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