Algebras of real-valued continuous functions in product spaces

Abstract

Let { X i : iϵI} be an arbitrary family of spaces, we say that the cartesian product X has the approximation property when C( X) coincides with the Algebra on X generated by the functions which depend on one variable. In this paper we study the problem of characterizing topologically when an arbitrary product space has the approximation property. We prove that if X is an uncountable pseudo-ℵ 1-compact P-space, then X× Y has the approximation property if, and only if, X× Y is pseudo-ℵ 1-compact. As a corollary we obtain the following characterization for P-spaces: Let X and Y be P-spaces, then X× Y has the approximation property if, and only if, X or Y is countable or X× Y is pseudo-ℵ 1-compact.