A little more on the product of two pseudocompact spaces

Abstract

( LÓDŹ) 0. Introduction. The main aim of this paper is to answer the question of when βX × βY is the Wallman compactification of X × Y with respect to the normal base consisting of the zero-sets of all those continuous real functions defined on X ×Y which are continuously extendable over βX ×βY . In passing, we shall obtain several new conditions which are necessary and sufficient for X × Y to be pseudocompact.To provide a framework for our discussion, let us recall that a normal base D for a Tikhonov space X is a base for the closed sets of X which is stable under finite unions and finite intersections and has the following properties: All the spaces considered below are assumed to be completely regular and Hausdorff. As usual, the symbol C(X) will stand for the algebra of continuous real functions defined on X, and C * (X) for the subalgebra of C(X) consisting of bounded functions.