10 - Continuous functions on product spaces

Summary

We begin this chapter with a proof that under suitable hypotheses a function f continuous on a product space X_I (or even on some of its dense subspaces) is effectively defined on a ‘small’ subproduct (in the sense that there are ‘small’ J ⊂ I and continuous g on X_J such that f = g°π_J). We apply this result, together with some of the product-space theorems of Chapter 3, to obtain an identification in concrete form of the Stone–Čech compactification, the Hewitt realcompactification, and the Dieudonné topological completion of certain completely regular Hausdorff spaces (these terms are defined in Appendix B). Such an opportunity occurs infrequently since the Stone–Čech compactification of a space X, whose elements are normally described as maximal filters in the class of zero-sets of X, is defined by an appeal to (a variant of) Zorn's lemma; it does not arise in practice as a readily identifiable, familiar space.

Among the specific results proved are these (cf. Corollary 10.7 and Theorem 10.17): in a product of metrizable spaces, and in a product of compact spaces, every ω⁺-Σ-space is C-embedded. We conclude with an example, one of several due to Ulmer, showing that there do exist ω⁺-Σ-products not C-embedded in X_I.

Definition. Let Y ⊂ X_I, J ⊂ I, let Z be a set and f : Y → Z.