Rings of functions determined by zero-sets

Abstract

Pseudocompactness and realcompactness can be defined in a more natural and more general setting than the usual one. One part of what is done here is simply to point out that much of the theory of rings of continuous functions applies without essential change in more general circumstances. The discussion includes, for example, analogues of X, vX, ^-ultrafilters, C(X) and C*(X\ but all for a zero-set space, instead of for a topological space.