Topological spaces in which Blumberg's theorem holds

Author:
H. E. White

Journal:
Proc. Amer. Math. Soc. **44** (1974), 454-462

MSC:
Primary 54C05

DOI:
https://doi.org/10.1090/S0002-9939-1974-0341379-3

MathSciNet review:
0341379

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Abstract: H. Blumberg proved that, if is a real-valued 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 pseudo-base if and only if is a Baire space. Then we identify some classes of topological spaces which have -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 -favorable, and pseudo-complete; the other is regular and hereditarily Lindelöf.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1974-0341379-3

Keywords:
Baire space,
pseudo-base,
-disjoint pseudo-base,
density topology

Article copyright:
© Copyright 1974
American Mathematical Society