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

TL;DR

This work analyzes cross-open and cross-uniform topologies on the space S(X×Y,Z) of separately continuous functions, proving their coincidence for pseudocompact X,Y and metrizable Z, and establishing a sharp, weight-based criterion for embedding compact spaces into S(X×Y,Z) when X,Y are infinite compacts and Z is metrizable and contains ℝ. By reducing to Eberlein compacts and bounding topological weights via sharp cellularity, the authors derive a complete description: a compact K embeds into S(X×Y,Z) if and only if $w(K)<\min\{c^♯(X),c^♯(Y)\}$. They further develop constructive embeddings using Tychonoff cubes and explicit witness functions, connecting the geometry of X and Y to the compact subspaces of S(X×Y,Z). The results advance understanding of the structure of compact subspaces in spaces of separately continuous functions and pose several open problems related to Rosenthal compacts and broader topological contexts.

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 $Y$.

Theorems & Definitions (23)

• Proposition 2.1
• proof
• Proposition 3.1
• proof
• Theorem 3.2
• proof
• Proposition 4.1
• proof
• Lemma 4.2
• proof