Topological spaces in which Blumberg’s theorem holds

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 $f|D$ 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.