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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Topological spaces in which Blumberg's theorem holds


Author: H. E. White
Journal: Proc. Amer. Math. Soc. 44 (1974), 454-462
MSC: Primary 54C05
MathSciNet review: 0341379
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Abstract: H. Blumberg proved that, if $ f$ is a real-valued function defined on the real line $ R$, then there is a dense subset $ D$ of $ R$ such that $ f\vert D$ is continuous. J. C. Bradford and C. Goffman showed [3] that this theorem holds for a metric space $ X$ if and only if $ X$ is a Baire space. In the present paper, we show that Blumberg's theorem holds for a topological space $ X$ having a $ \sigma $-disjoint pseudo-base if and only if $ X$ is a Baire space. Then we identify some classes of topological spaces which have $ \sigma $-disjoint pseudo-bases. Also, we show that a certain class of locally compact, Hausdorff spaces satisfies Blumberg's theorem. Finally, we describe two Baire spaces for which Blumberg's theorem does not hold. One is completely regular, Hausdorff, cocompact, strongly $ \alpha $-favorable, and pseudo-complete; the other is regular and hereditarily Lindelöf.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1974-0341379-3
PII: S 0002-9939(1974)0341379-3
Keywords: Baire space, pseudo-base, $ \sigma $-disjoint pseudo-base, density topology
Article copyright: © Copyright 1974 American Mathematical Society