Bounded Subsets and Weak Realcompactness Conditions

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. Mathematics Subject Classification (1991): 5450, 54F99, 54G20 Keywords: bounded set, zero set, (weak) bz-space, (weak) m-space, space of pseudo countable type, unbounded point, untractable point, special sets defined by functions, counterexamples, completely regular, study, Lie Quaestiones Mathematicae 24(2) 2001, 225-23