Stone-Čech Compactifications of Products

As is well known from the work of Tychonoff [10], Stone [8], and Čech [1], every completely regular space X can be imbedded as a dense subspace of a compact Hausdorff space β(X) in such a way that continuous (real valued and bounded) functions on X extend continuously to β(X); indeed the resulting compactification of X, the Stone-Čech com-pactification, is uniquely determined by just these properties. For a set of completely regular spaces, the naturally induced imbedding of their product PX α as a dense subspace of Pβ(X α ) yields a compactification of their product, and the question arises as to when one can identify(2) this with the Stone-Čech compactification. The main purpose of this paper is to show that aside from a trivial case, this identification is possible if and only if PX α is pseudo-compact(3) [5], i.e., if and only if every real valued continuous function on it is bounded, or, equivalently, every bounded continuous function assumes its bounds(4). A side result of the investigation is the fact that every product of uncountably many compact spaces, each having at least two points, is the Stone-Čech compactification of certain proper subspaces, yielding a fairly accessible body of nontrivial Stone-Čech compactifications. Finally we shall give several conditions sufficient to insure that a product of pseudo-compact spaces be pseudo-compact, and briefly discuss a related Question.