyle v}
2058:{\displaystyle u}
2038:{\displaystyle f}
2018:{\displaystyle U}
1971:{\displaystyle F}
1948:{\displaystyle f}
1862:{\displaystyle U}
1765:{\displaystyle X}
1741:{\displaystyle Y}
1721:{\displaystyle X}
1694:{\displaystyle X}
1674:{\displaystyle X}
1603:{\displaystyle L}
1396:, the (elliptic)
1295:Analogously: the
1173:minus the origin
1034:{\displaystyle X}
1009:{\displaystyle X}
968:: an open subset
954:{\displaystyle X}
838:{\displaystyle X}
814:{\displaystyle X}
799:fundamental group
790:{\displaystyle X}
770:{\displaystyle X}
580:{\displaystyle q}
560:{\displaystyle p}
364:{\displaystyle X}
225:{\displaystyle F}
127:{\displaystyle X}
99:{\displaystyle X}
85:topological space
61:fundamental group
36:topological space
16:(Redirected from
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970:
969:
943:
942:
890:
885:
884:
847:
846:
827:
826:
803:
802:
779:
778:
759:
758:
704:
703:
654:
653:
592:
591:
569:
568:
549:
548:
505:
504:
461:
460:
417:
416:
373:
372:
353:
352:
346:Euclidean plane
316:
311:
310:
289:
284:
283:
261:
260:
239:
234:
233:
214:
213:
186:
175:
174:
147:
136:
135:
116:
115:
88:
87:
69:
28:
23:
22:
15:
12:
11:
5:
2468:
2466:
2458:
2457:
2452:
2442:
2441:
2438:
2437:
2431:
2418:
2412:
2399:
2393:
2380:
2374:
2361:
2355:
2339:
2338:
2323:
2305:
2280:
2279:
2277:
2274:
2273:
2272:
2267:
2262:
2257:
2246:
2243:
2235:
2234:
2213:
2191:
2175:
2164:
2161:
2141:
2121:
2118:
2115:
2112:
2109:
2106:
2103:
2100:
2097:
2094:
2074:
2054:
2034:
2014:
1990:
1987:
1967:
1957:antiderivative
1944:
1919:
1915:
1912:
1909:
1906:
1886:
1882:
1858:
1823:
1820:
1817:
1814:
1811:
1807:
1784:
1781:
1761:
1737:
1717:
1690:
1670:
1657:
1643:
1640:
1639:
1638:
1624:
1620:
1599:
1585:
1572:
1550:
1538:
1526:
1523:
1520:
1517:
1514:
1491:
1487:
1483:
1480:
1477:
1474:
1471:
1448:
1445:
1442:
1439:
1428:
1425:Hilbert spaces
1413:
1390:
1376:
1371:
1354:
1343:
1340:
1337:
1315:
1311:
1293:
1279:
1274:
1250:
1245:
1223:
1220:
1217:
1214:
1194:
1191:
1188:
1185:
1182:
1160:
1155:
1131:
1126:
1096:
1093:
1086:
1047:
1044:
1030:
1018:Riemann sphere
1005:
984:
980:
977:
950:
926:
923:
920:
917:
914:
911:
906:
903:
900:
897:
893:
869:
866:
863:
860:
857:
854:
834:
810:
786:
766:
744:
741:
738:
735:
732:
729:
726:
723:
720:
717:
714:
711:
691:
688:
685:
682:
679:
676:
673:
670:
667:
664:
661:
641:
638:
635:
632:
629:
626:
623:
620:
617:
614:
611:
608:
605:
602:
599:
576:
556:
536:
533:
530:
527:
524:
521:
518:
515:
512:
492:
489:
486:
483:
480:
477:
474:
471:
468:
448:
445:
442:
439:
436:
433:
430:
427:
424:
404:
401:
398:
395:
392:
389:
386:
383:
380:
360:
348:respectively.
323:
319:
296:
292:
271:
268:
246:
242:
232:restricted to
221:
201:
198:
193:
189:
185:
182:
162:
159:
154:
150:
146:
143:
123:
109:
95:
68:
65:
52:path-connected
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
2467:
2456:
2453:
2451:
2448:
2447:
2445:
2434:
2432:0-85226-444-5
2428:
2424:
2419:
2415:
2413:0-387-95069-9
2409:
2405:
2400:
2396:
2394:3-540-43405-4
2390:
2386:
2381:
2377:
2375:0-387-90328-3
2371:
2367:
2362:
2358:
2356:0-387-94426-5
2352:
2348:
2343:
2342:
2334:
2330:
2326:
2320:
2316:
2309:
2306:
2295:
2291:
2285:
2282:
2275:
2271:
2268:
2266:
2263:
2261:
2258:
2252:
2249:
2248:
2244:
2242:
2240:
2232:
2228:
2180:
2176:
2162:
2159:
2139:
2119:
2113:
2107:
2104:
2098:
2092:
2072:
2052:
2032:
2012:
2004:
2003:line integral
1988:
1985:
1965:
1958:
1942:
1934:
1910:
1907:
1904:
1884:
1872:
1871:complex plane
1856:
1848:
1844:
1843:
1842:
1840:
1835:
1821:
1815:
1795:
1782:
1779:
1759:
1751:
1735:
1715:
1706:
1704:
1688:
1668:
1659:
1655:
1653:
1649:
1641:
1622:
1618:
1597:
1590:
1586:
1539:
1521:
1515:
1512:
1505:
1481:
1478:
1472:
1469:
1462:
1446:
1443:
1440:
1437:
1429:
1426:
1422:
1421:Banach spaces
1418:
1414:
1411:
1407:
1403:
1399:
1395:
1391:
1374:
1359:
1358:convex subset
1355:
1341:
1338:
1335:
1313:
1309:
1301:
1299:
1294:
1277:
1248:
1221:
1218:
1215:
1212:
1189:
1186:
1183:
1158:
1129:
1115:
1111:
1110:
1106:
1101:
1094:
1092:
1090:
1084:
1082:
1073:
1068:
1064:
1062:
1058:
1054:
1045:
1043:
1028:
1019:
1003:
978:
975:
967:
962:
948:
940:
921:
918:
915:
909:
901:
891:
883:
867:
864:
861:
858:
855:
852:
832:
825:. Similarly,
824:
808:
800:
784:
764:
755:
742:
736:
730:
727:
721:
718:
715:
709:
686:
680:
677:
671:
668:
665:
659:
639:
630:
627:
624:
618:
612:
609:
606:
600:
597:
590:
574:
554:
531:
525:
522:
516:
510:
487:
481:
478:
472:
466:
446:
437:
434:
431:
425:
422:
402:
393:
390:
387:
381:
378:
358:
349:
347:
343:
339:
321:
317:
294:
290:
269:
266:
244:
240:
219:
199:
191:
187:
183:
180:
160:
152:
148:
144:
141:
121:
113:
107:
93:
86:
78:
73:
66:
64:
62:
57:
53:
49:
45:
41:
37:
33:
19:
2422:
2406:. Springer.
2403:
2387:. Springer.
2384:
2368:. Springer.
2365:
2349:. Springer.
2346:
2314:
2308:
2297:. Retrieved
2293:
2284:
2236:
2203:(except for
1836:
1796:
1707:
1703:covering map
1660:
1645:
1410:Klein bottle
1402:Möbius strip
1297:
1077:
1060:
1056:
1049:
963:
756:
350:
336:denotes the
82:
76:
47:
43:
39:
29:
2294:ncatlab.org
2225:itself) is
1205:is not. If
1105:solid torus
880:the set of
340:and closed
338:unit circle
134:defined by
50:) if it is
44:1-connected
2444:Categories
2324:1419627228
2299:2017-09-17
2276:References
1642:Properties
1234:then both
652:such that
212:such that
106:is called
54:and every
38:is called
2333:712629429
2231:unit disk
2105:−
1914:→
1810:∖
1623:∗
1589:long line
1516:
1473:
1441:≥
1339:≥
1053:connected
979:⊆
910:
896:Π
882:morphisms
862:∈
637:→
619:×
444:→
400:→
342:unit disk
197:→
158:→
2245:See also
1648:manifold
1408:and the
1398:cylinder
1095:Examples
589:homotopy
547:), then
32:topology
2229:to the
1955:has an
1935:, then
1656:handles
937:in the
344:in the
2429:
2410:
2391:
2372:
2353:
2331:
2321:
1701:via a
1415:Every
1404:, the
1400:, the
1356:Every
1081:handle
1072:sphere
282:Here,
1931:is a
1652:genus
1394:torus
46:, or
2427:ISBN
2408:ISBN
2389:ISBN
2370:ISBN
2351:ISBN
2329:OCLC
2319:ISBN
2177:The
2152:and
2065:and
1897:and
1845:The
1752:and
1748:are
1728:and
1587:The
1459:the
1430:For
1423:and
1263:and
1216:>
1112:The
702:and
503:and
415:and
309:and
112:loop
56:path
42:(or
34:, a
2005:in
1978:on
1708:If
1360:of
1085:any
1059:or
964:In
941:of
892:Hom
801:of
259:is
114:in
77:not
30:In
2446::
2327:.
2292:.
2241:.
1705:.
1513:SU
1470:SO
1392:A
1342:2.
1091:.
1070:A
1063:.
83:A
2435:.
2416:.
2397:.
2378:.
2359:.
2335:.
2302:.
2233:.
2212:C
2190:C
2163:,
2160:v
2140:u
2120:.
2117:)
2114:u
2111:(
2108:F
2102:)
2099:v
2096:(
2093:F
2073:v
2053:u
2033:f
2013:U
1989:,
1986:U
1966:F
1943:f
1918:C
1911:U
1908::
1905:f
1885:,
1881:C
1857:U
1822:,
1819:}
1816:0
1813:{
1806:C
1783:.
1780:Y
1760:X
1736:Y
1716:X
1689:X
1669:X
1619:L
1598:L
1571:R
1549:R
1525:)
1522:n
1519:(
1490:)
1486:R
1482:,
1479:n
1476:(
1447:,
1444:2
1438:n
1427:.
1375:n
1370:R
1336:n
1314:n
1310:S
1298:n
1278:n
1273:R
1249:n
1244:R
1222:,
1219:2
1213:n
1193:)
1190:0
1187:,
1184:0
1181:(
1159:2
1154:R
1130:2
1125:R
1029:X
1004:X
983:C
976:X
949:X
925:)
922:y
919:,
916:x
913:(
905:)
902:X
899:(
868:,
865:X
859:y
856:,
853:x
833:X
809:X
785:X
765:X
743:.
740:)
737:x
734:(
731:q
728:=
725:)
722:1
719:,
716:x
713:(
710:F
690:)
687:x
684:(
681:p
678:=
675:)
672:0
669:,
666:x
663:(
660:F
640:X
634:]
631:1
628:,
625:0
622:[
616:]
613:1
610:,
607:0
604:[
601::
598:F
575:q
555:p
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532:1
529:(
526:q
523:=
520:)
517:1
514:(
511:p
491:)
488:0
485:(
482:q
479:=
476:)
473:0
470:(
467:p
447:X
441:]
438:1
435:,
432:0
429:[
426::
423:q
403:X
397:]
394:1
391:,
388:0
385:[
382::
379:p
359:X
322:2
318:D
295:1
291:S
270:.
267:f
245:1
241:S
220:F
200:X
192:2
188:D
184::
181:F
161:X
153:1
149:S
145::
142:f
122:X
94:X
20:)
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