51:
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
139:
263:
956:
925:
355:
874:
829:
999:
979:
894:
849:
805:
1239:
1101:
374:
370:
104:
1234:
750:
483:
312:
381:
399:
is closed if and only if it consists of all solutions to some system of polynomial equations. Since any such
228:
53:
56:
of an open set is closed and vice versa. In the following, it doesn't matter which definition is used.)
1045:
351:
28:
930:
899:
1006:
808:
528:
757:
of those topologies (the union of two topologies need not be a topology) but rather the topology
754:
1097:
777:
600:
453:
332:
729:
that is also closed under arbitrary intersections. That is, any collection of topologies on
1196:
1163:
1130:
1081:
1018:
854:
781:
769:
726:
388:
340:
814:
1002:
765:
308:
1024:
984:
964:
879:
834:
790:
362:
347:
32:
1168:
1151:
1093:
1021:, the coarsest topology on a set to make a family of mappings from that set continuous
713:). Intuitively, this makes sense: a finer topology should have smaller neighborhoods.
1228:
1089:
1077:
773:
1027:, the finest topology on a set to make a family of mappings into that set continuous
63:
of a topological space, since that is the standard meaning of the word "topology".
24:
1135:
1118:
1082:
366:
343:; this topology only admits the empty set and the whole space as open sets.
1201:
1184:
746:
531:
and therefore it is strongly open if and only if it is relatively open.)
516:
20:
59:
For definiteness the reader should think of a topology as the family of
758:
738:
48:
387:
may be equipped with either its usual (Euclidean) topology, or its
335:; this topology makes all subsets open. The coarsest topology on
534:
Two immediate corollaries of the above equivalent statements are
407:, the Zariski topology is strictly weaker than the ordinary one.
927:
is the trivial topology and the topology generated by the union
753:
of those topologies. The join, however, is not generally the
27:, the set of all possible topologies on a given set forms a
1152:"The lattice of topologies: Structure and complementation"
981:
has at least three elements, the lattice of topologies on
776:. In the case of topologies, the greatest element is the
47:
A topology on a set may be defined as the collection of
725:
together with the partial ordering relation ⊆ forms a
987:
967:
933:
902:
882:
857:
837:
817:
793:
356:
topologies on the set of operators on a
Hilbert space
354:
there are often a number of possible topologies. See
231:
107:
403:
also is a closed set in the ordinary sense, but not
993:
973:
950:
919:
888:
868:
843:
823:
799:
257:
133:
1156:Transactions of the American Mathematical Society
749:). The meet of a collection of topologies is the
433:. Then the following statements are equivalent:
1185:"The Lattice of all Topologies is Complemented"
580:remains open (resp. closed) if the topology on
8:
1117:Larson, Roland E.; Andima, Susan J. (1975).
134:{\displaystyle \tau _{1}\subseteq \tau _{2}}
1200:
1167:
1134:
986:
966:
932:
901:
881:
856:
836:
816:
792:
315:on the set of all possible topologies on
249:
236:
230:
125:
112:
106:
1215:
1069:
1056:with opposite meaning (Munkres, p. 78).
1037:
258:{\displaystyle \tau _{1}\neq \tau _{2}}
599:One can also compare topologies using
550:remains continuous if the topology on
1123:Rocky Mountain Journal of Mathematics
1119:"The lattice of topologies: A survey"
7:
1044:There are some authors, especially
787:The lattice of topologies on a set
721:The set of all topologies on a set
634:) be a local base for the topology
1088:(2nd ed.). Saddle River, NJ:
358:for some intricate relationships.
14:
1169:10.1090/S0002-9947-1966-0190893-2
764:Every complete lattice is also a
951:{\displaystyle \tau \cup \tau '}
920:{\displaystyle \tau \cap \tau '}
768:, which is to say that it has a
1189:Canadian Journal of Mathematics
1:
780:and the least element is the
1183:Van Rooij, A. C. M. (1968).
811:; that is, given a topology
517:strongly/relatively open map
37:comparison of the topologies
896:such that the intersection
617:be two topologies on a set
568:An open (resp. closed) map
429:be two topologies on a set
80:be two topologies on a set
1256:
958:is the discrete topology.
391:. In the latter, a subset
145:That is, every element of
1240:Comparison (mathematical)
695:) contains some open set
313:partial ordering relation
1136:10.1216/RMJ-1975-5-2-177
851:there exists a topology
1150:Steiner, A. K. (1966).
669:if and only if for all
327:The finest topology on
1202:10.4153/CJM-1968-079-9
995:
975:
952:
921:
890:
870:
869:{\displaystyle \tau '}
845:
825:
801:
259:
152:is also an element of
135:
996:
976:
953:
922:
891:
871:
846:
826:
824:{\displaystyle \tau }
802:
717:Lattice of topologies
373:and coarser than the
260:
136:
29:partially ordered set
23:and related areas of
16:Mathematical exercise
1048:, who use the terms
985:
965:
931:
900:
880:
855:
835:
815:
809:complemented lattice
791:
523:(The identity map id
382:complex vector space
229:
159:. Then the topology
105:
588:or the topology on
558:or the topology on
489:the identity map id
369:are finer than the
991:
971:
948:
917:
886:
866:
841:
821:
797:
601:neighborhood bases
255:
131:
1078:Munkres, James R.
994:{\displaystyle X}
974:{\displaystyle X}
889:{\displaystyle X}
844:{\displaystyle X}
800:{\displaystyle X}
778:discrete topology
538:A continuous map
333:discrete topology
222:If additionally
1247:
1235:General topology
1219:
1213:
1207:
1206:
1204:
1180:
1174:
1173:
1171:
1147:
1141:
1140:
1138:
1114:
1108:
1107:
1087:
1074:
1057:
1042:
1019:Initial topology
1005:, and hence not
1000:
998:
997:
992:
980:
978:
977:
972:
957:
955:
954:
949:
947:
926:
924:
923:
918:
916:
895:
893:
892:
887:
875:
873:
872:
867:
865:
850:
848:
847:
842:
830:
828:
827:
822:
806:
804:
803:
798:
782:trivial topology
727:complete lattice
677:, each open set
389:Zariski topology
363:polar topologies
341:trivial topology
277:strictly coarser
264:
262:
261:
256:
254:
253:
241:
240:
196:is said to be a
166:is said to be a
140:
138:
137:
132:
130:
129:
117:
116:
91:is contained in
35:can be used for
1255:
1254:
1250:
1249:
1248:
1246:
1245:
1244:
1225:
1224:
1223:
1222:
1214:
1210:
1182:
1181:
1177:
1149:
1148:
1144:
1116:
1115:
1111:
1104:
1076:
1075:
1071:
1066:
1061:
1060:
1043:
1039:
1034:
1015:
983:
982:
963:
962:
940:
929:
928:
909:
898:
897:
878:
877:
858:
853:
852:
833:
832:
813:
812:
789:
788:
766:bounded lattice
719:
708:
701:
690:
683:
668:
661:
642:
629:
616:
609:
526:
514:
503:
492:
481:
470:
459:
449:
442:
428:
421:
413:
375:strong topology
348:function spaces
325:
309:binary relation
303:
292:
285:
274:
245:
232:
227:
226:
218:
195:
188:
165:
158:
151:
121:
108:
103:
102:
97:
90:
79:
72:
45:
17:
12:
11:
5:
1253:
1251:
1243:
1242:
1237:
1227:
1226:
1221:
1220:
1218:, Theorem 3.1.
1208:
1175:
1162:(2): 379–398.
1142:
1129:(2): 177–198.
1109:
1102:
1068:
1067:
1065:
1062:
1059:
1058:
1036:
1035:
1033:
1030:
1029:
1028:
1025:Final topology
1022:
1014:
1011:
990:
970:
946:
943:
939:
936:
915:
912:
908:
905:
885:
864:
861:
840:
820:
796:
718:
715:
706:
699:
688:
681:
666:
659:
638:
625:
614:
607:
597:
596:
566:
524:
521:
520:
512:
501:
490:
487:
484:continuous map
479:
468:
457:
450:
447:
440:
426:
419:
412:
409:
350:and spaces of
324:
321:
301:
295:strictly finer
290:
283:
272:
266:
265:
252:
248:
244:
239:
235:
216:
193:
186:
163:
156:
149:
143:
142:
128:
124:
120:
115:
111:
95:
88:
77:
70:
44:
41:
33:order relation
15:
13:
10:
9:
6:
4:
3:
2:
1252:
1241:
1238:
1236:
1233:
1232:
1230:
1217:
1212:
1209:
1203:
1198:
1194:
1190:
1186:
1179:
1176:
1170:
1165:
1161:
1157:
1153:
1146:
1143:
1137:
1132:
1128:
1124:
1120:
1113:
1110:
1105:
1103:0-13-181629-2
1099:
1095:
1091:
1090:Prentice Hall
1086:
1085:
1079:
1073:
1070:
1063:
1055:
1051:
1047:
1041:
1038:
1031:
1026:
1023:
1020:
1017:
1016:
1012:
1010:
1008:
1004:
988:
968:
959:
944:
941:
937:
934:
913:
910:
906:
903:
883:
862:
859:
838:
818:
810:
794:
785:
783:
779:
775:
774:least element
771:
767:
762:
760:
756:
752:
748:
744:
740:
736:
732:
728:
724:
716:
714:
712:
705:
698:
694:
687:
680:
676:
672:
665:
658:
654:
650:
646:
641:
637:
633:
628:
624:
620:
613:
606:
602:
594:
591:
587:
583:
579:
575:
571:
567:
564:
561:
557:
553:
549:
545:
541:
537:
536:
535:
532:
530:
518:
511:
507:
500:
496:
488:
485:
478:
474:
467:
463:
455:
451:
446:
439:
436:
435:
434:
432:
425:
418:
410:
408:
406:
402:
398:
394:
390:
386:
383:
378:
376:
372:
371:weak topology
368:
364:
361:All possible
359:
357:
353:
349:
344:
342:
338:
334:
330:
322:
320:
318:
314:
310:
305:
300:
296:
289:
282:
278:
271:
250:
246:
242:
237:
233:
225:
224:
223:
220:
215:
211:
207:
203:
199:
192:
185:
181:
177:
173:
169:
162:
155:
148:
126:
122:
118:
113:
109:
101:
100:
99:
94:
87:
83:
76:
69:
64:
62:
57:
55:
50:
42:
40:
38:
34:
30:
26:
22:
1216:Steiner 1966
1211:
1192:
1188:
1178:
1159:
1155:
1145:
1126:
1122:
1112:
1083:
1072:
1053:
1049:
1040:
1007:distributive
960:
786:
763:
759:generated by
751:intersection
742:
734:
730:
722:
720:
710:
703:
696:
692:
685:
678:
674:
670:
663:
656:
655:= 1,2. Then
652:
648:
644:
639:
635:
631:
626:
622:
618:
611:
604:
598:
592:
589:
585:
581:
577:
573:
569:
562:
559:
555:
551:
547:
543:
539:
533:
522:
509:
505:
498:
494:
476:
472:
465:
461:
454:identity map
444:
437:
430:
423:
416:
414:
404:
400:
396:
392:
384:
379:
360:
345:
336:
328:
326:
316:
311:⊆ defines a
306:
298:
294:
287:
280:
276:
269:
267:
221:
213:
209:
205:
201:
197:
190:
183:
179:
175:
171:
167:
160:
153:
146:
144:
92:
85:
81:
74:
67:
65:
60:
58:
46:
36:
18:
1195:: 805–807.
1092:. pp.
961:If the set
761:the union.
25:mathematics
1229:Categories
1064:References
529:surjective
411:Properties
405:vice versa
84:such that
54:complement
43:Definition
942:τ
938:∪
935:τ
911:τ
907:∩
904:τ
860:τ
819:τ
493: : (
460: : (
367:dual pair
247:τ
243:≠
234:τ
123:τ
119:⊆
110:τ
61:open sets
1084:Topology
1080:(2000).
1046:analysts
1013:See also
1009:either.
945:′
914:′
863:′
770:greatest
747:supremum
741:) and a
621:and let
584:becomes
572: :
554:becomes
542: :
352:measures
323:Examples
210:topology
202:stronger
180:topology
21:topology
1003:modular
1001:is not
739:infimum
733:have a
593:coarser
556:coarser
515:) is a
482:) is a
339:is the
331:is the
268:we say
189:, and
176:smaller
168:coarser
49:subsets
31:. This
1100:
1054:strong
603:. Let
279:than
206:larger
172:weaker
1096:–78.
1032:Notes
807:is a
755:union
586:finer
563:finer
504:) → (
471:) → (
365:on a
297:than
212:than
198:finer
182:than
1098:ISBN
1052:and
1050:weak
772:and
745:(or
743:join
737:(or
735:meet
651:for
610:and
452:the
422:and
415:Let
380:The
307:The
286:and
73:and
66:Let
1197:doi
1164:doi
1160:122
1131:doi
876:on
831:on
702:in
684:in
643:at
527:is
395:of
346:In
293:is
275:is
204:or
174:or
19:In
1231::
1193:20
1191:.
1187:.
1158:.
1154:.
1125:.
1121:.
1094:77
784:.
673:∈
662:⊆
647:∈
576:→
546:→
508:,
497:,
475:,
464:,
456:id
443:⊆
377:.
319:.
304:.
219:.
208:)
178:)
98::
39:.
1205:.
1199::
1172:.
1166::
1139:.
1133::
1127:5
1106:.
989:X
969:X
884:X
839:X
795:X
731:X
723:X
711:x
709:(
707:2
704:B
700:2
697:U
693:x
691:(
689:1
686:B
682:1
679:U
675:X
671:x
667:2
664:τ
660:1
657:τ
653:i
649:X
645:x
640:i
636:τ
632:x
630:(
627:i
623:B
619:X
615:2
612:τ
608:1
605:τ
595:.
590:X
582:Y
578:Y
574:X
570:f
565:.
560:X
552:Y
548:Y
544:X
540:f
525:X
519:.
513:2
510:τ
506:X
502:1
499:τ
495:X
491:X
486:.
480:1
477:τ
473:X
469:2
466:τ
462:X
458:X
448:2
445:τ
441:1
438:τ
431:X
427:2
424:τ
420:1
417:τ
401:V
397:C
393:V
385:C
337:X
329:X
317:X
302:1
299:τ
291:2
288:τ
284:2
281:τ
273:1
270:τ
251:2
238:1
217:1
214:τ
200:(
194:2
191:τ
187:2
184:τ
170:(
164:1
161:τ
157:2
154:τ
150:1
147:τ
141:.
127:2
114:1
96:2
93:τ
89:1
86:τ
82:X
78:2
75:τ
71:1
68:τ
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