Topological Spaces in Which Blumberg's Theorem Holds

H. E. White
1974 Proceedings of the American Mathematical Society  
H. Blumberg proved that, if/is a real-valued function defined on the real line R, then there is a dense subset D of R such that/|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 rr-disjoint pseudo-base if and only if X is a Baire space. Then we identify some classes of topological spaces which have (/-disjoint
more » ... ave (/-disjoint pseudo-bases. Also, we show that a certain class of iocally 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 a-favorable, and pseudocomplete; the other is regular and hereditarily Lindelöf.
doi:10.2307/2040456 fatcat:ty2h4ccwlvbjtpulwqnd56tvju