Factoring functions on Cartesian products

Abstract

A function on a product space is said to depend on countably many coordinates if it can be written as a function defined on some countable subproduct composed with the projection onto that subproduct.It is shown, for X a completely regular Hausdorff space having uncountably many nontrivial factors, that each continuous real-valued function on X depends on countably many coordinates if and only if A" is pseudo-Xi-compact.It is also shown that a product space is pseudo-Kicompact if and only if each of its finite subproducts is.(This fact derives from a more general theorem which also shows, for example, that a product satisfies the countable chain condition if and only if each of its finite subproducts does.)All of these results are generalized in various ways. Introduction.A function/on a product space X=Y\aeA Xa is said to depend on countably many coordinates if, for some countable subset C of A, and some function g on Ylaec ^a>/is equal to g nc, where nc is the projection from A'to Ylaec Xa.The problem of determining conditions on a product space A'and a range space Y under which each continuous function from A" to Y will depend on countably many coordinates has developed a considerable literature which is very nicely summarized in [E], with one exception.That exception is the recent paper [V] by Vidossich in which it is shown that any uniformly continuous function from a subspace Z of a product space into a metric space can be factored as the composition of a projection onto some countable subproduct with a function uniformly continuous on the projection of Z.As applied to real-valued functions, all of these results, except that of Vidossich, are generalized by our main theorem, that if X is pseudo-K1-compact, then each continuous real-valued function on X depends upon countably many coordinates.If X is the product of uncountably many nontrivial completely regular Hausdorff spaces, then the converse also holds.(For any infinite cardinal X, a space is said to be pseudo-X-compact provided each locally finite family of open subsets has cardinality less than X; pseudocompact spaces, Lindelf