Metric spaces in which Blumberg's theorem holds

J. C. Bradford, Casper Goffman
1960 Proceedings of the American Mathematical Society  
l], showed that for every real function/ on the real line R there is a dense set DER such that/ is continuous on D relative to D. The purpose of this paper is to characterize the metric spaces in which Blumberg's theorem holds. Let 5 be a metric space and E a subset of 5. A point xES is said to be of the second category relative to E if every sphere of center x contains a subset of E which is of the second category in S. Otherwise, x is said to be of the first category relative to E. S is said
more » ... ve to E. S is said to be of homogeneous second category if every xES is of the second category relative to S. We prove the Theorem. If the metric space S is of homogeneous second category then for every real function f on S there is a dense DES such that f is continuous on D relative to D. Conversely, if S is not of homogeneous second category there is a real function f on S which is not continuous on any dense DES relative to D. Proof. We prove the converse first. Let 5 be a metric space which is not of homogeneous second category. There is then a sphere KES
doi:10.1090/s0002-9939-1960-0146310-1 fatcat:gavwvw3tund7hnrlnoleple3mu