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A space is locally
Hausdorff exactly if it can be written as a union of Hausdorff open subspaces. And in a locally Hausdorff space each point belongs to some Hausdorff dense open subspace.
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There are locally
Hausdorff spaces where a sequence has more than one limit. This can never happen for a Hausdorff space.
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576:. The converse is not true in general. For example, an infinite set with the
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819:, Lecture Notes in Math., vol. 1488, Springer, Berlin, pp. 298–305,
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Niefield, Susan B. (1991), "Weak products over a locally
Hausdorff locale",
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is locally
Hausdorff at every point, and is therefore
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and therefore the space is not locally
Hausdorff at
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671:is Hausdorff. This follows from the fact that if
944:
884:Proceedings of the American Mathematical Society
852:Cahiers de topologie et géométrie différentielle
306:is not Hausdorff, but it is locally Hausdorff.
880:"Manifolds: Hausdorffness versus homogeneity"
877:
878:Baillif, Mathieu; Gabard, Alexandre (2008).
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848:"A note on the locally Hausdorff property"
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622:that is locally Hausdorff at some point
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287:is locally Hausdorff (it is in fact
798:topological groups are Hausdorff).
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268:Examples and sufficient conditions
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591:Every locally Hausdorff space is
569:Every locally Hausdorff space is
515:any neighbourhood of it contains
302:of differentiable functions on a
469:is a Hausdorff neighbourhood of
588:that is not locally Hausdorff.
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1:
907:10.1090/S0002-9939-07-09100-9
801:
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817:Category theory (Como, 1990)
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331:particular point topology
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846:Niefield, S. B. (1983).
376:is locally Hausdorff at
693:{\displaystyle y\in G,}
644:{\displaystyle x\in G,}
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333:with particular point
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124:(completely Hausdorff)
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462:{\displaystyle \{p\}}
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304:differential manifold
285:line with two origins
277:is locally Hausdorff.
252:if every point has a
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492:For any other point
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291:) but not Hausdorff.
724:to itself carrying
329:be a set given the
142:(regular Hausdorff)
936:, Proposition 3.5.
924:, Proposition 3.4.
825:10.1007/BFb0084228
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760:{\displaystyle y,}
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551:{\displaystyle x.}
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508:{\displaystyle x,}
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485:{\displaystyle p.}
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443:and the singleton
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392:{\displaystyle p,}
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349:{\displaystyle p.}
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289:locally metrizable
240:, in the field of
195:(completely normal
177:(normal Hausdorff)
25:topological spaces
953:Separation axioms
780:{\displaystyle G}
737:{\displaystyle x}
717:{\displaystyle G}
664:{\displaystyle G}
620:topological group
611:{\displaystyle G}
578:cofinite topology
528:{\displaystyle p}
436:{\displaystyle X}
412:{\displaystyle p}
369:{\displaystyle X}
322:{\displaystyle X}
262:subspace topology
250:locally Hausdorff
246:topological space
234:
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215:(perfectly normal
20:Separation axioms
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890:(3): 1105–1111.
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275:Hausdorff space
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258:Hausdorff space
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34:classification
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934:Niefield 1983
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922:Niefield 1983
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702:homeomorphism
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254:neighbourhood
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897:math/0609098
887:
883:
873:
858:(1): 87–95.
855:
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841:
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597:
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568:
565:
249:
235:
112:completely T
52:(Kolmogorov)
912:, Lemma 4.2
868:, Lemma 3.2
296:etale space
238:mathematics
159:(Tychonoff)
88:(Hausdorff)
802:References
562:Properties
356:The space
260:under the
256:that is a
32:Kolmogorov
864:2681-2398
682:∈
633:∈
106:(Urysohn)
70:(Fréchet)
947:Category
298:for the
242:topology
833:1173020
226:History
152:3½
862:
831:
794:(and T
419:is an
399:since
273:Every
211:
191:
173:
138:
120:
99:½
84:
66:
48:
892:arXiv
704:from
651:then
618:is a
593:sober
586:space
580:is a
300:sheaf
860:ISSN
309:Let
294:The
283:The
244:, a
902:doi
888:136
821:doi
767:so
744:to
598:If
423:in
236:In
23:in
949::
900:.
886:.
882:.
856:24
854:.
850:.
829:MR
827:,
595:.
264:.
910:.
904::
894::
866:.
836:.
823::
796:1
791:1
789:T
775:G
755:,
752:y
732:x
712:G
688:,
685:G
679:y
659:G
639:,
636:G
630:x
606:G
584:1
582:T
573:1
571:T
546:.
543:x
523:p
503:,
500:x
480:.
477:p
457:}
454:p
451:{
431:X
407:p
387:,
384:p
364:X
344:.
341:p
317:X
206:6
203:T
186:5
183:T
168:4
165:T
148:T
133:3
130:T
115:2
97:2
94:T
79:2
76:T
61:1
58:T
43:0
40:T
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