Extension of mappings from the product of pseudocompact spaces

TL;DR

This paper explores conditions under which separately continuous functions on products of pseudocompact spaces extend to their Stone-Čech compactifications, linking the Namioka property, quasicontinuity, and extensions.

Contribution

It establishes the equivalence of several conditions for separately continuous functions on pseudocompact spaces and extends these results to products and algebraic structures.

Findings

Equivalence of Namioka property, quasicontinuity, and extension to Stone-Čech products.

Characterization of separately continuous functions on products of multiple pseudocompact spaces.

Application to groups and Mal'tsev spaces with separately continuous operations.

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.