297:
157:
1279:
1137:
1567:
1326:
843:
703:
1764:
1630:
1532:
1385:
363:
922:
496:
968:
1794:
1457:
1193:
1041:
1070:
739:
1346:
1213:
1686:, a function mapping finite sets of vertices to connected components of their complements. However, for locally finite graphs (graphs in which each vertex has finite
1162:
443:
396:
785:
1015:
628:
326:
1405:
988:
867:
759:
648:
599:
575:
551:
516:
463:
229:
209:
183:
92:
1796:— of any two of these maps exists a proper homotopy we say that they are equivalent and they define an equivalence class of proper rays. This set is called
1705:; this definition is insensitive to the choice of generating set. Every finitely-generated infinite group has either 1, 2, or infinitely many ends, and
1956:
1961:
1643:
has uncountably many ends, corresponding to the uncountably many different descending paths starting at the root. (This can be seen by letting
1706:
1908:
240:
1656:.) These ends can be thought of as the "leaves" of the infinite tree. In the end compactification, the set of ends has the topology of a
103:
1932:
1820:
881:
329:
48:
40:
1218:
1075:
1537:
602:
1690:), the ends defined in this way correspond one-for-one with the ends of topological spaces defined from the graph (
554:
1951:
1856:
790:
660:
1698:
1682:, an end is defined slightly differently, as an equivalence class of semi-infinite paths in the graph, or as a
1284:
43:
of the "ideal boundary" of the space. That is, each end represents a topologically distinct way to move to
1734:
1410:
The original definition above represents the special case where the direct system of compact subsets has a
1581:
399:
1687:
17:
1606:
1508:
1351:
335:
1683:
895:
468:
927:
186:
16:
This article is about the notion in general topology. For the construction in category theory, see
1777:
1440:
1171:
1020:
1881:
1767:
1670:
1046:
708:
518:. Such neighborhoods represent the neighborhoods of the corresponding point at infinity in the
1928:
1904:
1873:
1600:
1331:
1198:
578:
527:
95:
36:
415:
368:
1920:
1889:
1865:
1851:
1829:
1725:
1411:
889:
764:
189:
59:
1843:
993:
607:
304:
1893:
1839:
1469:
1915:
Scott, Peter; Wall, Terry; Wall, C. T. C. (1979). "Topological methods in group theory".
1815:
1498:). These ends are usually referred to as "infinity" and "minus infinity", respectively.
1142:
1718:
1676:
1390:
973:
852:
744:
655:
633:
584:
560:
536:
501:
448:
214:
194:
168:
162:
77:
1834:
1945:
1885:
1424:
870:
651:
1702:
1679:
1924:
1640:
28:
1729:
1721:
1657:
1877:
1435:
1428:
402:
44:
24:
1869:
877:
522:(this "compactification" is not always compact; the topological space
292:{\displaystyle U_{1}\supseteq U_{2}\supseteq U_{3}\supseteq \cdots ,}
152:{\displaystyle K_{1}\subseteq K_{2}\subseteq K_{3}\subseteq \cdots }
1588:
is equal to the number of connected components of the boundary of
405:
between the sets of ends associated with any two such sequences.
1818:(2003), "Graph-theoretical versus topological ends of graphs",
1043:
and they are compatible with maps induced by inclusions) then
365:. The number of ends does not depend on the specific sequence
58:
The notion of an end of a topological space was introduced by
1774:: more precisely, if between the restriction —to the subset
1709:
provides a decomposition for groups with more than one end.
1854:(1931), "Über die Enden topologischer Räume und Gruppen",
533:
The definition of ends given above applies only to spaces
47:
within the space. Adding a point at each end yields a
1274:{\displaystyle \pi _{0}(X\setminus \varphi ^{-1}(K'))}
1780:
1737:
1609:
1569:
has only one unbounded component for any compact set
1540:
1511:
1443:
1393:
1354:
1334:
1287:
1221:
1201:
1174:
1145:
1078:
1049:
1023:
996:
976:
930:
898:
855:
793:
767:
747:
711:
663:
636:
610:
587:
563:
539:
504:
471:
451:
418:
371:
338:
307:
243:
217:
197:
171:
106:
80:
1472:, then the two ends are the sequences of open sets
1132:{\displaystyle \varphi _{*}(x_{\varphi ^{-1}(K')})}
741:denotes the set of connected components of a space
1788:
1758:
1624:
1561:
1526:
1451:
1399:
1379:
1340:
1320:
1273:
1207:
1187:
1156:
1131:
1064:
1035:
1009:
982:
962:
916:
861:
837:
779:
753:
733:
697:
642:
622:
593:
581:). However, it can be generalized as follows: let
569:
545:
510:
490:
457:
437:
390:
357:
320:
291:
223:
203:
177:
151:
86:
1701:are defined to be the ends of the corresponding
1562:{\displaystyle \mathbb {R} ^{n}\smallsetminus K}
1584:, then the number of ends of the interior of
8:
876:Under this definition, the set of ends is a
692:
664:
617:
611:
1691:
601:be any topological space, and consider the
63:
1017:in the family is a connected component of
838:{\displaystyle \pi _{0}(Y)\to \pi _{0}(Z)}
698:{\displaystyle \{\pi _{0}(X\setminus K)\}}
1864:, Springer Berlin / Heidelberg: 692–713,
1833:
1782:
1781:
1779:
1744:
1740:
1739:
1736:
1616:
1612:
1611:
1608:
1547:
1543:
1542:
1539:
1518:
1514:
1513:
1510:
1445:
1444:
1442:
1392:
1359:
1353:
1333:
1292:
1286:
1245:
1226:
1220:
1200:
1179:
1173:
1144:
1101:
1096:
1083:
1077:
1048:
1022:
1001:
995:
975:
954:
944:
929:
897:
854:
820:
798:
792:
766:
746:
716:
710:
671:
662:
635:
609:
586:
562:
538:
503:
482:
470:
450:
426:
417:
379:
370:
349:
337:
312:
306:
274:
261:
248:
242:
216:
196:
170:
137:
124:
111:
105:
79:
1505: > 1, then Euclidean space
1321:{\displaystyle \pi _{0}(Y\setminus K')}
1304:
1238:
1027:
683:
342:
1707:Stallings theorem about ends of groups
1459:has two ends. For example, if we let
1759:{\displaystyle \mathbb {R} ^{+}\to X}
1652:be the complete binary tree of depth
7:
1494: = (−∞, −
51:of the original space, known as the
1901:Topological methods in group theory
1724:, the ends can be characterized as
1534:has only one end. This is because
14:
1957:Properties of topological spaces
1625:{\displaystyle \mathbb {R} ^{2}}
1527:{\displaystyle \mathbb {R} ^{n}}
1380:{\displaystyle \varphi ^{-1}(K)}
358:{\displaystyle X\setminus K_{n}}
1821:Journal of Combinatorial Theory
1164:ranges over compact subsets of
917:{\displaystyle \varphi :X\to Y}
1962:Compactification (mathematics)
1750:
1374:
1368:
1315:
1298:
1268:
1265:
1254:
1232:
1126:
1121:
1110:
1089:
1059:
1053:
951:
937:
908:
882:category of topological spaces
832:
826:
813:
810:
804:
771:
728:
722:
689:
677:
491:{\displaystyle V\supset U_{n}}
432:
419:
385:
372:
1:
1835:10.1016/S0095-8956(02)00034-5
1641:infinite complete binary tree
1603:emanating from the origin in
963:{\displaystyle x=(x_{K})_{K}}
1925:10.1017/CBO9781107325449.007
1789:{\displaystyle \mathbb {N} }
1452:{\displaystyle \mathbb {R} }
1348:is used to ensure that each
1188:{\displaystyle \varphi _{*}}
1036:{\displaystyle X\setminus K}
654:. There is a corresponding
398:of compact sets; there is a
161:is an ascending sequence of
1903:, GTM-243 (2008), Springer
1065:{\displaystyle \varphi (x)}
884:, where morphisms are only
734:{\displaystyle \pi _{0}(Y)}
39:are, roughly speaking, the
1978:
1668:
555:exhaustion by compact sets
15:
1857:Mathematische Zeitschrift
1665:Ends of graphs and groups
761:, and each inclusion map
408:Using this definition, a
1917:Homological Group Theory
1699:finitely generated group
1341:{\displaystyle \varphi }
1208:{\displaystyle \varphi }
888:continuous maps, to the
873:of this inverse system.
526:has to be connected and
1692:Diestel & Kühn 2003
1423:The set of ends of any
438:{\displaystyle (U_{i})}
391:{\displaystyle (K_{i})}
1790:
1760:
1626:
1582:manifold with boundary
1563:
1528:
1453:
1401:
1381:
1342:
1322:
1275:
1209:
1195:is the map induced by
1189:
1158:
1133:
1066:
1037:
1011:
984:
964:
918:
863:
839:
781:
780:{\displaystyle Y\to Z}
755:
735:
699:
644:
630:of compact subsets of
624:
595:
571:
547:
512:
492:
459:
439:
392:
359:
322:
293:
225:
205:
179:
153:
88:
1791:
1761:
1627:
1564:
1529:
1454:
1402:
1382:
1343:
1323:
1276:
1210:
1190:
1159:
1134:
1067:
1038:
1012:
1010:{\displaystyle x_{K}}
985:
965:
919:
869:is defined to be the
864:
840:
782:
756:
736:
700:
645:
625:
623:{\displaystyle \{K\}}
596:
572:
548:
513:
493:
460:
440:
393:
360:
323:
321:{\displaystyle U_{n}}
294:
226:
206:
180:
154:
89:
18:End (category theory)
1919:. pp. 137–204.
1778:
1735:
1713:Ends of a CW complex
1607:
1538:
1509:
1441:
1391:
1352:
1332:
1285:
1219:
1199:
1172:
1143:
1076:
1047:
1021:
994:
974:
928:
924:is a proper map and
896:
853:
791:
765:
745:
709:
661:
634:
608:
585:
561:
537:
520:end compactification
502:
469:
449:
416:
369:
336:
305:
241:
215:
195:
169:
104:
78:
60:Hans Freudenthal
53:end compactification
41:connected components
1814:Diestel, Reinhard;
1576:More generally, if
990:(i.e. each element
787:induces a function
330:connected component
235:for every sequence
98:, and suppose that
1870:10.1007/BF01174375
1786:
1756:
1671:End (graph theory)
1622:
1559:
1524:
1449:
1397:
1377:
1338:
1318:
1271:
1205:
1185:
1157:{\displaystyle K'}
1154:
1129:
1062:
1033:
1007:
980:
960:
914:
859:
835:
777:
751:
731:
695:
640:
620:
591:
567:
543:
508:
488:
455:
435:
388:
355:
318:
289:
221:
201:
175:
149:
84:
1909:978-0-387-74611-1
1852:Freudenthal, Hans
1400:{\displaystyle X}
983:{\displaystyle X}
892:. Explicitly, if
862:{\displaystyle X}
754:{\displaystyle Y}
643:{\displaystyle X}
594:{\displaystyle X}
570:{\displaystyle X}
546:{\displaystyle X}
528:locally connected
511:{\displaystyle n}
458:{\displaystyle V}
224:{\displaystyle X}
204:{\displaystyle X}
178:{\displaystyle X}
96:topological space
87:{\displaystyle X}
37:topological space
1969:
1952:General topology
1938:
1899:Ross Geoghegan,
1896:
1846:
1837:
1795:
1793:
1792:
1787:
1785:
1765:
1763:
1762:
1757:
1749:
1748:
1743:
1726:homotopy classes
1631:
1629:
1628:
1623:
1621:
1620:
1615:
1568:
1566:
1565:
1560:
1552:
1551:
1546:
1533:
1531:
1530:
1525:
1523:
1522:
1517:
1458:
1456:
1455:
1450:
1448:
1412:cofinal sequence
1406:
1404:
1403:
1398:
1386:
1384:
1383:
1378:
1367:
1366:
1347:
1345:
1344:
1339:
1328:. Properness of
1327:
1325:
1324:
1319:
1314:
1297:
1296:
1280:
1278:
1277:
1272:
1264:
1253:
1252:
1231:
1230:
1214:
1212:
1211:
1206:
1194:
1192:
1191:
1186:
1184:
1183:
1163:
1161:
1160:
1155:
1153:
1138:
1136:
1135:
1130:
1125:
1124:
1120:
1109:
1108:
1088:
1087:
1071:
1069:
1068:
1063:
1042:
1040:
1039:
1034:
1016:
1014:
1013:
1008:
1006:
1005:
989:
987:
986:
981:
969:
967:
966:
961:
959:
958:
949:
948:
923:
921:
920:
915:
890:category of sets
868:
866:
865:
860:
844:
842:
841:
836:
825:
824:
803:
802:
786:
784:
783:
778:
760:
758:
757:
752:
740:
738:
737:
732:
721:
720:
704:
702:
701:
696:
676:
675:
649:
647:
646:
641:
629:
627:
626:
621:
600:
598:
597:
592:
576:
574:
573:
568:
553:that possess an
552:
550:
549:
544:
517:
515:
514:
509:
497:
495:
494:
489:
487:
486:
464:
462:
461:
456:
444:
442:
441:
436:
431:
430:
397:
395:
394:
389:
384:
383:
364:
362:
361:
356:
354:
353:
327:
325:
324:
319:
317:
316:
298:
296:
295:
290:
279:
278:
266:
265:
253:
252:
230:
228:
227:
222:
210:
208:
207:
202:
184:
182:
181:
176:
158:
156:
155:
150:
142:
141:
129:
128:
116:
115:
93:
91:
90:
85:
49:compactification
1977:
1976:
1972:
1971:
1970:
1968:
1967:
1966:
1942:
1941:
1935:
1914:
1850:
1813:
1810:
1776:
1775:
1738:
1733:
1732:
1715:
1673:
1667:
1651:
1610:
1605:
1604:
1541:
1536:
1535:
1512:
1507:
1506:
1493:
1480:
1470:closed interval
1467:
1439:
1438:
1420:
1389:
1388:
1355:
1350:
1349:
1330:
1329:
1307:
1288:
1283:
1282:
1257:
1241:
1222:
1217:
1216:
1197:
1196:
1175:
1170:
1169:
1146:
1141:
1140:
1113:
1097:
1092:
1079:
1074:
1073:
1045:
1044:
1019:
1018:
997:
992:
991:
972:
971:
950:
940:
926:
925:
894:
893:
851:
850:
816:
794:
789:
788:
763:
762:
743:
742:
712:
707:
706:
667:
659:
658:
632:
631:
606:
605:
583:
582:
559:
558:
535:
534:
500:
499:
478:
467:
466:
447:
446:
445:is an open set
422:
414:
413:
375:
367:
366:
345:
334:
333:
308:
303:
302:
299:
270:
257:
244:
239:
238:
213:
212:
193:
192:
167:
166:
163:compact subsets
159:
133:
120:
107:
102:
101:
76:
75:
72:
21:
12:
11:
5:
1975:
1973:
1965:
1964:
1959:
1954:
1944:
1943:
1940:
1939:
1933:
1912:
1897:
1848:
1828:(1): 197–206,
1809:
1806:
1784:
1755:
1752:
1747:
1742:
1719:path connected
1714:
1711:
1697:The ends of a
1669:Main article:
1666:
1663:
1662:
1661:
1647:
1637:
1619:
1614:
1593:
1574:
1558:
1555:
1550:
1545:
1521:
1516:
1499:
1489:
1485:, ∞) and
1481: = (
1476:
1463:
1447:
1432:
1419:
1416:
1396:
1387:is compact in
1376:
1373:
1370:
1365:
1362:
1358:
1337:
1317:
1313:
1310:
1306:
1303:
1300:
1295:
1291:
1270:
1267:
1263:
1260:
1256:
1251:
1248:
1244:
1240:
1237:
1234:
1229:
1225:
1204:
1182:
1178:
1152:
1149:
1128:
1123:
1119:
1116:
1112:
1107:
1104:
1100:
1095:
1091:
1086:
1082:
1072:is the family
1061:
1058:
1055:
1052:
1032:
1029:
1026:
1004:
1000:
979:
957:
953:
947:
943:
939:
936:
933:
913:
910:
907:
904:
901:
858:
834:
831:
828:
823:
819:
815:
812:
809:
806:
801:
797:
776:
773:
770:
750:
730:
727:
724:
719:
715:
694:
691:
688:
685:
682:
679:
674:
670:
666:
656:inverse system
652:inclusion maps
639:
619:
616:
613:
590:
566:
542:
507:
485:
481:
477:
474:
454:
434:
429:
425:
421:
387:
382:
378:
374:
352:
348:
344:
341:
315:
311:
288:
285:
282:
277:
273:
269:
264:
260:
256:
251:
247:
237:
220:
200:
174:
148:
145:
140:
136:
132:
127:
123:
119:
114:
110:
100:
83:
71:
68:
27:, a branch of
13:
10:
9:
6:
4:
3:
2:
1974:
1963:
1960:
1958:
1955:
1953:
1950:
1949:
1947:
1936:
1934:9781107325449
1930:
1926:
1922:
1918:
1913:
1910:
1906:
1902:
1898:
1895:
1891:
1887:
1883:
1879:
1875:
1871:
1867:
1863:
1859:
1858:
1853:
1849:
1845:
1841:
1836:
1831:
1827:
1823:
1822:
1817:
1816:Kühn, Daniela
1812:
1811:
1807:
1805:
1803:
1799:
1773:
1769:
1753:
1745:
1731:
1727:
1723:
1720:
1712:
1710:
1708:
1704:
1700:
1695:
1693:
1689:
1685:
1681:
1678:
1672:
1664:
1659:
1655:
1650:
1646:
1642:
1638:
1635:
1617:
1602:
1598:
1595:The union of
1594:
1591:
1587:
1583:
1580:is a compact
1579:
1575:
1572:
1556:
1553:
1548:
1519:
1504:
1500:
1497:
1492:
1488:
1484:
1479:
1475:
1471:
1466:
1462:
1437:
1433:
1430:
1426:
1425:compact space
1422:
1421:
1417:
1415:
1413:
1408:
1394:
1371:
1363:
1360:
1356:
1335:
1311:
1308:
1301:
1293:
1289:
1261:
1258:
1249:
1246:
1242:
1235:
1227:
1223:
1202:
1180:
1176:
1167:
1150:
1147:
1117:
1114:
1105:
1102:
1098:
1093:
1084:
1080:
1056:
1050:
1030:
1024:
1002:
998:
977:
970:is an end of
955:
945:
941:
934:
931:
911:
905:
902:
899:
891:
887:
883:
879:
874:
872:
871:inverse limit
856:
848:
829:
821:
817:
807:
799:
795:
774:
768:
748:
725:
717:
713:
686:
680:
672:
668:
657:
653:
637:
614:
604:
603:direct system
588:
580:
564:
556:
540:
531:
529:
525:
521:
505:
483:
479:
475:
472:
452:
427:
423:
411:
406:
404:
401:
380:
376:
350:
346:
339:
331:
313:
309:
286:
283:
280:
275:
271:
267:
262:
258:
254:
249:
245:
236:
234:
218:
198:
191:
188:
172:
164:
146:
143:
138:
134:
130:
125:
121:
117:
112:
108:
99:
97:
81:
69:
67:
65:
61:
56:
54:
50:
46:
42:
38:
34:
30:
26:
19:
1916:
1900:
1861:
1855:
1825:
1824:, Series B,
1819:
1801:
1797:
1771:
1716:
1703:Cayley graph
1696:
1680:graph theory
1674:
1653:
1648:
1644:
1633:
1596:
1589:
1585:
1577:
1570:
1502:
1495:
1490:
1486:
1482:
1477:
1473:
1464:
1460:
1409:
1165:
885:
875:
846:
532:
523:
519:
410:neighborhood
409:
407:
300:
232:
160:
73:
57:
52:
32:
22:
1730:proper maps
847:set of ends
579:hemicompact
301:where each
29:mathematics
1946:Categories
1894:0002.05603
1808:References
1722:CW-complex
1658:Cantor set
557:(that is,
465:such that
412:of an end
70:Definition
1886:120965216
1878:0025-5874
1766:, called
1751:→
1599:distinct
1554:∖
1436:real line
1429:empty set
1361:−
1357:φ
1336:φ
1305:∖
1290:π
1247:−
1243:φ
1239:∖
1224:π
1203:φ
1181:∗
1177:φ
1103:−
1099:φ
1085:∗
1081:φ
1051:φ
1028:∖
909:→
900:φ
880:from the
818:π
814:→
796:π
772:→
714:π
684:∖
669:π
498:for some
476:⊃
403:bijection
343:∖
284:⋯
281:⊇
268:⊇
255:⊇
187:interiors
147:⋯
144:⊆
131:⊆
118:⊆
1677:infinite
1418:Examples
1312:′
1262:′
1151:′
1118:′
845:. Then
705:, where
577:must be
231:has one
211:. Then
45:infinity
25:topology
1844:1967888
1468:be the
1427:is the
878:functor
400:natural
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1798:an end
1717:For a
1688:degree
1139:where
886:proper
185:whose
31:, the
1882:S2CID
1684:haven
1636:ends.
1215:from
328:is a
190:cover
94:be a
35:of a
1929:ISBN
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1768:rays
1639:The
1632:has
1601:rays
1434:The
1168:and
650:and
74:Let
64:1931
33:ends
1921:doi
1890:Zbl
1866:doi
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1800:of
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