1064:
185:
957:
786:
293:
922:
503:
417:
221:
1020:
338:
883:
640:
993:
859:
451:
601:
575:
109:
812:
248:
705:
681:
532:
73:
1134:
1025:
116:
646:; however, the above topology does not capture the notion of convergence associated to the operator, one should consider a
1159:
931:
44:
713:
608:
32:
258:
891:
456:
345:
193:
998:
647:
304:
864:
613:
962:
828:
36:
430:
1097:
Prague: Academia, Publishing House of the
Czechoslovak Academy of Sciences, 1966, Theorem 14 A.9
822:
296:
580:
1130:
886:
643:
925:
541:
1141:
78:
794:
230:
1074:
690:
666:
517:
58:
1153:
642:
is closed. The collection of all open sets generated by the preclosure operator is a
40:
684:
224:
20:
885:
with respect to which the sequential closure operator is defined, the
252:
The preclosure operator has to satisfy the following properties:
1098:
43:. That is, a preclosure operator obeys only three of the four
1048:
1032:
1004:
938:
906:
870:
199:
163:
144:
1059:{\displaystyle {\mathcal {T}}_{\text{seq}}={\mathcal {T}}.}
180:{\displaystyle _{p}:{\mathcal {P}}(X)\to {\mathcal {P}}(X)}
1028:
1001:
965:
934:
894:
867:
831:
797:
716:
693:
669:
616:
583:
544:
520:
459:
433:
348:
307:
261:
233:
196:
119:
81:
61:
1146:, Comment. Math. Univ. Carolinae 33 (1992), 303–309.
1058:
1014:
987:
951:
916:
877:
853:
806:
780:
699:
675:
634:
595:
569:
526:
497:
445:
411:
332:
287:
242:
215:
179:
103:
67:
284:
1093:Eduard Čech, Zdeněk Frolík, Miroslav Katětov,
8:
775:
736:
952:{\displaystyle {\mathcal {T}}_{\text{seq}}}
861:is a preclosure operator. Given a topology
1047:
1046:
1037:
1031:
1030:
1027:
1003:
1002:
1000:
979:
964:
943:
937:
936:
933:
905:
904:
893:
869:
868:
866:
845:
830:
796:
727:
715:
692:
668:
615:
582:
555:
543:
519:
489:
470:
458:
432:
403:
384:
365:
347:
324:
306:
272:
260:
232:
198:
197:
195:
162:
161:
143:
142:
133:
118:
95:
80:
60:
781:{\displaystyle _{p}=\{x\in X:d(x,A)=0\}}
607:(with respect to the preclosure) if its
1086:
626:
423:The last axiom implies the following:
281:
265:
39:, except that it is not required to be
35:between subsets of a set, similar to a
1143:Bourbaki's Fixpoint Lemma reconsidered
1117:, AMS, Contemporary Mathematics, 2009.
1111:An Initiation into Convergence Theory
7:
1125:A.V. Arkhangelskii, L.S.Pontryagin,
1113:, in F. Mynard, E. Pearl (editors),
538:(with respect to the preclosure) if
288:{\displaystyle _{p}=\varnothing \!}
1129:, (1990) Springer-Verlag, Berlin.
917:{\displaystyle (X,{\mathcal {T}})}
498:{\displaystyle _{p}\subseteq _{p}}
412:{\displaystyle _{p}=_{p}\cup _{p}}
14:
216:{\displaystyle {\mathcal {P}}(X)}
419:(Preservation of binary unions).
1015:{\displaystyle {\mathcal {T}},}
333:{\displaystyle A\subseteq _{p}}
55:A preclosure operator on a set
976:
966:
911:
895:
878:{\displaystyle {\mathcal {T}}}
842:
832:
766:
754:
724:
717:
635:{\displaystyle A=X\setminus U}
552:
545:
486:
479:
467:
460:
400:
393:
381:
374:
362:
349:
321:
314:
269:
262:
210:
204:
174:
168:
158:
155:
149:
130:
120:
92:
82:
1:
988:{\displaystyle _{\text{seq}}}
854:{\displaystyle _{\text{seq}}}
928:if and only if the topology
446:{\displaystyle A\subseteq B}
37:topological closure operator
1176:
596:{\displaystyle U\subset X}
295:(Preservation of nullary
45:Kuratowski closure axioms
1060:
1016:
989:
953:
918:
879:
855:
808:
782:
701:
677:
636:
597:
571:
570:{\displaystyle _{p}=A}
528:
499:
447:
413:
334:
289:
244:
217:
181:
105:
69:
1061:
1017:
990:
954:
919:
880:
856:
809:
783:
702:
678:
637:
598:
572:
529:
500:
448:
414:
335:
290:
245:
218:
182:
106:
70:
29:Čech closure operator
1026:
999:
963:
932:
892:
865:
829:
795:
714:
691:
667:
614:
581:
542:
518:
457:
431:
346:
305:
259:
231:
194:
117:
104:{\displaystyle _{p}}
79:
59:
791:is a preclosure on
25:preclosure operator
1127:General Topology I
1095:Topological spaces
1056:
1012:
985:
949:
914:
875:
851:
823:sequential closure
807:{\displaystyle X.}
804:
778:
697:
673:
632:
593:
567:
524:
495:
443:
409:
330:
285:
243:{\displaystyle X.}
240:
213:
177:
101:
65:
1160:Closure operators
1040:
982:
974:
971:
946:
887:topological space
848:
840:
837:
817:Sequential spaces
700:{\displaystyle X}
676:{\displaystyle d}
527:{\displaystyle A}
128:
125:
90:
87:
68:{\displaystyle X}
1167:
1140:B. Banascheski,
1118:
1107:
1101:
1091:
1065:
1063:
1062:
1057:
1052:
1051:
1042:
1041:
1038:
1036:
1035:
1021:
1019:
1018:
1013:
1008:
1007:
994:
992:
991:
986:
984:
983:
980:
972:
969:
958:
956:
955:
950:
948:
947:
944:
942:
941:
926:sequential space
923:
921:
920:
915:
910:
909:
884:
882:
881:
876:
874:
873:
860:
858:
857:
852:
850:
849:
846:
838:
835:
813:
811:
810:
805:
787:
785:
784:
779:
732:
731:
706:
704:
703:
698:
682:
680:
679:
674:
641:
639:
638:
633:
602:
600:
599:
594:
576:
574:
573:
568:
560:
559:
533:
531:
530:
525:
504:
502:
501:
496:
494:
493:
475:
474:
452:
450:
449:
444:
418:
416:
415:
410:
408:
407:
389:
388:
370:
369:
339:
337:
336:
331:
329:
328:
294:
292:
291:
286:
277:
276:
249:
247:
246:
241:
222:
220:
219:
214:
203:
202:
186:
184:
183:
178:
167:
166:
148:
147:
138:
137:
126:
123:
110:
108:
107:
102:
100:
99:
88:
85:
74:
72:
71:
66:
16:Closure operator
1175:
1174:
1170:
1169:
1168:
1166:
1165:
1164:
1150:
1149:
1122:
1121:
1115:Beyond Topology
1108:
1104:
1092:
1088:
1083:
1071:
1029:
1024:
1023:
997:
996:
975:
961:
960:
935:
930:
929:
890:
889:
863:
862:
841:
827:
826:
819:
793:
792:
723:
712:
711:
689:
688:
665:
664:
661:
656:
612:
611:
579:
578:
551:
540:
539:
516:
515:
512:
485:
466:
455:
454:
429:
428:
399:
380:
361:
344:
343:
320:
303:
302:
268:
257:
256:
229:
228:
192:
191:
129:
115:
114:
91:
77:
76:
57:
56:
53:
17:
12:
11:
5:
1173:
1171:
1163:
1162:
1152:
1151:
1148:
1147:
1138:
1120:
1119:
1102:
1085:
1084:
1082:
1079:
1078:
1077:
1070:
1067:
1055:
1050:
1045:
1034:
1011:
1006:
978:
968:
940:
913:
908:
903:
900:
897:
872:
844:
834:
818:
815:
803:
800:
789:
788:
777:
774:
771:
768:
765:
762:
759:
756:
753:
750:
747:
744:
741:
738:
735:
730:
726:
722:
719:
696:
672:
660:
657:
655:
652:
631:
628:
625:
622:
619:
592:
589:
586:
566:
563:
558:
554:
550:
547:
523:
511:
508:
507:
506:
492:
488:
484:
481:
478:
473:
469:
465:
462:
442:
439:
436:
421:
420:
406:
402:
398:
395:
392:
387:
383:
379:
376:
373:
368:
364:
360:
357:
354:
351:
341:
340:(Extensivity);
327:
323:
319:
316:
313:
310:
300:
283:
280:
275:
271:
267:
264:
239:
236:
212:
209:
206:
201:
188:
187:
176:
173:
170:
165:
160:
157:
154:
151:
146:
141:
136:
132:
122:
98:
94:
84:
64:
52:
49:
15:
13:
10:
9:
6:
4:
3:
2:
1172:
1161:
1158:
1157:
1155:
1145:
1144:
1139:
1136:
1135:3-540-18178-4
1132:
1128:
1124:
1123:
1116:
1112:
1106:
1103:
1099:
1096:
1090:
1087:
1080:
1076:
1073:
1072:
1068:
1066:
1053:
1043:
1009:
959:generated by
927:
901:
898:
888:
824:
816:
814:
801:
798:
772:
769:
763:
760:
757:
751:
748:
745:
742:
739:
733:
728:
720:
710:
709:
708:
694:
686:
670:
658:
653:
651:
649:
645:
629:
623:
620:
617:
610:
606:
590:
587:
584:
564:
561:
556:
548:
537:
521:
509:
490:
482:
476:
471:
463:
440:
437:
434:
426:
425:
424:
404:
396:
390:
385:
377:
371:
366:
358:
355:
352:
342:
325:
317:
311:
308:
301:
298:
278:
273:
255:
254:
253:
250:
237:
234:
226:
207:
171:
152:
139:
134:
113:
112:
111:
96:
62:
50:
48:
46:
42:
38:
34:
30:
26:
22:
1142:
1126:
1114:
1110:
1109:S. Dolecki,
1105:
1094:
1089:
1022:that is, if
995:is equal to
820:
790:
662:
604:
535:
513:
422:
251:
189:
54:
28:
24:
18:
1075:Eduard Čech
650:, instead.
648:pretopology
1081:References
659:Premetrics
609:complement
51:Definition
41:idempotent
825:operator
743:∈
685:premetric
627:∖
588:⊂
577:. A set
477:⊆
438:⊆
391:∪
356:∪
312:⊆
282:∅
266:∅
225:power set
159:→
75:is a map
1154:Category
1069:See also
654:Examples
644:topology
510:Topology
453:implies
21:topology
707:, then
223:is the
1133:
973:
970:
839:
836:
663:Given
536:closed
514:A set
297:unions
190:where
127:
124:
89:
86:
924:is a
31:is a
1131:ISBN
821:The
605:open
23:, a
1039:seq
981:seq
945:seq
847:seq
687:on
603:is
534:is
427:4.
227:of
33:map
27:or
19:In
1156::
683:a
299:);
47:.
1137:.
1100:.
1054:.
1049:T
1044:=
1033:T
1010:,
1005:T
977:]
967:[
939:T
912:)
907:T
902:,
899:X
896:(
871:T
843:]
833:[
802:.
799:X
776:}
773:0
770:=
767:)
764:A
761:,
758:x
755:(
752:d
749::
746:X
740:x
737:{
734:=
729:p
725:]
721:A
718:[
695:X
671:d
630:U
624:X
621:=
618:A
591:X
585:U
565:A
562:=
557:p
553:]
549:A
546:[
522:A
505:.
491:p
487:]
483:B
480:[
472:p
468:]
464:A
461:[
441:B
435:A
405:p
401:]
397:B
394:[
386:p
382:]
378:A
375:[
372:=
367:p
363:]
359:B
353:A
350:[
326:p
322:]
318:A
315:[
309:A
279:=
274:p
270:]
263:[
238:.
235:X
211:)
208:X
205:(
200:P
175:)
172:X
169:(
164:P
156:)
153:X
150:(
145:P
140::
135:p
131:]
121:[
97:p
93:]
83:[
63:X
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