Supports of continuous functions

Abstract

Gillman and Jerison have shown that when X is a realcompact space, every function in C ( X ) C(X) that belongs to all the free maximal ideals has compact support. A space with the latter property will be called μ \mu -compact. In this paper we give several characterizations of μ \mu -compact spaces and also introduce and study a related class of spaces, the ψ \psi -compact spaces; these are spaces X with the property that every function in C ( X ) C(X) with pseudocompact support has compact support. It is shown that every realcompact space is ψ \psi -compact and every ψ \psi -compact space is μ \mu -compact. A family F \mathcal {F} of subsets of a space X is said to be stable if every function in C ( X ) C(X) is bounded on some member of F \mathcal {F} . We show that a completely regular Hausdorff space is realcompact if and only if every stable family of closed subsets with the finite intersection property has nonempty intersection. We adopt this condition as the definition of realcompactness for arbitrary (not necessarily completely regular Hausdorff) spaces, determine some of the properties of these realcompact spaces, and construct a realcompactification of an arbitrary space.