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 $X$ is pseudo-${\aleph _1}$-compact. It is also shown that a product space is pseudo-${\aleph _1}$-compact 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.