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.

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, 225-235