Topological spaces in which Blumberg's theorem holds
Author:
H. E. White
Journal:
Proc. Amer. Math. Soc. 44 (1974), 454462
MSC:
Primary 54C05
MathSciNet review:
0341379
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Abstract: H. Blumberg proved that, if is a realvalued function defined on the real line , then there is a dense subset of such that is continuous. J. C. Bradford and C. Goffman showed [3] that this theorem holds for a metric space if and only if is a Baire space. In the present paper, we show that Blumberg's theorem holds for a topological space having a disjoint pseudobase if and only if is a Baire space. Then we identify some classes of topological spaces which have disjoint pseudobases. 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 favorable, and pseudocomplete; the other is regular and hereditarily Lindelöf.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939197403413793
PII:
S 00029939(1974)03413793
Keywords:
Baire space,
pseudobase,
disjoint pseudobase,
density topology
Article copyright:
© Copyright 1974
American Mathematical Society
