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.
176:
156:
71:
985:
332:. Second-countability is therefore a rather restrictive property on a topological space, requiring only a separation axiom to imply metrizability.
1011:
539:
topology than the above space. It is a separable metric space (consider the set of rational points), and hence is second-countable.
410:
76:
619:
973:
968:
592:
310:
17:
963:
276:
Second-countability implies certain other topological properties. Specifically, every second-countable space is
865:
241:
radii and whose centers have rational coordinates. This restricted set is countable and still forms a basis.
388:
873:
1006:
358:
535:/~ is not first-countable at the coset of the identified points and hence also not second-countable.
181:
672:
210:, the property of being second-countable restricts the number of open sets that a space can have.
958:
944:
325:
527:
by identifying the left ends of the intervals - that is, identify 0 ~ 2 ~ 4 ~ … ~ 2k and so on.
893:
691:
679:
652:
612:
888:
735:
662:
250:
296:
on the real line is first-countable, separable, and Lindelöf, but not second-countable. For
883:
835:
809:
657:
543:
372:
of a second-countable space is second-countable, although uncountable products need not be.
293:
8:
730:
352:
345:
934:
928:
898:
878:
799:
789:
667:
647:
207:
161:
141:
56:
394:
Any base for a second-countable space has a countable subfamily which is still a base.
923:
916:
782:
740:
605:
588:
524:
43:
948:
285:
696:
642:
307:, sequential compactness, and countable compactness are all equivalent properties.
281:
292:
has a countable subcover). The reverse implications do not hold. For example, the
755:
750:
314:
277:
238:
222:
50:
938:
397:
Every collection of disjoint open sets in a second-countable space is countable.
845:
777:
531:
is second-countable, as a countable union of second-countable spaces. However,
270:
1000:
855:
765:
745:
369:
317:
304:
840:
760:
706:
297:
214:
229:) with its usual topology is second-countable. Although the usual base of
850:
380:
329:
234:
218:
794:
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321:
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254:
819:
230:
47:
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,
73:
is second-countable if there exists some countable collection
237:, one can restrict to the collection of all open balls with
178:
can be written as an union of elements of some subfamily of
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
515:
202:. A second-countable space is said to satisfy the
194:
170:
150:
126:
65:
998:
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.
104:
90:
981:
954:
620:
606:
583:John G. Hocking and Gail S. Young (1961).
257:. Given a base for a topology and a point
523:. Define an equivalence relation and a
261:, the set of all basis sets containing
53:. More explicitly, a topological space
14:
999:
407:Consider the disjoint countable union
324:. It follows that every such space is
601:
375:The topology of a second-countable T
313:states that every second-countable,
335:
221:are second-countable. For example,
24:
187:
119:
82:
25:
1023:
587:Corrected reprint, Dover, 1988.
1012:Properties of topological spaces
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953:
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911:
705:
563:Willard, theorem 16.11, p. 112
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420:
195:{\displaystyle {\mathcal {U}}}
13:
1:
570:
311:Urysohn's metrization theorem
244:
158:such that any open subset of
627:
389:cardinality of the continuum
204:second axiom of countability
7:
401:
10:
1028:
874:Banach fixed-point theorem
40:completely separable space
907:
864:
828:
714:
703:
635:
550:
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:
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:
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578:General Topology
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336:Other properties
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133:
131:
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38:, also called a
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18:Second-countable
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1016:
997:
996:
995:
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921:
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899:Urysohn's lemma
860:
824:
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673:low-dimensional
631:
626:
573:
568:
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553:
412:
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378:
338:
247:
223:Euclidean space
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28:
23:
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15:
12:
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5:
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846:Winding number
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778:homotopy group
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376:
373:
368:Any countable
366:
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349:
341:A continuous,
337:
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271:discrete space
246:
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147:
121:
116:
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62:
26:
9:
6:
4:
3:
2:
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1004:
1002:
987:
979:
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962:
961:
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946:
942:
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936:
932:
930:
925:
920:
918:
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895:
892:
890:
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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:
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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:
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723:
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693:
692:Set-theoretic
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683:
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653:Combinatorial
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630:
623:
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593:0-486-65676-4
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339:
333:
331:
327:
323:
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318:regular space
316:
312:
308:
306:
301:
299:
298:metric spaces
295:
291:
287:
283:
279:
274:
272:
268:
264:
260:
256:
252:
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236:
232:
228:
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211:
209:
206:. Like other
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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
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391:).
621:e
614:t
607:v
533:X
529:X
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496:k
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