924:
707:
945:
913:
982:
955:
935:
538:
The above space is not homeomorphic to the same set of equivalence classes endowed with the obvious metric: i.e. regular
Euclidean distance for two points in the same interval, and the sum of the distances to the left hand point for points not in the same interval -- yielding a strictly coarser
521:
269:. Thus, if one has a countable base for a topology then one has a countable local base at every point, and hence every second-countable space is also a first-countable space. However any uncountable
132:
200:
300:, however, the properties of being second-countable, separable, and Lindelöf are all equivalent. Therefore, the lower limit topology on the real line is not metrizable.
985:
176:
156:
71:
1011:
332:. Second-countability is therefore a rather restrictive property on a topological space, requiring only a separation axiom to imply metrizability.
619:
973:
968:
592:
539:
topology than the above space. It is a separable metric space (consider the set of rational points), and hence is second-countable.
310:
963:
410:
76:
865:
388:
873:
1006:
358:
276:
Second-countability implies certain other topological properties. Specifically, every second-countable space is
944:
672:
241:
radii and whose centers have rational coordinates. This restricted set is countable and still forms a basis.
958:
325:
893:
888:
691:
679:
652:
612:
735:
662:
250:
923:
883:
835:
809:
657:
543:
293:
535:/~ is not first-countable at the coset of the identified points and hence also not second-countable.
181:
934:
730:
352:
345:
210:, the property of being second-countable restricts the number of open sets that a space can have.
928:
878:
799:
789:
667:
647:
207:
898:
527:
by identifying the left ends of the intervals - that is, identify 0 ~ 2 ~ 4 ~ … ~ 2k and so on.
916:
782:
740:
605:
588:
524:
285:
43:
696:
642:
296:
on the real line is first-countable, separable, and Lindelöf, but not second-countable. For
281:
372:
of a second-countable space is second-countable, although uncountable products need not be.
755:
750:
314:
277:
238:
222:
50:
845:
777:
270:
161:
141:
56:
394:
Any base for a second-countable space has a countable subfamily which is still a base.
1000:
855:
765:
745:
369:
317:
304:
948:
840:
760:
706:
307:, sequential compactness, and countable compactness are all equivalent properties.
297:
214:
17:
292:
has a countable subcover). The reverse implications do not hold. For example, the
938:
850:
397:
Every collection of disjoint open sets in a second-countable space is countable.
380:
329:
234:
218:
531:
is second-countable, as a countable union of second-countable spaces. However,
794:
725:
684:
321:
289:
254:
819:
230:
47:
229:) with its usual topology is second-countable. Although the usual base of
804:
772:
721:
628:
342:
135:
31:
580:, (1970) Addison-Wesley Publishing Company, Reading Massachusetts.
361:
of second-countable spaces need not be second-countable; however,
601:
73:
is second-countable if there exists some countable collection
237:, one can restrict to the collection of all open balls with
187:
178:
can be written as an union of elements of some subfamily of
82:
597:
303:
In second-countable spaces—as in metric spaces—
253:. A space is first-countable if each point has a countable
516:{\displaystyle X=\cup \cup \cup \dots \cup \cup \dotsb }
127:{\displaystyle {\mathcal {U}}=\{U_{i}\}_{i=1}^{\infty }}
413:
184:
164:
144:
79:
59:
27:
Topological space whose topology has a countable base
864:
828:
714:
635:
515:
202:. A second-countable space is said to satisfy the
194:
170:
150:
126:
65:
546:is not second-countable, but is first-countable.
355:of a second-countable space is second-countable.
348:of a second-countable space is second-countable.
249:Second-countability is a stronger notion than
613:
273:is first-countable but not second-countable.
8:
104:
90:
981:
954:
620:
606:
598:
583:John G. Hocking and Gail S. Young (1961).
257:. Given a base for a topology and a point
412:
186:
185:
183:
163:
143:
118:
107:
97:
81:
80:
78:
58:
556:
523:. Define an equivalence relation and a
261:, the set of all basis sets containing
53:. More explicitly, a topological space
407:Consider the disjoint countable union
324:. It follows that every such space is
7:
375:The topology of a second-countable T
313:states that every second-countable,
221:are second-countable. For example,
119:
25:
587:Corrected reprint, Dover, 1988.
1012:Properties of topological spaces
980:
953:
943:
933:
922:
912:
911:
705:
563:Willard, theorem 16.11, p. 112
504:
480:
468:
456:
450:
438:
432:
420:
195:{\displaystyle {\mathcal {U}}}
1:
311:Urysohn's metrization theorem
158:such that any open subset of
389:cardinality of the continuum
204:second axiom of countability
1028:
874:Banach fixed-point theorem
40:completely separable space
907:
703:
929:Mathematics portal
829:Metrics and properties
815:Second-countable space
517:
383:less than or equal to
265:forms a local base at
196:
172:
152:
128:
67:
36:second-countable space
518:
365:quotients always are.
197:
173:
153:
129:
68:
46:whose topology has a
884:Invariance of domain
836:Euler characteristic
810:Bundle (mathematics)
411:
294:lower limit topology
182:
162:
142:
77:
57:
894:Tychonoff's theorem
889:Poincaré conjecture
643:General (point-set)
208:countability axioms
123:
18:Second countability
879:De Rham cohomology
800:Polyhedral complex
790:Simplicial complex
513:
251:first-countability
192:
168:
148:
124:
103:
63:
994:
993:
783:fundamental group
576:Stephen Willard,
525:quotient topology
326:completely normal
280:(has a countable
171:{\displaystyle T}
151:{\displaystyle T}
66:{\displaystyle T}
44:topological space
16:(Redirected from
1019:
1007:General topology
984:
983:
957:
956:
947:
937:
927:
926:
915:
914:
709:
622:
615:
608:
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578:General Topology
564:
561:
522:
520:
519:
514:
336:Other properties
201:
199:
198:
193:
191:
190:
177:
175:
174:
169:
157:
155:
154:
149:
133:
131:
130:
125:
122:
117:
102:
101:
86:
85:
72:
70:
69:
64:
38:, also called a
21:
1027:
1026:
1022:
1021:
1020:
1018:
1017:
1016:
997:
996:
995:
990:
921:
903:
899:Urysohn's lemma
860:
824:
710:
701:
673:low-dimensional
631:
626:
573:
568:
567:
562:
558:
553:
409:
408:
404:
378:
338:
247:
223:Euclidean space
180:
179:
160:
159:
140:
139:
93:
75:
74:
55:
54:
28:
23:
22:
15:
12:
11:
5:
1025:
1023:
1015:
1014:
1009:
999:
998:
992:
991:
989:
988:
978:
977:
976:
971:
966:
951:
941:
931:
919:
908:
905:
904:
902:
901:
896:
891:
886:
881:
876:
870:
868:
862:
861:
859:
858:
853:
848:
846:Winding number
843:
838:
832:
830:
826:
825:
823:
822:
817:
812:
807:
802:
797:
792:
787:
786:
785:
780:
778:homotopy group
770:
769:
768:
763:
758:
753:
748:
738:
733:
728:
718:
716:
712:
711:
704:
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689:
688:
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639:
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633:
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627:
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617:
610:
602:
596:
595:
581:
572:
569:
566:
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552:
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540:
536:
512:
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461:
458:
455:
452:
449:
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443:
440:
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434:
431:
428:
425:
422:
419:
416:
403:
400:
399:
398:
395:
392:
376:
373:
368:Any countable
366:
356:
349:
341:A continuous,
337:
334:
271:discrete space
246:
243:
189:
167:
147:
121:
116:
113:
110:
106:
100:
96:
92:
89:
84:
62:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
1024:
1013:
1010:
1008:
1005:
1004:
1002:
987:
979:
975:
972:
970:
967:
965:
962:
961:
960:
952:
950:
946:
942:
940:
936:
932:
930:
925:
920:
918:
910:
909:
906:
900:
897:
895:
892:
890:
887:
885:
882:
880:
877:
875:
872:
871:
869:
867:
863:
857:
856:Orientability
854:
852:
849:
847:
844:
842:
839:
837:
834:
833:
831:
827:
821:
818:
816:
813:
811:
808:
806:
803:
801:
798:
796:
793:
791:
788:
784:
781:
779:
776:
775:
774:
771:
767:
764:
762:
759:
757:
754:
752:
749:
747:
744:
743:
742:
739:
737:
734:
732:
729:
727:
723:
720:
719:
717:
713:
708:
698:
695:
693:
692:Set-theoretic
690:
686:
683:
682:
681:
678:
674:
671:
670:
669:
666:
664:
661:
659:
656:
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653:Combinatorial
651:
649:
646:
644:
641:
640:
638:
634:
630:
623:
618:
616:
611:
609:
604:
603:
600:
594:
593:0-486-65676-4
590:
586:
582:
579:
575:
574:
570:
560:
557:
550:
545:
541:
537:
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483:
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396:
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382:
374:
371:
367:
364:
360:
357:
354:
350:
347:
344:
340:
339:
335:
333:
331:
327:
323:
319:
318:regular space
316:
312:
308:
306:
301:
299:
298:metric spaces
295:
291:
287:
283:
279:
274:
272:
268:
264:
260:
256:
252:
244:
242:
240:
236:
232:
228:
224:
220:
216:
211:
209:
206:. Like other
205:
165:
145:
137:
114:
111:
108:
98:
94:
87:
60:
52:
49:
45:
41:
37:
33:
19:
986:Publications
851:Chern number
841:Betti number
814:
724: /
715:Key concepts
663:Differential
584:
577:
559:
532:
528:
384:
362:
309:
302:
284:subset) and
275:
266:
262:
258:
248:
226:
217:" spaces in
215:well-behaved
212:
203:
39:
35:
29:
949:Wikiversity
866:Key results
381:cardinality
330:paracompact
328:as well as
305:compactness
235:uncountable
219:mathematics
138:subsets of
1001:Categories
795:CW complex
736:Continuity
726:Closed set
685:cohomology
571:References
379:space has
322:metrizable
290:open cover
255:local base
245:Properties
231:open balls
974:geometric
969:algebraic
820:Cobordism
756:Hausdorff
751:connected
668:Geometric
658:Continuum
648:Algebraic
585:Topology.
544:long line
511:⋯
508:∪
478:∪
475:⋯
472:∪
454:∪
436:∪
359:Quotients
315:Hausdorff
278:separable
120:∞
48:countable
939:Wikibook
917:Category
805:Manifold
773:Homotopy
731:Interior
722:Open set
680:Homology
629:Topology
402:Examples
353:subspace
286:Lindelöf
239:rational
32:topology
964:general
766:uniform
746:compact
697:Digital
370:product
288:(every
42:, is a
959:Topics
761:metric
636:Fields
591:
351:Every
213:Many "
741:Space
551:Notes
387:(the
346:image
282:dense
589:ISBN
542:The
363:open
343:open
136:open
51:base
34:, a
320:is
233:is
134:of
30:In
1003::
391:).
621:e
614:t
607:v
533:X
529:X
505:]
502:1
499:+
496:k
493:2
490:,
487:k
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481:[
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466:5
463:,
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457:[
451:]
448:3
445:,
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439:[
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430:1
427:,
424:0
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418:=
415:X
385:c
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188:U
166:T
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109:i
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