360:
650:
734:
839:
1026:
685:
543:
146:
879:
585:
270:
787:
176:
998:
210:
972:
941:
914:
503:
472:
441:
414:
387:
237:
754:
1128:
1594:
1562:
1599:
1526:
1500:
596:
97:
1095:
291:
1545:
1231:
1091:
602:
1045:
692:
1148:, countable compactness, sequential compactness, limit point compactness and compactness are all equivalent.
653:
78:
1540:
796:
1212:
1156:
1134:
1137:, countable compactness and sequential compactness are equivalent. More generally, the same holds for
1011:
1130:
with the product topology is compact and hence countably compact; but it is not sequentially compact.
599:
of the sequence. Otherwise, every value in the sequence occurs only finitely many times and the set
1241:
1077:
670:
528:
131:
1577:
1550:
844:
550:
242:
1160:
759:
1558:
1536:
1522:
1496:
1488:
1246:
1191:
1187:
1164:
1152:
1070:
32:
161:
1333:
1145:
1138:
977:
189:
950:
919:
892:
481:
450:
419:
392:
365:
215:
1554:
1201:
1198:
1183:
In a countably compact space, every locally finite family of nonempty subsets is finite.
1004:
is not an accumulation point of the sequence after all. This contradiction proves (1).
1049:
739:
1101:
1588:
1518:
1510:
1236:
1205:
1177:
1063:
1462:"Is the Product of a Compact Space and a Countably Compact Space Countably Compact?"
1218:
The product of a compact space and a countably compact space is countably compact.
17:
1275:
1094:
is countably compact. The converse does not hold. For example, the product of
28:
1030:
Conditions (1) and (4) are easily seen to be equivalent by taking complements.
1276:"General topology - Does sequential compactness imply countable compactness?"
111:
with an empty intersection has a finite subfamily with an empty intersection.
1461:
1431:
1384:
1221:
The product of two countably compact spaces need not be countably compact.
1084:
1337:
1173:
The continuous image of a countably compact space is countably compact.
663:
is then an accumulation point of the sequence, as is easily checked.
1170:
Closed subspaces of a countably compact space are countably compact.
736:
is a countable open cover without a finite subcover. Then for each
1087:, countable compactness and limit point compactness are equivalent.
1052:) is an example of a countably compact space that is not compact.
55:
if it satisfies any of the following equivalent conditions:
1069:
A countably compact space is compact if and only if it is
1190:
is compact. More generally, every countably compact
1104:
1014:
980:
953:
922:
895:
847:
799:
762:
742:
695:
673:
605:
553:
531:
484:
453:
422:
395:
368:
294:
245:
218:
192:
164:
134:
39:
if every countable open cover has a finite subcover.
1151:
The example of the set of all real numbers with the
595:
that occurs infinitely many times, that value is an
1122:
1020:
992:
966:
935:
908:
873:
833:
781:
748:
728:
679:
644:
579:
537:
497:
466:
435:
408:
381:
355:{\displaystyle O_{F}=\cup \{O_{x}:O_{x}\cap A=F\}}
354:
264:
231:
204:
170:
140:
1432:"Prove that a countably compact, first countable
280:an ω-accumulation point. For every finite subset
1385:"Countably compact paracompact space is compact"
107:(4) Every countable family of closed subsets of
447:. Since there are countably many of them, the
1553:reprint of 1978 ed.). Berlin, New York:
1379:
1377:
645:{\displaystyle A=\{x_{n}:n\in \mathbb {N} \}}
8:
1288:
723:
696:
639:
612:
349:
311:
178:-accumulation point. By taking a subset of
729:{\displaystyle \{O_{n}:n\in \mathbb {N} \}}
1332:(Thesis), University of British Columbia,
115:
1580:reprint of 1970 ed.), Addison-Wesley
1409:
1397:
1368:
1103:
1013:
979:
958:
952:
927:
921:
900:
894:
865:
855:
846:
825:
815:
804:
798:
767:
761:
741:
719:
718:
703:
694:
672:
635:
634:
619:
604:
571:
561:
552:
530:
513:), so finitely many of them cannot cover
489:
483:
458:
452:
427:
421:
400:
394:
373:
367:
331:
318:
299:
293:
250:
244:
223:
217:
191:
163:
133:
1309:Steen & Seebach, Example 105, p, 125
1211:Every normal countably compact space is
1258:
118:
7:
1421:Steen & Seebach, Figure 7, p. 25
834:{\displaystyle \cup _{i=1}^{n}O_{i}}
521:. This contradiction proves (2).
272:is finite (possibly empty), since
59:(1) Every countable open cover of
25:
1176:Every countably compact space is
1076:Every countably compact space is
947:that does not contain any of the
182:if necessary, we can assume that
1595:Properties of topological spaces
1328:Kremsater, Terry Philip (1972),
1021:{\displaystyle \Leftrightarrow }
591:. If the sequence has a value
474:form a countable open cover of
1117:
1105:
1015:
862:
848:
674:
568:
554:
532:
135:
1:
1495:. Heldermann Verlag, Berlin.
1451:Willard, problem 17F, p. 125
1359:Willard, problem 17F, p. 125
1350:Willard, problem 17F, p. 125
1318:Willard, problem 17G, p. 125
1167:imply countable compactness.
680:{\displaystyle \Rightarrow }
538:{\displaystyle \Rightarrow }
141:{\displaystyle \Rightarrow }
1546:Counterexamples in Topology
874:{\displaystyle (x_{n})_{n}}
580:{\displaystyle (x_{n})_{n}}
547:Suppose (2) holds, and let
509:in a finite subset (namely
265:{\displaystyle O_{x}\cap A}
1616:
1572:Willard, Stephen (2004) ,
1474:Engelking, example 3.10.19
1300:Steen & Seebach, p. 20
1265:Steen & Seebach, p. 19
1232:Sequentially compact space
1092:sequentially compact space
881:has an accumulation point
782:{\displaystyle x_{n}\in X}
652:is infinite and so has an
389:is a subset of one of the
212:has an open neighbourhood
1600:Compactness (mathematics)
1046:first uncountable ordinal
154:is an infinite subset of
1330:Sequential space methods
1289:Steen & Seebach 1995
1197:Every countably compact
1186:Every countably compact
1098:-many closed intervals
171:{\displaystyle \omega }
1541:Seebach, J. Arthur Jr.
1135:first-countable spaces
1124:
1022:
994:
993:{\displaystyle n>k}
968:
937:
910:
875:
835:
783:
756:we can choose a point
750:
730:
689:Suppose (3) holds and
681:
646:
581:
539:
499:
468:
437:
410:
383:
356:
266:
233:
206:
205:{\displaystyle x\in X}
172:
150:Suppose (1) holds and
142:
63:has a finite subcover.
43:Equivalent definitions
1371:, Theorem 3.10.3(ii).
1213:collectionwise normal
1125:
1066:is countably compact.
1023:
995:
969:
967:{\displaystyle x_{n}}
943:is a neighborhood of
938:
936:{\displaystyle O_{k}}
911:
909:{\displaystyle O_{k}}
876:
836:
784:
751:
731:
682:
647:
582:
540:
500:
498:{\displaystyle O_{F}}
469:
467:{\displaystyle O_{F}}
438:
436:{\displaystyle O_{F}}
411:
409:{\displaystyle O_{F}}
384:
382:{\displaystyle O_{x}}
357:
267:
234:
232:{\displaystyle O_{x}}
207:
173:
143:
1291:, example 42, p. 68.
1102:
1012:
978:
951:
920:
893:
845:
797:
760:
740:
693:
671:
654:ω-accumulation point
603:
551:
529:
482:
451:
420:
393:
366:
292:
243:
216:
190:
186:is countable. Every
162:
132:
120:Proof of equivalence
79:ω-accumulation point
47:A topological space
1242:Limit point compact
1155:shows that neither
1078:limit point compact
820:
66:(2) Every infinite
1537:Steen, Lynn Arthur
1489:Engelking, Ryszard
1338:10.14288/1.0080490
1120:
1018:
990:
964:
933:
906:
871:
831:
800:
779:
746:
726:
677:
642:
597:accumulation point
577:
535:
495:
464:
433:
406:
379:
352:
262:
229:
202:
168:
138:
98:accumulation point
1564:978-0-486-68735-3
1439:space is regular"
1400:, Theorem 5.1.20.
1192:metacompact space
1188:paracompact space
1157:local compactness
1153:standard topology
1146:metrizable spaces
1139:sequential spaces
1035:
1034:
749:{\displaystyle n}
587:be a sequence in
53:countably compact
37:countably compact
33:topological space
18:Countably compact
16:(Redirected from
1607:
1581:
1574:General Topology
1568:
1532:
1517:(2nd ed.).
1506:
1493:General Topology
1475:
1472:
1466:
1465:
1458:
1452:
1449:
1443:
1442:
1428:
1422:
1419:
1413:
1412:, Theorem 5.3.2.
1407:
1401:
1395:
1389:
1388:
1381:
1372:
1366:
1360:
1357:
1351:
1348:
1342:
1340:
1325:
1319:
1316:
1310:
1307:
1301:
1298:
1292:
1286:
1280:
1279:
1272:
1266:
1263:
1129:
1127:
1126:
1123:{\displaystyle }
1121:
1027:
1025:
1024:
1019:
999:
997:
996:
991:
973:
971:
970:
965:
963:
962:
942:
940:
939:
934:
932:
931:
915:
913:
912:
907:
905:
904:
880:
878:
877:
872:
870:
869:
860:
859:
841:. The sequence
840:
838:
837:
832:
830:
829:
819:
814:
788:
786:
785:
780:
772:
771:
755:
753:
752:
747:
735:
733:
732:
727:
722:
708:
707:
686:
684:
683:
678:
651:
649:
648:
643:
638:
624:
623:
586:
584:
583:
578:
576:
575:
566:
565:
544:
542:
541:
536:
504:
502:
501:
496:
494:
493:
473:
471:
470:
465:
463:
462:
442:
440:
439:
434:
432:
431:
415:
413:
412:
407:
405:
404:
388:
386:
385:
380:
378:
377:
361:
359:
358:
353:
336:
335:
323:
322:
304:
303:
271:
269:
268:
263:
255:
254:
238:
236:
235:
230:
228:
227:
211:
209:
208:
203:
177:
175:
174:
169:
147:
145:
144:
139:
116:
21:
1615:
1614:
1610:
1609:
1608:
1606:
1605:
1604:
1585:
1584:
1571:
1565:
1555:Springer-Verlag
1535:
1529:
1509:
1503:
1487:
1484:
1479:
1478:
1473:
1469:
1460:
1459:
1455:
1450:
1446:
1438:
1430:
1429:
1425:
1420:
1416:
1408:
1404:
1396:
1392:
1383:
1382:
1375:
1367:
1363:
1358:
1354:
1349:
1345:
1327:
1326:
1322:
1317:
1313:
1308:
1304:
1299:
1295:
1287:
1283:
1274:
1273:
1269:
1264:
1260:
1255:
1228:
1202:first-countable
1165:paracompactness
1100:
1099:
1059:
1041:
1036:
1010:
1009:
976:
975:
954:
949:
948:
923:
918:
917:
896:
891:
890:
861:
851:
843:
842:
821:
795:
794:
763:
758:
757:
738:
737:
699:
691:
690:
669:
668:
615:
601:
600:
567:
557:
549:
548:
527:
526:
485:
480:
479:
454:
449:
448:
423:
418:
417:
396:
391:
390:
369:
364:
363:
327:
314:
295:
290:
289:
246:
241:
240:
219:
214:
213:
188:
187:
160:
159:
130:
129:
121:
45:
23:
22:
15:
12:
11:
5:
1613:
1611:
1603:
1602:
1597:
1587:
1586:
1583:
1582:
1569:
1563:
1533:
1527:
1507:
1501:
1483:
1480:
1477:
1476:
1467:
1453:
1444:
1436:
1423:
1414:
1410:Engelking 1989
1402:
1398:Engelking 1989
1390:
1373:
1369:Engelking 1989
1361:
1352:
1343:
1341:, Theorem 1.20
1320:
1311:
1302:
1293:
1281:
1267:
1257:
1256:
1254:
1251:
1250:
1249:
1247:Lindelöf space
1244:
1239:
1234:
1227:
1224:
1223:
1222:
1219:
1216:
1209:
1195:
1184:
1181:
1174:
1171:
1168:
1149:
1142:
1131:
1119:
1116:
1113:
1110:
1107:
1088:
1081:
1074:
1067:
1058:
1055:
1054:
1053:
1050:order topology
1040:
1037:
1033:
1032:
1017:
989:
986:
983:
961:
957:
930:
926:
903:
899:
868:
864:
858:
854:
850:
828:
824:
818:
813:
810:
807:
803:
778:
775:
770:
766:
745:
725:
721:
717:
714:
711:
706:
702:
698:
676:
641:
637:
633:
630:
627:
622:
618:
614:
611:
608:
574:
570:
564:
560:
556:
534:
492:
488:
461:
457:
430:
426:
403:
399:
376:
372:
351:
348:
345:
342:
339:
334:
330:
326:
321:
317:
313:
310:
307:
302:
298:
261:
258:
253:
249:
226:
222:
201:
198:
195:
167:
137:
123:
122:
119:
114:
113:
112:
105:
86:
64:
44:
41:
24:
14:
13:
10:
9:
6:
4:
3:
2:
1612:
1601:
1598:
1596:
1593:
1592:
1590:
1579:
1575:
1570:
1566:
1560:
1556:
1552:
1548:
1547:
1542:
1538:
1534:
1530:
1528:0-13-181629-2
1524:
1520:
1519:Prentice Hall
1516:
1512:
1511:James Munkres
1508:
1504:
1502:3-88538-006-4
1498:
1494:
1490:
1486:
1485:
1481:
1471:
1468:
1463:
1457:
1454:
1448:
1445:
1440:
1435:
1427:
1424:
1418:
1415:
1411:
1406:
1403:
1399:
1394:
1391:
1386:
1380:
1378:
1374:
1370:
1365:
1362:
1356:
1353:
1347:
1344:
1339:
1335:
1331:
1324:
1321:
1315:
1312:
1306:
1303:
1297:
1294:
1290:
1285:
1282:
1277:
1271:
1268:
1262:
1259:
1252:
1248:
1245:
1243:
1240:
1238:
1237:Compact space
1235:
1233:
1230:
1229:
1225:
1220:
1217:
1214:
1210:
1207:
1203:
1200:
1196:
1193:
1189:
1185:
1182:
1179:
1178:pseudocompact
1175:
1172:
1169:
1166:
1162:
1161:σ-compactness
1158:
1154:
1150:
1147:
1143:
1140:
1136:
1132:
1114:
1111:
1108:
1097:
1093:
1089:
1086:
1082:
1079:
1075:
1072:
1068:
1065:
1064:compact space
1061:
1060:
1056:
1051:
1047:
1043:
1042:
1038:
1031:
1029:
1005:
1003:
987:
984:
981:
959:
955:
946:
928:
924:
901:
897:
888:
884:
866:
856:
852:
826:
822:
816:
811:
808:
805:
801:
792:
776:
773:
768:
764:
743:
715:
712:
709:
704:
700:
688:
664:
662:
658:
655:
631:
628:
625:
620:
616:
609:
606:
598:
594:
590:
572:
562:
558:
546:
522:
520:
516:
512:
508:
490:
486:
478:. But every
477:
459:
455:
446:
428:
424:
401:
397:
374:
370:
346:
343:
340:
337:
332:
328:
324:
319:
315:
308:
305:
300:
296:
287:
283:
279:
275:
259:
256:
251:
247:
224:
220:
199:
196:
193:
185:
181:
165:
157:
153:
149:
125:
124:
117:
110:
106:
103:
99:
95:
91:
87:
84:
80:
76:
72:
69:
65:
62:
58:
57:
56:
54:
50:
42:
40:
38:
34:
30:
19:
1573:
1544:
1514:
1492:
1470:
1456:
1447:
1433:
1426:
1417:
1405:
1393:
1364:
1355:
1346:
1329:
1323:
1314:
1305:
1296:
1284:
1270:
1261:
1007:
1006:
1001:
944:
916:. But then
886:
882:
790:
666:
665:
660:
656:
592:
588:
524:
523:
518:
517:, let alone
514:
510:
506:
475:
444:
285:
281:
277:
273:
183:
179:
155:
151:
127:
126:
108:
101:
93:
89:
82:
74:
70:
67:
60:
52:
48:
46:
36:
26:
1194:is compact.
889:is in some
29:mathematics
1589:Categories
1482:References
1057:Properties
1048:(with the
505:intersect
239:such that
88:(3) Every
51:is called
35:is called
1543:(1995) .
1204:space is
1199:Hausdorff
1096:continuum
1085:T1 spaces
1016:⇔
885:and that
802:∪
774:∈
716:∈
675:⇒
632:∈
533:⇒
416:, so the
338:∩
309:∪
257:∩
197:∈
166:ω
136:⇒
1515:Topology
1513:(1999).
1491:(1989).
1226:See also
1071:Lindelöf
1039:Examples
789:that is
659:. That
362:. Every
158:without
90:sequence
1206:regular
288:define
96:has an
77:has an
1561:
1525:
1499:
1090:Every
1062:Every
443:cover
1578:Dover
1551:Dover
1253:Notes
1000:, so
974:with
1559:ISBN
1523:ISBN
1497:ISBN
1163:nor
1159:nor
1144:For
1133:For
1083:For
1044:The
1028:(1):
1008:(4)
985:>
687:(1):
667:(3)
545:(3):
525:(2)
148:(2):
128:(1)
1334:doi
793:in
791:not
284:of
278:not
276:is
100:in
92:in
81:in
73:in
68:set
27:In
1591::
1557:.
1539:;
1521:.
1376:^
31:a
1576:(
1567:.
1549:(
1531:.
1505:.
1464:.
1441:.
1437:2
1434:T
1387:.
1336::
1278:.
1215:.
1208:.
1180:.
1141:.
1118:]
1115:1
1112:,
1109:0
1106:[
1080:.
1073:.
1002:x
988:k
982:n
960:n
956:x
945:x
929:k
925:O
902:k
898:O
887:x
883:x
867:n
863:)
857:n
853:x
849:(
827:i
823:O
817:n
812:1
809:=
806:i
777:X
769:n
765:x
744:n
724:}
720:N
713:n
710::
705:n
701:O
697:{
661:x
657:x
640:}
636:N
629:n
626::
621:n
617:x
613:{
610:=
607:A
593:x
589:X
573:n
569:)
563:n
559:x
555:(
519:X
515:A
511:F
507:A
491:F
487:O
476:X
460:F
456:O
445:X
429:F
425:O
402:F
398:O
375:x
371:O
350:}
347:F
344:=
341:A
333:x
329:O
325::
320:x
316:O
312:{
306:=
301:F
297:O
286:A
282:F
274:x
260:A
252:x
248:O
225:x
221:O
200:X
194:x
184:A
180:A
156:X
152:A
109:X
104:.
102:X
94:X
85:.
83:X
75:X
71:A
61:X
49:X
20:)
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