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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 βX,νX,
z-ultrafilters, C(X) and C∗(X), but all for a zero-set space, instead of for a
topological space.
There is another respect, besides greater generality, in which the theory of
zero-set spaces differs from that of topological spaces. Using the definitions of
subspace and product space which are obvious and natural for zero-set spaces,
this paper obtains, for such spaces, a number of results which are known
to fail for topological spaces. Most notably, a product of any number of
pseudocompact zero-set spaces is pseudocompact, even though the product of just
two pseudocompact topological spaces may fail to be pseudocompact. Also a
countable union of realcompact subspaces of a zero-set space is realcompact;
again the corresponding statement does not hold even for two topological
subspaces.
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