BOUNDED SUBSETS AND WEAK REALCOMPACTNESS CONDITIONS

Abstract

A subset A of X is bounded if every continuous real-valued function on X is bounded on A. A completely regular Hausdorff space X is said to have the bz-property if every bounded subset of X is contained in a bounded zero subset of X. In this paper, we study the bz-property and its relation to other well known topological properties. We also introduce some new topological properties, all weaker than realcompactness, that are related to the bz-property. The origin of the bz-property lies in a measure-theoretic problem.