Extension of mappings from the product of pseudocompact spaces

Abstract

Let $X$ and $Y$ be pseudocompact spaces and let the function $\Phi :X\times Y\to \mathbb{R}$ be separately continuous. The following conditions are equivalent: (1) there is a dense $G_{\delta }$ subset of $D\subset Y$ so that $\Phi$ is continuous at every point of $X\times D$ (Namioka property); (2) $\Phi$ is quasicontinuous; (3) $\Phi$ extends to a separately continuous function on $\beta X\times \beta Y$. 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-Cech extensions of spaces. These results are used to study groups and Mal'tsev spaces with separately continuous operations.

Evgenii Reznichenko Extension of mappings from the product of pseudocompact spaces Topology and its Applications Available online 4 November 2022, 108329 In Press, Journal Pre-proof Topology and its Applications