© 1996 Copyright by Melvin Heriksen and R.G. Woods. All rights reserved.

Question

Cardinality of the ring of continuous functions on a product of two Tychonoff spaces Melvin Heriksen and R.G. Woods

As usual, let C(X) denote the ring of continuous functions on a topological space X, C^*(X) its subring of bounded elements, and let |S| denote the cardinality of set S. Consider the following assertion about Tychonoff spaces X and Y

It can be shown with the aid of the Stone-Weierstrass theorem that (*) holds if X and Y are compact. Indeed, using this latter and the fact that |C(X)| is always equal to |C^*(X)| = |C(\beta(X))| (where \beta(X) denotes the Stone-Cech compactification of X), it follows that (*) holds whenever |\beta(X ×Y)| = |\betaX| ·|\beta(Y)|; in particular when \beta(X ×Y) = \beta(X) ×\beta(Y). So, by a well known theorem of Glicksberg, (*) holds when X ×Y is pseudocompact; that is when C(X ×Y) = C^*(X ×Y).

Is there a characterization of those pairs of Tychonoff spaces for which (*) holds? Assuming a set-theoretic hypothesis a bit weaker than GCH, we are able to find a pair of Tychonoff spaces for which (*) fails, but whether there is a pair of such spaces in ZFC remains open.