62:
which are considered to be "open". (An alternative definition is that it is the collection of subsets which are considered "closed". These two ways of defining the topology are essentially equivalent because the
150:
274:
967:
936:
885:
840:
1010:
990:
905:
860:
816:
366:
1250:
1112:
17:
385:
115:
381:
410:
is closed if and only if it consists of all solutions to some system of polynomial equations. Since any such
239:
67:
of an open set is closed and vice versa. In the following, it doesn't matter which definition is used.)
941:
910:
1245:
761:
494:
323:
392:
768:
of those topologies (the union of two topologies need not be a topology) but rather the topology
64:
1056:
740:
that is also closed under arbitrary intersections. That is, any collection of topologies on
362:
39:
1092:
865:
1017:
825:
819:
8:
539:
995:
975:
890:
845:
801:
765:
1179:
1162:
1104:
1032:, the coarsest topology on a set to make a family of mappings from that set continuous
724:). Intuitively, this makes sense: a finer topology should have smaller neighborhoods.
1108:
788:
611:
464:
343:
1038:, the finest topology on a set to make a family of mappings into that set continuous
1207:
1174:
1141:
1029:
792:
780:
737:
399:
351:
74:
of a topological space, since that is the standard meaning of the word "topology".
1013:
776:
319:
1035:
373:
358:
43:
1239:
1146:
1129:
1100:
1093:
1088:
784:
354:; this topology only admits the empty set and the whole space as open sets.
1212:
1195:
542:
and therefore it is strongly open if and only if it is relatively open.)
35:
70:
For definiteness the reader should think of a topology as the family of
377:
757:
527:
398:
may be equipped with either its usual (Euclidean) topology, or its
31:
346:; this topology makes all subsets open. The coarsest topology on
769:
749:
545:
Two immediate corollaries of the above equivalent statements are
418:, the Zariski topology is strictly weaker than the ordinary one.
938:
is the trivial topology and the topology generated by the union
59:
1220:
764:
of those topologies. The join, however, is not generally the
38:, the set of all possible topologies on a given set forms a
1163:"The lattice of topologies: Structure and complementation"
992:
has at least three elements, the lattice of topologies on
787:. In the case of topologies, the greatest element is the
58:
A topology on a set may be defined as the collection of
736:
together with the partial ordering relation ⊆ forms a
998:
978:
944:
913:
893:
868:
848:
828:
804:
367:
topologies on the set of operators on a
Hilbert space
365:
there are often a number of possible topologies. See
242:
118:
414:
also is a closed set in the ordinary sense, but not
1004:
984:
961:
930:
899:
879:
854:
834:
810:
268:
144:
1167:Transactions of the American Mathematical Society
760:). The meet of a collection of topologies is the
1237:
444:. Then the following statements are equivalent:
1196:"The Lattice of all Topologies is Complemented"
591:remains open (resp. closed) if the topology on
1128:Larson, Roland E.; Andima, Susan J. (1975).
1127:
145:{\displaystyle \tau _{1}\subseteq \tau _{2}}
1211:
1193:
1178:
1145:
326:on the set of all possible topologies on
727:
1226:
1160:
1087:
1067:with opposite meaning (Munkres, p. 78).
269:{\displaystyle \tau _{1}\neq \tau _{2}}
14:
1238:
1081:
610:One can also compare topologies using
561:remains continuous if the topology on
1134:Rocky Mountain Journal of Mathematics
1130:"The lattice of topologies: A survey"
1055:There are some authors, especially
798:The lattice of topologies on a set
732:The set of all topologies on a set
645:) be a local base for the topology
24:
1099:(2nd ed.). Saddle River, NJ:
369:for some intricate relationships.
25:
1262:
1180:10.1090/S0002-9947-1966-0190893-2
775:Every complete lattice is also a
962:{\displaystyle \tau \cup \tau '}
931:{\displaystyle \tau \cap \tau '}
779:, which is to say that it has a
1200:Canadian Journal of Mathematics
1187:
1154:
1121:
1049:
13:
1:
1074:
791:and the least element is the
421:
53:
1194:Van Rooij, A. C. M. (1968).
822:; that is, given a topology
528:strongly/relatively open map
48:comparison of the topologies
7:
1023:
907:such that the intersection
628:be two topologies on a set
579:An open (resp. closed) map
440:be two topologies on a set
333:
91:be two topologies on a set
10:
1267:
969:is the discrete topology.
402:. In the latter, a subset
156:That is, every element of
1251:Comparison (mathematical)
706:) contains some open set
324:partial ordering relation
1147:10.1216/RMJ-1975-5-2-177
1042:
862:there exists a topology
1161:Steiner, A. K. (1966).
680:if and only if for all
338:The finest topology on
1213:10.4153/CJM-1968-079-9
1006:
986:
963:
932:
901:
881:
880:{\displaystyle \tau '}
856:
836:
812:
270:
163:is also an element of
146:
1007:
987:
964:
933:
902:
882:
857:
837:
835:{\displaystyle \tau }
813:
728:Lattice of topologies
384:and coarser than the
271:
147:
40:partially ordered set
34:and related areas of
27:Mathematical exercise
1059:, who use the terms
996:
976:
942:
911:
891:
866:
846:
826:
820:complemented lattice
802:
534:(The identity map id
393:complex vector space
240:
170:. Then the topology
116:
599:or the topology on
569:or the topology on
500:the identity map id
380:are finer than the
1002:
982:
959:
928:
897:
877:
852:
832:
808:
612:neighborhood bases
266:
142:
1089:Munkres, James R.
1005:{\displaystyle X}
985:{\displaystyle X}
900:{\displaystyle X}
855:{\displaystyle X}
811:{\displaystyle X}
789:discrete topology
549:A continuous map
344:discrete topology
233:If additionally
18:Coarsest topology
16:(Redirected from
1258:
1246:General topology
1230:
1224:
1218:
1217:
1215:
1191:
1185:
1184:
1182:
1158:
1152:
1151:
1149:
1125:
1119:
1118:
1098:
1085:
1068:
1053:
1030:Initial topology
1016:, and hence not
1011:
1009:
1008:
1003:
991:
989:
988:
983:
968:
966:
965:
960:
958:
937:
935:
934:
929:
927:
906:
904:
903:
898:
886:
884:
883:
878:
876:
861:
859:
858:
853:
841:
839:
838:
833:
817:
815:
814:
809:
793:trivial topology
738:complete lattice
688:, each open set
400:Zariski topology
374:polar topologies
352:trivial topology
288:strictly coarser
275:
273:
272:
267:
265:
264:
252:
251:
207:is said to be a
177:is said to be a
151:
149:
148:
143:
141:
140:
128:
127:
102:is contained in
46:can be used for
21:
1266:
1265:
1261:
1260:
1259:
1257:
1256:
1255:
1236:
1235:
1234:
1233:
1225:
1221:
1192:
1188:
1159:
1155:
1126:
1122:
1115:
1086:
1082:
1077:
1072:
1071:
1054:
1050:
1045:
1026:
997:
994:
993:
977:
974:
973:
951:
943:
940:
939:
920:
912:
909:
908:
892:
889:
888:
869:
867:
864:
863:
847:
844:
843:
827:
824:
823:
803:
800:
799:
777:bounded lattice
730:
719:
712:
701:
694:
679:
672:
653:
640:
627:
620:
537:
525:
514:
503:
492:
481:
470:
460:
453:
439:
432:
424:
386:strong topology
359:function spaces
336:
320:binary relation
314:
303:
296:
285:
260:
256:
247:
243:
241:
238:
237:
229:
206:
199:
176:
169:
162:
136:
132:
123:
119:
117:
114:
113:
108:
101:
90:
83:
56:
28:
23:
22:
15:
12:
11:
5:
1264:
1254:
1253:
1248:
1232:
1231:
1229:, Theorem 3.1.
1219:
1186:
1173:(2): 379–398.
1153:
1140:(2): 177–198.
1120:
1113:
1079:
1078:
1076:
1073:
1070:
1069:
1047:
1046:
1044:
1041:
1040:
1039:
1036:Final topology
1033:
1025:
1022:
1001:
981:
957:
954:
950:
947:
926:
923:
919:
916:
896:
875:
872:
851:
831:
807:
729:
726:
717:
710:
699:
692:
677:
670:
649:
636:
625:
618:
608:
607:
577:
535:
532:
531:
523:
512:
501:
498:
495:continuous map
490:
479:
468:
461:
458:
451:
437:
430:
423:
420:
361:and spaces of
335:
332:
312:
306:strictly finer
301:
294:
283:
277:
276:
263:
259:
255:
250:
246:
227:
204:
197:
174:
167:
160:
154:
153:
139:
135:
131:
126:
122:
106:
99:
88:
81:
55:
52:
44:order relation
26:
9:
6:
4:
3:
2:
1263:
1252:
1249:
1247:
1244:
1243:
1241:
1228:
1223:
1214:
1209:
1205:
1201:
1197:
1190:
1181:
1176:
1172:
1168:
1164:
1157:
1148:
1143:
1139:
1135:
1131:
1124:
1116:
1114:0-13-181629-2
1110:
1106:
1102:
1101:Prentice Hall
1097:
1096:
1090:
1084:
1080:
1066:
1062:
1058:
1052:
1048:
1037:
1034:
1031:
1028:
1027:
1021:
1019:
1015:
999:
979:
970:
955:
952:
948:
945:
924:
921:
917:
914:
894:
873:
870:
849:
829:
821:
805:
796:
794:
790:
786:
785:least element
782:
778:
773:
771:
767:
763:
759:
755:
751:
747:
743:
739:
735:
725:
723:
716:
709:
705:
698:
691:
687:
683:
676:
669:
665:
661:
657:
652:
648:
644:
639:
635:
631:
624:
617:
613:
605:
602:
598:
594:
590:
586:
582:
578:
575:
572:
568:
564:
560:
556:
552:
548:
547:
546:
543:
541:
529:
522:
518:
511:
507:
499:
496:
489:
485:
478:
474:
466:
462:
457:
450:
447:
446:
445:
443:
436:
429:
419:
417:
413:
409:
405:
401:
397:
394:
389:
387:
383:
382:weak topology
379:
375:
372:All possible
370:
368:
364:
360:
355:
353:
349:
345:
341:
331:
329:
325:
321:
316:
311:
307:
300:
293:
289:
282:
261:
257:
253:
248:
244:
236:
235:
234:
231:
226:
222:
218:
214:
210:
203:
196:
192:
188:
184:
180:
173:
166:
159:
137:
133:
129:
124:
120:
112:
111:
110:
105:
98:
94:
87:
80:
75:
73:
68:
66:
61:
51:
49:
45:
41:
37:
33:
19:
1227:Steiner 1966
1222:
1203:
1199:
1189:
1170:
1166:
1156:
1137:
1133:
1123:
1094:
1083:
1064:
1060:
1051:
1018:distributive
971:
797:
774:
770:generated by
762:intersection
753:
745:
741:
733:
731:
721:
714:
707:
703:
696:
689:
685:
681:
674:
667:
666:= 1,2. Then
663:
659:
655:
650:
646:
642:
637:
633:
629:
622:
615:
609:
603:
600:
596:
592:
588:
584:
580:
573:
570:
566:
562:
558:
554:
550:
544:
533:
520:
516:
509:
505:
487:
483:
476:
472:
465:identity map
455:
448:
441:
434:
427:
425:
415:
411:
407:
403:
395:
390:
371:
356:
347:
339:
337:
327:
322:⊆ defines a
317:
309:
305:
298:
291:
287:
280:
278:
232:
224:
220:
216:
212:
208:
201:
194:
190:
186:
182:
178:
171:
164:
157:
155:
103:
96:
92:
85:
78:
76:
71:
69:
57:
47:
29:
1206:: 805–807.
1103:. pp.
972:If the set
772:the union.
36:mathematics
1240:Categories
1075:References
540:surjective
422:Properties
416:vice versa
95:such that
65:complement
54:Definition
953:τ
949:∪
946:τ
922:τ
918:∩
915:τ
871:τ
830:τ
504: : (
471: : (
378:dual pair
258:τ
254:≠
245:τ
134:τ
130:⊆
121:τ
72:open sets
1095:Topology
1091:(2000).
1057:analysts
1024:See also
1020:either.
956:′
925:′
874:′
781:greatest
758:supremum
752:) and a
632:and let
595:becomes
583: :
565:becomes
553: :
363:measures
334:Examples
221:topology
213:stronger
191:topology
32:topology
1014:modular
1012:is not
750:infimum
744:have a
604:coarser
567:coarser
526:) is a
493:) is a
350:is the
342:is the
279:we say
200:, and
187:smaller
179:coarser
60:subsets
42:. This
1111:
1065:strong
614:. Let
290:than
217:larger
183:weaker
1107:–78.
1043:Notes
818:is a
766:union
597:finer
574:finer
515:) → (
482:) → (
376:on a
308:than
223:than
209:finer
193:than
1109:ISBN
1063:and
1061:weak
783:and
756:(or
754:join
748:(or
746:meet
662:for
621:and
463:the
433:and
426:Let
391:The
318:The
297:and
84:and
77:Let
1208:doi
1175:doi
1171:122
1142:doi
887:on
842:on
713:in
695:in
654:at
538:is
406:of
357:In
304:is
286:is
215:or
185:or
30:In
1242::
1204:20
1202:.
1198:.
1169:.
1165:.
1136:.
1132:.
1105:77
795:.
684:∈
673:⊆
658:∈
587:→
557:→
519:,
508:,
486:,
475:,
467:id
454:⊆
388:.
330:.
315:.
230:.
219:)
189:)
109::
50:.
1216:.
1210::
1183:.
1177::
1150:.
1144::
1138:5
1117:.
1000:X
980:X
895:X
850:X
806:X
742:X
734:X
722:x
720:(
718:2
715:B
711:2
708:U
704:x
702:(
700:1
697:B
693:1
690:U
686:X
682:x
678:2
675:τ
671:1
668:τ
664:i
660:X
656:x
651:i
647:τ
643:x
641:(
638:i
634:B
630:X
626:2
623:τ
619:1
616:τ
606:.
601:X
593:Y
589:Y
585:X
581:f
576:.
571:X
563:Y
559:Y
555:X
551:f
536:X
530:.
524:2
521:τ
517:X
513:1
510:τ
506:X
502:X
497:.
491:1
488:τ
484:X
480:2
477:τ
473:X
469:X
459:2
456:τ
452:1
449:τ
442:X
438:2
435:τ
431:1
428:τ
412:V
408:C
404:V
396:C
348:X
340:X
328:X
313:1
310:τ
302:2
299:τ
295:2
292:τ
284:1
281:τ
262:2
249:1
228:1
225:τ
211:(
205:2
202:τ
198:2
195:τ
181:(
175:1
172:τ
168:2
165:τ
161:1
158:τ
152:.
138:2
125:1
107:2
104:τ
100:1
97:τ
93:X
89:2
86:τ
82:1
79:τ
20:)
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