Compact subspaces of the space of separately continuous functions with the cross-uniform topology

TL;DR

This paper investigates the properties of certain topologies on the space of separately continuous functions, establishing conditions under which these topologies coincide and characterizing when compact spaces embed into these function spaces.

Contribution

It introduces and compares the cross-open and cross-uniform topologies on separately continuous functions, proving their equivalence under specific conditions and characterizing embeddings of compact spaces.

Findings

The cross-open and cross-uniform topologies coincide for pseudocompact spaces.

A compact space embeds into the function space if its weight is less than the sharp cellularity of the domain spaces.

The paper provides necessary and sufficient conditions for embeddings of compact spaces into these function spaces.

Abstract

We consider two natural topologies on the space $S(X\times Y,Z)$ of all separately continuous functions defined on the product of two topological spaces $X$ and $Y$ and ranged into a topological or metric space $X$. These topologies are the cross-open topology and the cross-uniform topology. We show that these topologies coincides if $X$ and $Y$ are pseudocompacts and $Z$ is a metric space. We prove that a compact space $K$ embeds into $S(X\times Y,Z)$ for infinite compacts $X$, $Y$ and a metrizable space $Z\supseteq\mathbb{R}$ if and only if the weight of $K$ is less than the sharp cellularity of both spaces $X$ and $Y$.