Extension of mappings from the product of pseudocompact spaces

Abstract

Let X and Y be pseudocompact spaces and let a function Φ:X×Y→R be separately continuous. The following conditions are equivalent: (1) there is a dense Gδ subset of D⊂Y such that Φ is continuous at every point of X×D (Namioka property); (2) Φ is quasicontinuous; (3) Φ extends to a separately continuous function on βX×βY. This theorem makes it possible to combine studies of the Namioka property and generalizations of the Eberlein-Grothendieck theorem on the precompactness of subsets of function spaces. We also obtain a characterization of separately continuous functions on the product of several pseudocompact spaces extending to separately continuous functions on products of Stone–Čech extensions of spaces. These results are used to study groups and Mal'tsev spaces with separately continuous operations.