Factoring functions on Cartesian products

N. Noble, Milton Ulmer
1972 Transactions of the American Mathematical Society  
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
more » ... ach 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.
doi:10.1090/s0002-9947-1972-0288721-2 fatcat:5m2m6oj6cnaopm5qynvwbsjvai