On the sequential closure of the set of continuous functions in the space of separately continuous functions

TL;DR

This paper proves that in certain topological spaces, every separately continuous function with a zero-dimensional image can be approximated by jointly continuous functions within a specific topology.

Contribution

It establishes a sequential closure result for the set of continuous functions in the space of separately continuous functions under layer-wise uniform convergence.

Findings

Separately continuous functions with zero-dimensional images are limits of continuous functions.

The result applies to functions between separable metrizable spaces and a metrizable topological group.

The topology considered is generated by layer-wise uniform convergence.

Abstract

For separable metrizable spaces $X,Y$ and a metrizable topological group $Z$ by $S(X\times Y,Z)$ we denote the space of all separately continuous functions $f:X\times Y\to Z$ endowed with the topology of layer-wise uniform convergence, generated by the subbase consisting of the sets $[K_X\times K_Y,U]=\{f\in S(X\times Y,Z):f(K_X\times K_Y)\subset U\}$, where $U$ is an open subset of $Z$ and $K_X\subset X$, $K_Y\subset Y$ are compact sets one of which is a singleton. We prove that every separately continuous function $f:X\times Y\to Z$ with zero-dimensional image $f(X\times Y)$ is a limit of a sequence of jointly continuous functions in the topology of layer-wise uniform convergence.