88:
80:
317:
460:
194:
1041:
584:
508:
349:
127:
58:
955:
925:
628:
608:
206:
1110:
361:
518:
The comb space and the deleted comb space have some interesting topological properties mostly related to the notion of
1138:
707:
1081:
69:
1143:
143:
1071:
530:
519:
996:
1002:
545:
469:
325:
103:
34:
691:
938:
908:
1066:
587:
526:
1106:
613:
593:
352:
138:
130:
28:
758:
1051:
679:
637:
65:
1076:
1061:
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958:
826:
1132:
1102:
1094:
630:
the closure of E, where E is connected, the deleted comb space is also connected.
675:
992:
671:
20:
87:
355:
is known as the comb space. The deleted comb space, D, is defined by:
999:
onto a point for every choice of basepoint that lies in the segment
636:
3. The deleted comb space is not path connected since there is no
86:
78:
312:{\displaystyle (\{0\}\times )\cup (K\times )\cup (\times \{0\})}
61:
79:
455:{\displaystyle \{(0,1)\}\cup (K\times )\cup (\times \{0\})}
64:. The comb space has properties that serve as a number of
985:} is both open and closed in . This is a contradiction.
1005:
941:
911:
616:
596:
548:
472:
364:
328:
209:
146:
106:
37:
1035:
949:
919:
622:
602:
578:
502:
454:
343:
311:
188:
121:
52:
825:is connected since it is a basis element for the
72:has similar properties to the comb space. The
466:This is the comb space with the line segment
8:
1012:
1006:
555:
549:
536:2. The deleted comb space, D, is connected:
525:1. The comb space, C, is path connected and
479:
473:
446:
440:
383:
365:
303:
297:
219:
213:
183:
147:
706:} is open, we proceed as follows: Choose a
189:{\displaystyle \{1/n~|~n\in \mathbb {N} \}}
650:= (0, 1) to the point (0, 0) in
1004:
943:
942:
940:
913:
912:
910:
615:
595:
547:
471:
363:
335:
331:
330:
327:
208:
179:
178:
164:
153:
145:
113:
109:
108:
105:
44:
40:
39:
36:
1122:Encyclopedic Dictionary of Mathematics
969:); contradicting the connectedness of
91:The intricated double comb for r=3/4.
7:
586:. E is also path connected and the
1120:Kiyosi Itô (ed.). "Connectedness".
821:} will then be open. We know that
662:be this path. We shall prove that
76:is a variation on the comb space.
14:
1036:{\displaystyle \{0\}\times (0,1]}
579:{\displaystyle \{0\}\times (0,1]}
503:{\displaystyle \{0\}\times [0,1)}
1124:. Mathematical Society of Japan.
995:to a point but does not admit a
542:Let E be the comb space without
529:, but not locally contractible,
344:{\displaystyle \mathbb {R} ^{2}}
122:{\displaystyle \mathbb {R} ^{2}}
53:{\displaystyle \mathbb {R} ^{2}}
1030:
1018:
573:
561:
497:
485:
449:
434:
422:
419:
413:
410:
398:
389:
380:
368:
306:
291:
279:
276:
270:
267:
255:
246:
240:
237:
225:
210:
165:
16:Pathological topological space
1:
682:of this set. Clearly we have
646:Suppose there is a path from
590:of E is the comb space. As E
19:In mathematics, particularly
950:{\displaystyle \mathbb {R} }
920:{\displaystyle \mathbb {R} }
322:considered as a subspace of
721:that doesn’t intersect the
1160:
997:strong deformation retract
658: : →
893:) does not intersect the
729:is an arbitrary point in
837:) is connected. Suppose
623:{\displaystyle \subset }
603:{\displaystyle \subset }
533:, or locally connected.
1082:Topologist's sine curve
1072:Locally connected space
793:} which will mean that
690:} is closed in by the
70:topologist's sine curve
1037:
951:
921:
757:) is open, there is a
678:in contradicting the
624:
604:
580:
531:locally path connected
514:Topological properties
504:
456:
345:
313:
190:
123:
92:
84:
54:
1038:
991:4. The comb space is
952:
922:
877: + 1) <
797:is an open subset of
640:from (0,1) to (0,0):
625:
605:
581:
505:
457:
346:
314:
191:
124:
90:
82:
55:
1003:
939:
909:
614:
594:
546:
470:
362:
326:
207:
144:
104:
35:
935:, +∞) ×
901:= (−∞,
845:) contains a point
1139:Topological spaces
1067:List of topologies
1033:
947:
917:
781:. We assert that
620:
600:
576:
500:
452:
351:equipped with the
341:
309:
186:
119:
93:
85:
74:deleted comb space
50:
865:) must belong to
777:) is a subset of
353:subspace topology
171:
163:
131:standard topology
96:Formal definition
83:Topologist's comb
60:that resembles a
1151:
1144:Trees (topology)
1125:
1116:
1101:(2nd ed.).
1042:
1040:
1039:
1034:
956:
954:
953:
948:
946:
926:
924:
923:
918:
916:
897:-axis, the sets
829:on . Therefore,
698:. To prove that
629:
627:
626:
621:
609:
607:
606:
601:
585:
583:
582:
577:
509:
507:
506:
501:
461:
459:
458:
453:
350:
348:
347:
342:
340:
339:
334:
318:
316:
315:
310:
195:
193:
192:
187:
182:
169:
168:
161:
157:
128:
126:
125:
120:
118:
117:
112:
59:
57:
56:
51:
49:
48:
43:
27:is a particular
1159:
1158:
1154:
1153:
1152:
1150:
1149:
1148:
1129:
1128:
1119:
1113:
1093:
1090:
1052:Connected space
1048:
1001:
1000:
937:
936:
907:
906:
813:was arbitrary,
725:–axis. Suppose
612:
611:
592:
591:
544:
543:
516:
468:
467:
360:
359:
329:
324:
323:
205:
204:
142:
141:
107:
102:
101:
98:
66:counterexamples
38:
33:
32:
17:
12:
11:
5:
1157:
1155:
1147:
1146:
1141:
1131:
1130:
1127:
1126:
1117:
1111:
1089:
1086:
1085:
1084:
1079:
1077:Order topology
1074:
1069:
1064:
1062:Infinite broom
1059:
1057:Hedgehog space
1054:
1047:
1044:
1032:
1029:
1026:
1023:
1020:
1017:
1014:
1011:
1008:
989:
988:
987:
986:
977:). Therefore,
945:
915:
827:order topology
634:
633:
632:
631:
619:
599:
575:
572:
569:
566:
563:
560:
557:
554:
551:
515:
512:
499:
496:
493:
490:
487:
484:
481:
478:
475:
464:
463:
451:
448:
445:
442:
439:
436:
433:
430:
427:
424:
421:
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415:
412:
409:
406:
403:
400:
397:
394:
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388:
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382:
379:
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338:
333:
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308:
305:
302:
299:
296:
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281:
278:
275:
272:
269:
266:
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260:
257:
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248:
245:
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236:
233:
230:
227:
224:
221:
218:
215:
212:
185:
181:
177:
174:
167:
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156:
152:
149:
116:
111:
97:
94:
47:
42:
15:
13:
10:
9:
6:
4:
3:
2:
1156:
1145:
1142:
1140:
1137:
1136:
1134:
1123:
1118:
1114:
1112:0-13-181629-2
1108:
1104:
1103:Prentice Hall
1100:
1096:
1095:James Munkres
1092:
1091:
1087:
1083:
1080:
1078:
1075:
1073:
1070:
1068:
1065:
1063:
1060:
1058:
1055:
1053:
1050:
1049:
1045:
1043:
1027:
1024:
1021:
1015:
1009:
998:
994:
984:
980:
976:
972:
968:
964:
960:
934:
930:
904:
900:
896:
892:
888:
884:
880:
876:
873:such that 1/(
872:
868:
864:
860:
856:
852:
848:
844:
840:
836:
832:
828:
824:
820:
816:
812:
808:
805:} containing
804:
800:
796:
792:
788:
784:
780:
776:
772:
768:
764:
760:
756:
752:
749:. Then since
748:
744:
740:
736:
732:
728:
724:
720:
716:
712:
709:
708:neighbourhood
705:
701:
697:
693:
689:
685:
681:
680:connectedness
677:
673:
669:
665:
661:
657:
653:
649:
645:
644:
643:
642:
641:
639:
617:
597:
589:
570:
567:
564:
558:
552:
541:
540:
539:
538:
537:
534:
532:
528:
523:
521:
520:connectedness
513:
511:
494:
491:
488:
482:
476:
443:
437:
431:
428:
425:
416:
407:
404:
401:
395:
392:
386:
377:
374:
371:
358:
357:
356:
354:
336:
300:
294:
288:
285:
282:
273:
264:
261:
258:
252:
249:
243:
234:
231:
228:
222:
216:
203:
202:
201:
199:
175:
172:
158:
154:
150:
140:
136:
132:
114:
95:
89:
81:
77:
75:
71:
67:
63:
45:
30:
26:
22:
1121:
1098:
990:
982:
978:
974:
970:
966:
962:
957:will form a
932:
928:
902:
898:
894:
890:
886:
882:
878:
874:
870:
866:
862:
858:
854:
850:
846:
842:
838:
834:
830:
822:
818:
814:
810:
806:
802:
798:
794:
790:
786:
782:
778:
774:
770:
766:
762:
754:
750:
746:
742:
738:
737:}. Clearly,
734:
730:
726:
722:
718:
714:
710:
703:
699:
695:
687:
683:
667:
663:
659:
655:
651:
647:
635:
535:
527:contractible
524:
517:
465:
321:
200:defined by:
197:
134:
99:
73:
24:
18:
849:other than
765:containing
1133:Categories
1088:References
959:separation
769:such that
692:continuity
670:} is both
196:. The set
25:comb space
1016:×
993:homotopic
905:) ×
869:. Choose
713:(open in
618:⊂
598:⊂
559:×
510:deleted.
483:×
438:×
417:∪
396:×
387:∪
295:×
274:∪
253:×
244:∪
223:×
176:∈
129:with its
100:Consider
1099:Topology
1097:(1999).
1046:See also
885:. Since
853:. Then
809:. Since
761:element
717:) about
133:and let
29:subspace
21:topology
881:< 1/
861:,
588:closure
137:be the
1109:
676:closed
654:. Let
170:
162:
68:. The
857:= (1/
789:) = {
759:basis
1107:ISBN
927:and
745:) =
674:and
672:open
638:path
62:comb
23:, a
961:on
931:= (
694:of
139:set
31:of
1135::
1105:.
610:D
522:.
1115:.
1031:]
1028:1
1025:,
1022:0
1019:(
1013:}
1010:0
1007:{
983:p
981:{
979:f
975:U
973:(
971:f
967:U
965:(
963:f
944:R
933:r
929:B
914:R
903:r
899:A
895:x
891:U
889:(
887:f
883:n
879:r
875:n
871:r
867:D
863:z
859:n
855:s
851:p
847:s
843:U
841:(
839:f
835:U
833:(
831:f
823:U
819:p
817:{
815:f
811:x
807:x
803:p
801:{
799:f
795:U
791:p
787:U
785:(
783:f
779:V
775:U
773:(
771:f
767:x
763:U
755:V
753:(
751:f
747:p
743:x
741:(
739:f
735:p
733:{
731:f
727:x
723:x
719:p
715:R
711:V
704:p
702:{
700:f
696:f
688:p
686:{
684:f
668:p
666:{
664:f
660:D
656:f
652:D
648:p
574:]
571:1
568:,
565:0
562:(
556:}
553:0
550:{
498:)
495:1
492:,
489:0
486:[
480:}
477:0
474:{
462:.
450:)
447:}
444:0
441:{
435:]
432:1
429:,
426:0
423:[
420:(
414:)
411:]
408:1
405:,
402:0
399:[
393:K
390:(
384:}
381:)
378:1
375:,
372:0
369:(
366:{
337:2
332:R
307:)
304:}
301:0
298:{
292:]
289:1
286:,
283:0
280:[
277:(
271:)
268:]
265:1
262:,
259:0
256:[
250:K
247:(
241:)
238:]
235:1
232:,
229:0
226:[
220:}
217:0
214:{
211:(
198:C
184:}
180:N
173:n
166:|
159:n
155:/
151:1
148:{
135:K
115:2
110:R
46:2
41:R
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