368:
containing a point intersects the closure of a set, it necessarily intersects the set itself, hence a neighborhood can intersect at most the same number of closures (it may intersect fewer, since two distinct, indeed disjoint, sets can have the same closure). The converse, however, can fail if the
267:, meaning that every point of the space belongs to only finitely many sets in the collection. Point finiteness is a strictly weaker notion, as illustrated by the collection of intervals
990:
755:
902:
650:
544:
415:
393:
327:
216:
129:
1078:
1024:
305:
167:
805:
251:
835:
618:
571:
1258:
1238:
1218:
1198:
1178:
1158:
1138:
1118:
1098:
1044:
926:
855:
779:
690:
670:
591:
510:
486:
466:
446:
347:
190:
41:
109:
collection of subsets of a topological space is locally finite. Infinite collections can also be locally finite: for example, the collection of subsets of
395:
the collection of all open sets is not locally finite, but the collection of all closures of these sets is locally finite (since the only closures are
1574:
428:
is not closed in general. However, the union of a locally finite collection of closed sets is closed. To see this we note that if
1304:
1537:
1509:
1490:
935:
94:
448:
is a point outside the union of this locally finite collection of closed sets, we merely choose a neighbourhood
48:
707:
370:
860:
1569:
1564:
623:
515:
398:
376:
310:
199:
112:
1276:, is countable. This is proved by a similar argument as in the result above for compact spaces.
1331:
1273:
1049:
995:
196:
collection of subsets need not be locally finite, as shown by the collection of all subsets of
270:
134:
87:
784:
221:
813:
596:
549:
8:
1529:
1243:
1223:
1203:
1183:
1163:
1143:
1123:
1103:
1083:
1029:
911:
840:
808:
764:
675:
655:
576:
495:
471:
451:
431:
332:
193:
175:
26:
1543:
1533:
1505:
1486:
1478:
354:
350:
79:
1327:
1269:
546:
thus giving an index to each of these sets. Then for each set, choose an open set
1519:
1320:
1316:
83:
758:
254:
71:
488:
that intersects this collection at only finitely many of these sets. Define a
1558:
1525:
1312:
701:
264:
1547:
361:
59:
1308:
905:
425:
106:
52:
692:
that does not intersect the union of this collection of closed sets.
489:
418:
357:, and this is the reason why every paracompact space is metacompact.
1383:
1455:
929:
365:
63:
1371:
1443:
1407:
360:
If a collection of sets is locally finite, the collection of the
170:
1395:
1359:
75:
20:
1140:
intersects only a finite number of subsets in the collection
329:, which is point finite, but not locally finite at the point
369:
closures of the sets are not distinct. For example, in the
364:
is also locally finite. The reason for this is that if an
1334:, every Ï-locally finite collection of sets is countable.
1300:
if it is a countable union of locally finite collections.
1284:
A collection of subsets of a topological space is called
1349:
1347:
593:
that doesn't intersect it. The intersection of all such
1419:
1303:
The Ï-locally finite notion is a key ingredient in the
1279:
1431:
1344:
1246:
1226:
1206:
1186:
1166:
1146:
1126:
1106:
1086:
1052:
1032:
998:
938:
914:
863:
843:
816:
787:
767:
710:
678:
658:
626:
599:
579:
552:
518:
498:
474:
454:
434:
401:
379:
335:
313:
273:
224:
202:
178:
137:
115:
97:
has different meanings in other mathematical fields.
29:
349:. The two concepts are used in the definitions of
1252:
1232:
1212:
1192:
1172:
1152:
1132:
1112:
1092:
1072:
1038:
1018:
984:
920:
896:
849:
837:that intersects a finite number of the subsets in
829:
799:
773:
749:
684:
664:
644:
612:
585:
565:
538:
504:
480:
460:
440:
409:
387:
341:
321:
299:
245:
210:
184:
161:
123:
55:only finitely many of the sets in the collection.
35:
1556:
1080:intersects only a finite number of subsets in
1026:intersects only a finite number of subsets in
1268:Every locally finite collection of sets in a
700:Every locally finite collection of sets in a
979:
939:
891:
864:
744:
717:
1307:, which states that a topological space is
985:{\displaystyle \{U_{k_{n}}|n\in 1\dots n\}}
263:Every locally finite collection of sets is
100:
1477:
1449:
1425:
1401:
1389:
1377:
403:
381:
315:
204:
117:
1517:
1499:
1461:
1437:
1413:
1365:
1353:
1557:
1100:. Since this union is the whole space
1263:
492:map from the collection of sets that
82:. It is fundamental in the study of
1280:Countably locally finite collections
695:
13:
1305:NagataâSmirnov metrization theorem
750:{\displaystyle G=\{G_{a}|a\in A\}}
14:
1586:
47:if each point in the space has a
1575:Properties of topological spaces
897:{\displaystyle \{U_{x}|x\in X\}}
1504:(2nd ed.), Prentice Hall,
960:
878:
857:. Clearly the family of sets:
761:of subsets of a compact space
731:
294:
274:
240:
225:
156:
138:
1:
1485:. Berlin: Heldermann Verlag.
1471:
1319:, and has a Ï-locally finite
928:, and therefore has a finite
645:{\displaystyle 1\leq i\leq k}
1337:
539:{\displaystyle {1,\dots ,k}}
410:{\displaystyle \mathbb {R} }
388:{\displaystyle \mathbb {R} }
322:{\displaystyle \mathbb {R} }
211:{\displaystyle \mathbb {R} }
124:{\displaystyle \mathbb {R} }
7:
1464:, p. 250 Theorem 40.3.
10:
1591:
1518:Willard, Stephen (2004) .
1500:Munkres, James R. (2000),
1180:is composed of subsets of
371:finite complement topology
1368:, p. 245 Lemma 39.1.
1073:{\displaystyle U_{k_{i}}}
1019:{\displaystyle U_{k_{i}}}
1296:countably locally finite
1046:, the union of all such
672:, is a neighbourhood of
704:is finite. Indeed, let
300:{\displaystyle (0,1/n)}
162:{\displaystyle (n,n+2)}
101:Examples and properties
23:of a topological space
1332:second-countable space
1274:second-countable space
1254:
1234:
1214:
1194:
1174:
1154:
1134:
1114:
1094:
1074:
1040:
1020:
986:
922:
898:
851:
831:
801:
800:{\displaystyle x\in X}
775:
751:
686:
666:
646:
614:
587:
567:
540:
506:
482:
462:
442:
424:An arbitrary union of
411:
389:
362:closures of these sets
343:
323:
301:
247:
246:{\displaystyle (-n,n)}
212:
186:
163:
125:
37:
1330:, in particular in a
1311:if and only if it is
1272:, in particular in a
1255:
1235:
1215:
1195:
1175:
1155:
1135:
1115:
1095:
1075:
1041:
1021:
987:
923:
899:
852:
832:
830:{\displaystyle U_{x}}
802:
776:
752:
687:
667:
647:
615:
613:{\displaystyle U_{i}}
588:
568:
566:{\displaystyle U_{i}}
541:
507:
483:
463:
443:
412:
390:
344:
324:
302:
248:
213:
187:
164:
126:
88:topological dimension
38:
1244:
1224:
1204:
1184:
1164:
1144:
1124:
1104:
1084:
1050:
1030:
996:
936:
912:
861:
841:
814:
785:
765:
757:be a locally finite
708:
676:
656:
624:
597:
577:
550:
516:
496:
472:
452:
432:
399:
377:
333:
311:
271:
222:
200:
176:
135:
113:
27:
1392:, Corollary 1.1.12.
93:Note that the term
16:Topological concept
1530:Dover Publications
1479:Engelking, Ryszard
1416:, Definition 20.2.
1264:In Lindelöf spaces
1250:
1230:
1210:
1190:
1170:
1150:
1130:
1120:, it follows that
1110:
1090:
1070:
1036:
1016:
982:
918:
894:
847:
827:
809:open neighbourhood
797:
771:
747:
682:
662:
642:
610:
583:
563:
536:
502:
478:
458:
438:
407:
385:
339:
319:
297:
243:
208:
182:
159:
121:
33:
1539:978-0-486-43479-7
1380:, Theorem 1.1.13.
1253:{\displaystyle G}
1233:{\displaystyle X}
1213:{\displaystyle G}
1193:{\displaystyle X}
1173:{\displaystyle G}
1153:{\displaystyle G}
1133:{\displaystyle X}
1113:{\displaystyle X}
1093:{\displaystyle G}
1039:{\displaystyle G}
921:{\displaystyle X}
850:{\displaystyle G}
781:. For each point
774:{\displaystyle X}
696:In compact spaces
685:{\displaystyle x}
665:{\displaystyle V}
652:intersected with
586:{\displaystyle x}
505:{\displaystyle V}
481:{\displaystyle x}
461:{\displaystyle V}
441:{\displaystyle x}
355:metacompact space
351:paracompact space
342:{\displaystyle 0}
185:{\displaystyle n}
80:topological space
70:is a property of
36:{\displaystyle X}
1582:
1570:General topology
1565:Families of sets
1551:
1521:General Topology
1514:
1496:
1483:General topology
1465:
1459:
1453:
1452:, Theorem 4.4.7.
1447:
1441:
1435:
1429:
1423:
1417:
1411:
1405:
1399:
1393:
1387:
1381:
1375:
1369:
1363:
1357:
1351:
1298:
1297:
1290:
1289:
1288:Ï-locally finite
1259:
1257:
1256:
1251:
1239:
1237:
1236:
1231:
1219:
1217:
1216:
1211:
1200:every member of
1199:
1197:
1196:
1191:
1179:
1177:
1176:
1171:
1159:
1157:
1156:
1151:
1139:
1137:
1136:
1131:
1119:
1117:
1116:
1111:
1099:
1097:
1096:
1091:
1079:
1077:
1076:
1071:
1069:
1068:
1067:
1066:
1045:
1043:
1042:
1037:
1025:
1023:
1022:
1017:
1015:
1014:
1013:
1012:
991:
989:
988:
983:
963:
958:
957:
956:
955:
927:
925:
924:
919:
903:
901:
900:
895:
881:
876:
875:
856:
854:
853:
848:
836:
834:
833:
828:
826:
825:
806:
804:
803:
798:
780:
778:
777:
772:
756:
754:
753:
748:
734:
729:
728:
691:
689:
688:
683:
671:
669:
668:
663:
651:
649:
648:
643:
619:
617:
616:
611:
609:
608:
592:
590:
589:
584:
572:
570:
569:
564:
562:
561:
545:
543:
542:
537:
535:
511:
509:
508:
503:
487:
485:
484:
479:
467:
465:
464:
459:
447:
445:
444:
439:
416:
414:
413:
408:
406:
394:
392:
391:
386:
384:
348:
346:
345:
340:
328:
326:
325:
320:
318:
306:
304:
303:
298:
290:
252:
250:
249:
244:
217:
215:
214:
209:
207:
191:
189:
188:
183:
168:
166:
165:
160:
130:
128:
127:
122:
120:
68:local finiteness
42:
40:
39:
34:
19:A collection of
1590:
1589:
1585:
1584:
1583:
1581:
1580:
1579:
1555:
1554:
1540:
1512:
1493:
1474:
1469:
1468:
1460:
1456:
1448:
1444:
1436:
1432:
1424:
1420:
1412:
1408:
1404:, Lemma 5.1.24.
1400:
1396:
1388:
1384:
1376:
1372:
1364:
1360:
1352:
1345:
1340:
1295:
1294:
1287:
1286:
1282:
1266:
1245:
1242:
1241:
1225:
1222:
1221:
1220:must intersect
1205:
1202:
1201:
1185:
1182:
1181:
1165:
1162:
1161:
1145:
1142:
1141:
1125:
1122:
1121:
1105:
1102:
1101:
1085:
1082:
1081:
1062:
1058:
1057:
1053:
1051:
1048:
1047:
1031:
1028:
1027:
1008:
1004:
1003:
999:
997:
994:
993:
959:
951:
947:
946:
942:
937:
934:
933:
913:
910:
909:
877:
871:
867:
862:
859:
858:
842:
839:
838:
821:
817:
815:
812:
811:
786:
783:
782:
766:
763:
762:
730:
724:
720:
709:
706:
705:
698:
677:
674:
673:
657:
654:
653:
625:
622:
621:
604:
600:
598:
595:
594:
578:
575:
574:
557:
553:
551:
548:
547:
519:
517:
514:
513:
497:
494:
493:
473:
470:
469:
453:
450:
449:
433:
430:
429:
402:
400:
397:
396:
380:
378:
375:
374:
334:
331:
330:
314:
312:
309:
308:
286:
272:
269:
268:
223:
220:
219:
203:
201:
198:
197:
177:
174:
173:
136:
133:
132:
116:
114:
111:
110:
103:
84:paracompactness
28:
25:
24:
17:
12:
11:
5:
1588:
1578:
1577:
1572:
1567:
1553:
1552:
1538:
1515:
1510:
1497:
1491:
1473:
1470:
1467:
1466:
1454:
1450:Engelking 1989
1442:
1440:, p. 245.
1430:
1428:, p. 280.
1426:Engelking 1989
1418:
1406:
1402:Engelking 1989
1394:
1390:Engelking 1989
1382:
1378:Engelking 1989
1370:
1358:
1356:, p. 244.
1342:
1341:
1339:
1336:
1328:Lindelöf space
1281:
1278:
1270:Lindelöf space
1265:
1262:
1249:
1229:
1209:
1189:
1169:
1149:
1129:
1109:
1089:
1065:
1061:
1056:
1035:
1011:
1007:
1002:
981:
978:
975:
972:
969:
966:
962:
954:
950:
945:
941:
917:
893:
890:
887:
884:
880:
874:
870:
866:
846:
824:
820:
796:
793:
790:
770:
746:
743:
740:
737:
733:
727:
723:
719:
716:
713:
697:
694:
681:
661:
641:
638:
635:
632:
629:
607:
603:
582:
560:
556:
534:
531:
528:
525:
522:
512:intersects to
501:
477:
457:
437:
405:
383:
338:
317:
296:
293:
289:
285:
282:
279:
276:
255:natural number
242:
239:
236:
233:
230:
227:
206:
181:
158:
155:
152:
149:
146:
143:
140:
119:
102:
99:
95:locally finite
45:locally finite
43:is said to be
32:
15:
9:
6:
4:
3:
2:
1587:
1576:
1573:
1571:
1568:
1566:
1563:
1562:
1560:
1549:
1545:
1541:
1535:
1531:
1527:
1526:Mineola, N.Y.
1523:
1522:
1516:
1513:
1511:0-13-181629-2
1507:
1503:
1498:
1494:
1492:3-88538-006-4
1488:
1484:
1480:
1476:
1475:
1463:
1458:
1451:
1446:
1439:
1434:
1427:
1422:
1415:
1410:
1403:
1398:
1391:
1386:
1379:
1374:
1367:
1362:
1355:
1350:
1348:
1343:
1335:
1333:
1329:
1324:
1322:
1318:
1314:
1310:
1306:
1301:
1299:
1291:
1277:
1275:
1271:
1261:
1247:
1227:
1207:
1187:
1167:
1147:
1127:
1107:
1087:
1063:
1059:
1054:
1033:
1009:
1005:
1000:
992:. Since each
976:
973:
970:
967:
964:
952:
948:
943:
931:
915:
907:
888:
885:
882:
872:
868:
844:
822:
818:
810:
794:
791:
788:
768:
760:
741:
738:
735:
725:
721:
714:
711:
703:
702:compact space
693:
679:
659:
639:
636:
633:
630:
627:
605:
601:
580:
558:
554:
532:
529:
526:
523:
520:
499:
491:
475:
455:
435:
427:
422:
420:
372:
367:
363:
358:
356:
352:
336:
291:
287:
283:
280:
277:
266:
261:
259:
256:
237:
234:
231:
228:
195:
179:
172:
153:
150:
147:
144:
141:
108:
98:
96:
91:
89:
85:
81:
77:
73:
69:
65:
61:
56:
54:
50:
49:neighbourhood
46:
30:
22:
1520:
1501:
1482:
1462:Munkres 2000
1457:
1445:
1438:Munkres 2000
1433:
1421:
1414:Willard 2004
1409:
1397:
1385:
1373:
1366:Munkres 2000
1361:
1354:Munkres 2000
1325:
1302:
1293:
1285:
1283:
1267:
1160:. And since
807:, choose an
699:
423:
359:
265:point finite
262:
257:
218:of the form
131:of the form
104:
92:
67:
60:mathematical
57:
44:
18:
1260:is finite.
573:containing
426:closed sets
72:collections
1559:Categories
1472:References
1309:metrizable
906:open cover
53:intersects
1338:Citations
1317:Hausdorff
974:…
968:∈
886:∈
792:∈
739:∈
637:≤
631:≤
527:…
490:bijective
419:empty set
229:−
194:countable
62:field of
1502:Topology
1481:(1989).
930:subcover
417:and the
366:open set
64:topology
1313:regular
1240:, thus
171:integer
169:for an
76:subsets
58:In the
21:subsets
1548:115240
1546:
1536:
1508:
1489:
904:is an
759:family
253:for a
107:finite
1326:In a
78:of a
51:that
1544:OCLC
1534:ISBN
1506:ISBN
1487:ISBN
1321:base
620:for
353:and
192:. A
86:and
1292:or
908:of
468:of
421:).
373:on
307:in
74:of
1561::
1542:.
1532:.
1528::
1524:.
1346:^
1323:.
1315:,
932::
260:.
105:A
90:.
66:,
1550:.
1495:.
1248:G
1228:X
1208:G
1188:X
1168:G
1148:G
1128:X
1108:X
1088:G
1064:i
1060:k
1055:U
1034:G
1010:i
1006:k
1001:U
980:}
977:n
971:1
965:n
961:|
953:n
949:k
944:U
940:{
916:X
892:}
889:X
883:x
879:|
873:x
869:U
865:{
845:G
823:x
819:U
795:X
789:x
769:X
745:}
742:A
736:a
732:|
726:a
722:G
718:{
715:=
712:G
680:x
660:V
640:k
634:i
628:1
606:i
602:U
581:x
559:i
555:U
533:k
530:,
524:,
521:1
500:V
476:x
456:V
436:x
404:R
382:R
337:0
316:R
295:)
292:n
288:/
284:1
281:,
278:0
275:(
258:n
241:)
238:n
235:,
232:n
226:(
205:R
180:n
157:)
154:2
151:+
148:n
145:,
142:n
139:(
118:R
31:X
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