553:
478:
1322:
1192:
1282:
1355:
1235:
928:
1152:
1119:
1072:
1031:
990:
671:
954:
714:
609:
473:
426:
363:
270:
157:
829:
805:
781:
734:
629:
586:
450:
383:
340:
320:
292:
227:
134:
110:
86:
66:
1450:
1033:
or the whole set. The open sets are the complements of the closed sets; namely, each open set consists of all but a finite number of pairs
905:
as the product of two compact spaces; alternatively, it is compact because each nonempty open set contains all but finitely many points.
166:
These arise naturally when generalizing structures on finite sets to infinite sets, particularly on infinite products, as in the
1421:
452:
as open sets. As a consequence, in the cofinite topology, the only closed subsets are finite sets, or the whole of
957:
887:
548:{\displaystyle {\mathcal {T}}=\{A\subseteq X:A=\varnothing {\mbox{ or }}X\setminus A{\mbox{ is finite}}\}.}
1287:
1157:
1404:
230:
1455:
1257:
1327:
748:
17:
1197:
911:
1127:
1094:
184:
89:
1399:
1036:
995:
302:
not generated by a single element of the algebra) if and only if there exists an infinite set
1252:
637:
1431:
848:
741:
966:
647:
8:
869:
865:
566:
234:
936:
696:
591:
455:
408:
345:
252:
139:
1409:
1381:
902:
814:
790:
766:
719:
614:
571:
435:
368:
325:
305:
277:
212:
119:
95:
71:
51:
1417:
1395:
1239:
The analog without requiring that cofinitely many factors are the whole space is the
683:
402:
1372:
1122:
1088:
558:
365:
In this case, the non-principal ultrafilter is the set of all cofinite subsets of
1427:
1413:
832:
641:
1360:
898:
808:
299:
1444:
840:
760:
159:
If the complement is not finite, but is countable, then one says the set is
1359:
The analog without requiring that cofinitely many summands are zero is the
1240:
844:
238:
193:
908:
For an example of the countable double-pointed cofinite topology, the set
931:
737:
295:
160:
38:
961:
562:
113:
31:
864:
is the cofinite topology with every point doubled; that is, it is the
429:
891:
880:
873:
752:
690:
847:
because no two nonempty open sets are disjoint (that is, it is
45:
787:
axiom if and only if it contains the cofinite topology. If
1384: – List of concrete topologies and topological spaces
992:. The closed sets are the unions of finitely many pairs
233:, which means that it is closed under the operations of
759:; that is, it is the smallest topology for which every
533:
517:
1330:
1290:
1260:
1200:
1160:
1130:
1097:
1039:
998:
969:
939:
914:
817:
793:
769:
722:
699:
686:
of the cofinite topology is also a cofinite topology.
650:
635:) is the cofinite topology. The same is true for any
617:
594:
574:
557:
This topology occurs naturally in the context of the
481:
458:
438:
411:
371:
348:
328:
308:
280:
255:
215:
142:
122:
98:
74:
54:
1377:
Pages displaying wikidata descriptions as a fallback
930:
of integers can be given a topology such that every
855:
807:is finite then the cofinite topology is simply the
241:, and complementation. This Boolean algebra is the
192:
is consistent with its use in other terms such as "
1349:
1316:
1276:
1229:
1186:
1146:
1113:
1066:
1025:
984:
948:
922:
823:
799:
775:
728:
708:
665:
623:
603:
580:
547:
467:
444:
420:
377:
357:
334:
314:
286:
264:
221:
151:
128:
104:
80:
60:
1442:
342:is isomorphic to the finite–cofinite algebra on
897:since topologically distinguishable points are
27:Being a subset whose complement is a finite set
183:" to describe a property possessed by a set's
1412:reprint of 1978 ed.), Berlin, New York:
763:is closed. In fact, an arbitrary topology on
1394:
539:
492:
229:that are either finite or cofinite forms a
136:contains all but finitely many elements of
747:Separation: The cofinite topology is the
274:In the other direction, a Boolean algebra
916:
693:contains all but finitely many points of
588:are zero on finite sets, or the whole of
475:Symbolically, one writes the topology as
831:is not finite then this topology is not
886:, since the points of each doublet are
14:
1443:
1451:Basic concepts in infinite set theory
167:
1317:{\displaystyle \alpha _{i}\in M_{i}}
1187:{\displaystyle U_{i}\subseteq X_{i}}
388:
1091:on a product of topological spaces
1082:
644:; it is not true, for example, for
204:
24:
868:of the cofinite topology with the
484:
25:
1467:
1077:
872:on a two-element set. It is not
526:
513:
405:that can be defined on every set
862:double-pointed cofinite topology
856:Double-pointed cofinite topology
1277:{\displaystyle \bigoplus M_{i}}
958:topologically indistinguishable
888:topologically indistinguishable
171:
1350:{\displaystyle \alpha _{i}=0.}
13:
1:
1388:
1246:
1194:is open, and cofinitely many
676:
1230:{\displaystyle U_{i}=X_{i}.}
923:{\displaystyle \mathbb {Z} }
432:and all cofinite subsets of
7:
1405:Counterexamples in Topology
1366:
1147:{\displaystyle \prod U_{i}}
1114:{\displaystyle \prod X_{i}}
294:has a unique non-principal
10:
1472:
399:finite complement topology
209:The set of all subsets of
29:
689:Compactness: Since every
1067:{\displaystyle 2n,2n+1,}
1026:{\displaystyle 2n,2n+1,}
611:the Zariski topology on
177:This use of the prefix "
30:Not to be confused with
565:in one variable over a
245:finite–cofinite algebra
1400:Seebach, J. Arthur Jr.
1375: – frechet filter
1351:
1324:where cofinitely many
1318:
1278:
1231:
1188:
1148:
1115:
1068:
1027:
986:
950:
924:
825:
801:
777:
730:
710:
667:
625:
605:
582:
549:
469:
446:
422:
397:(sometimes called the
379:
359:
336:
316:
288:
266:
223:
153:
130:
106:
82:
62:
1352:
1319:
1279:
1253:direct sum of modules
1232:
1189:
1149:
1116:
1074:or is the empty set.
1069:
1028:
987:
951:
925:
826:
802:
778:
731:
711:
668:
626:
606:
583:
550:
470:
447:
428:It has precisely the
423:
380:
360:
337:
317:
289:
267:
224:
154:
131:
107:
83:
63:
1328:
1288:
1258:
1251:The elements of the
1198:
1158:
1128:
1095:
1037:
996:
985:{\displaystyle 2n+1}
967:
937:
912:
815:
791:
767:
742:sequentially compact
720:
697:
666:{\displaystyle XY=0}
648:
615:
592:
572:
479:
456:
436:
409:
369:
346:
326:
306:
278:
253:
213:
140:
120:
96:
72:
52:
960:from the following
890:. It is, however,
870:indiscrete topology
866:topological product
1396:Steen, Lynn Arthur
1382:List of topologies
1347:
1314:
1274:
1227:
1184:
1144:
1111:
1064:
1023:
982:
949:{\displaystyle 2n}
946:
920:
821:
797:
773:
726:
709:{\displaystyle X,}
706:
663:
621:
604:{\displaystyle K,}
601:
578:
545:
537:
521:
468:{\displaystyle X.}
465:
442:
421:{\displaystyle X.}
418:
375:
358:{\displaystyle X.}
355:
332:
312:
284:
265:{\displaystyle X.}
262:
219:
152:{\displaystyle X.}
149:
126:
116:. In other words,
102:
78:
58:
1423:978-0-486-68735-3
824:{\displaystyle X}
809:discrete topology
800:{\displaystyle X}
776:{\displaystyle X}
749:coarsest topology
729:{\displaystyle X}
684:subspace topology
682:Subspaces: Every
624:{\displaystyle K}
581:{\displaystyle K}
536:
520:
445:{\displaystyle X}
395:cofinite topology
389:Cofinite topology
378:{\displaystyle X}
335:{\displaystyle A}
315:{\displaystyle X}
287:{\displaystyle A}
222:{\displaystyle X}
129:{\displaystyle A}
105:{\displaystyle X}
81:{\displaystyle A}
61:{\displaystyle X}
16:(Redirected from
1463:
1456:General topology
1436:(See example 18)
1434:
1378:
1356:
1354:
1353:
1348:
1340:
1339:
1323:
1321:
1320:
1315:
1313:
1312:
1300:
1299:
1283:
1281:
1280:
1275:
1273:
1272:
1236:
1234:
1233:
1228:
1223:
1222:
1210:
1209:
1193:
1191:
1190:
1185:
1183:
1182:
1170:
1169:
1153:
1151:
1150:
1145:
1143:
1142:
1120:
1118:
1117:
1112:
1110:
1109:
1089:product topology
1083:Product topology
1073:
1071:
1070:
1065:
1032:
1030:
1029:
1024:
991:
989:
988:
983:
955:
953:
952:
947:
929:
927:
926:
921:
919:
901:. The space is
830:
828:
827:
822:
806:
804:
803:
798:
782:
780:
779:
774:
735:
733:
732:
727:
715:
713:
712:
707:
672:
670:
669:
664:
630:
628:
627:
622:
610:
608:
607:
602:
587:
585:
584:
579:
559:Zariski topology
554:
552:
551:
546:
538:
534:
522:
518:
488:
487:
474:
472:
471:
466:
451:
449:
448:
443:
427:
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384:
382:
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364:
362:
361:
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341:
339:
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333:
321:
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293:
291:
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285:
271:
269:
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263:
247:
246:
228:
226:
225:
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205:Boolean algebras
168:product topology
158:
156:
155:
150:
135:
133:
132:
127:
111:
109:
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103:
87:
85:
84:
79:
67:
65:
64:
59:
21:
1471:
1470:
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1414:Springer-Verlag
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1285:
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1259:
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1249:
1218:
1214:
1205:
1201:
1199:
1196:
1195:
1178:
1174:
1165:
1161:
1159:
1156:
1155:
1138:
1134:
1129:
1126:
1125:
1105:
1101:
1096:
1093:
1092:
1085:
1080:
1038:
1035:
1034:
997:
994:
993:
968:
965:
964:
938:
935:
934:
915:
913:
910:
909:
895:
884:
877:
858:
836:
816:
813:
812:
792:
789:
788:
786:
783:satisfies the T
768:
765:
764:
756:
751:satisfying the
721:
718:
717:
698:
695:
694:
679:
649:
646:
645:
642:algebraic curve
631:(considered as
616:
613:
612:
593:
590:
589:
573:
570:
569:
535: is finite
532:
516:
483:
482:
480:
477:
476:
457:
454:
453:
437:
434:
433:
410:
407:
406:
391:
370:
367:
366:
347:
344:
343:
327:
324:
323:
307:
304:
303:
279:
276:
275:
254:
251:
250:
244:
243:
231:Boolean algebra
214:
211:
210:
207:
141:
138:
137:
121:
118:
117:
97:
94:
93:
73:
70:
69:
53:
50:
49:
35:
28:
23:
22:
15:
12:
11:
5:
1469:
1459:
1458:
1453:
1439:
1438:
1422:
1390:
1387:
1386:
1385:
1379:
1373:Fréchet filter
1368:
1365:
1361:direct product
1346:
1343:
1338:
1334:
1311:
1307:
1303:
1298:
1294:
1284:are sequences
1271:
1267:
1263:
1248:
1245:
1226:
1221:
1217:
1213:
1208:
1204:
1181:
1177:
1173:
1168:
1164:
1141:
1137:
1133:
1108:
1104:
1100:
1084:
1081:
1079:
1078:Other examples
1076:
1063:
1060:
1057:
1054:
1051:
1048:
1045:
1042:
1022:
1019:
1016:
1013:
1010:
1007:
1004:
1001:
981:
978:
975:
972:
945:
942:
918:
893:
882:
875:
857:
854:
853:
852:
849:hyperconnected
834:
820:
796:
784:
772:
754:
745:
725:
705:
702:
687:
678:
675:
673:in the plane.
662:
659:
656:
653:
620:
600:
597:
577:
544:
541:
531:
528:
525:
519: or
515:
512:
509:
506:
503:
500:
497:
494:
491:
486:
464:
461:
441:
417:
414:
390:
387:
374:
354:
351:
331:
311:
300:maximal filter
283:
261:
258:
218:
206:
203:
197:
188:
181:
148:
145:
125:
101:
77:
57:
26:
9:
6:
4:
3:
2:
1468:
1457:
1454:
1452:
1449:
1448:
1446:
1437:
1433:
1429:
1425:
1419:
1415:
1411:
1407:
1406:
1401:
1397:
1393:
1392:
1383:
1380:
1374:
1371:
1370:
1364:
1362:
1357:
1344:
1341:
1336:
1332:
1309:
1305:
1301:
1296:
1292:
1269:
1265:
1261:
1254:
1244:
1242:
1237:
1224:
1219:
1215:
1211:
1206:
1202:
1179:
1175:
1171:
1166:
1162:
1139:
1135:
1131:
1124:
1106:
1102:
1098:
1090:
1075:
1061:
1058:
1055:
1052:
1049:
1046:
1043:
1040:
1020:
1017:
1014:
1011:
1008:
1005:
1002:
999:
979:
976:
973:
970:
963:
959:
943:
940:
933:
906:
904:
900:
896:
889:
885:
878:
871:
867:
863:
850:
846:
842:
838:
818:
810:
794:
770:
762:
761:singleton set
758:
750:
746:
743:
739:
723:
703:
700:
692:
688:
685:
681:
680:
674:
660:
657:
654:
651:
643:
640:
639:
634:
618:
598:
595:
575:
568:
564:
560:
555:
542:
529:
523:
510:
507:
504:
501:
498:
495:
489:
462:
459:
439:
431:
415:
412:
404:
400:
396:
386:
372:
352:
349:
329:
309:
301:
297:
281:
272:
259:
256:
248:
240:
236:
232:
216:
202:
200:
198:
195:
191:
189:
186:
182:
179:
175:
173:
169:
164:
162:
146:
143:
123:
115:
99:
91:
75:
55:
47:
44:
40:
33:
19:
1435:
1403:
1358:
1250:
1241:box topology
1238:
1086:
907:
861:
859:
833:Hausdorff (T
636:
632:
556:
398:
394:
392:
298:(that is, a
273:
242:
239:intersection
208:
194:
185:
178:
176:
165:
68:is a subset
42:
36:
932:even number
638:irreducible
633:affine line
563:polynomials
296:ultrafilter
161:cocountable
39:mathematics
1445:Categories
1389:References
1247:Direct sum
962:odd number
716:the space
677:Properties
322:such that
199:meagre set
172:direct sum
114:finite set
90:complement
32:cofinality
1402:(1995) ,
1333:α
1302:∈
1293:α
1262:⨁
1172:⊆
1132:∏
1099:∏
899:separated
527:∖
514:∅
499:⊆
430:empty set
48:of a set
1367:See also
691:open set
561:. Since
403:topology
190:mplement
43:cofinite
18:Cofinite
1432:0507446
903:compact
841:regular
738:compact
401:) is a
1430:
1420:
1154:where
845:normal
88:whose
46:subset
1410:Dover
1123:basis
811:. If
757:axiom
567:field
235:union
112:is a
1418:ISBN
1121:has
1087:The
860:The
740:and
393:The
201:".
41:, a
956:is
879:or
843:or
736:is
249:on
174:.
170:or
92:in
37:In
1447::
1428:MR
1426:,
1416:,
1398:;
1363:.
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1243:.
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839:,
385:.
237:,
196:co
187:co
180:co
163:.
1408:(
1342:=
1337:i
1310:i
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1297:i
1270:i
1266:M
1225:.
1220:i
1216:X
1212:=
1207:i
1203:U
1180:i
1176:X
1167:i
1163:U
1140:i
1136:U
1107:i
1103:X
1062:,
1059:1
1056:+
1053:n
1050:2
1047:,
1044:n
1041:2
1021:,
1018:1
1015:+
1012:n
1009:2
1006:,
1003:n
1000:2
980:1
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894:0
892:R
883:1
881:T
876:0
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835:2
819:X
795:X
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744:.
724:X
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655:Y
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540:}
530:A
524:X
511:=
508:A
505::
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493:{
490:=
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460:X
440:X
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413:X
373:X
353:.
350:X
330:A
310:X
282:A
260:.
257:X
217:X
147:.
144:X
124:A
100:X
76:A
56:X
34:.
20:)
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