The cross-topology and Lebesgue triples

TL;DR

This paper investigates the properties of the cross topology on product spaces, demonstrating the existence of separately continuous functions that are not pointwise limits of continuous functions, and identifying conditions under which such limits exist.

Contribution

It establishes the existence of separately continuous functions not approximable by continuous functions in certain spaces and characterizes when pointwise limits of continuous functions are guaranteed.

Findings

Existence of separately continuous functions not approximable by continuous functions in certain spaces.

Every separately continuous function is a pointwise limit of continuous functions on specific product spaces.

Results apply to spaces including all Euclidean spaces and certain zero-dimensional metrizable spaces.

Abstract

The cross topology $\gamma$ on a product of topological spaces $X$ and $Y$ is the collection of all sets $G\subseteq X\times Y$ such that the intersection of $G$ with every vertical line and every horizontal line is an open subset of either vertical or horizontal line, respectively. For spaces $X$ and $Y$ from a wide class, which includes all spaces $\mathbb R^n$, we prove that there exists a separately continuous mapping $f:X\times Y\to (X\times Y,\gamma)$ which is not a pointwise limit of a sequence of continuous functions. Also we prove that every separately continuous mapping is a pointwise limit of a sequence of continuous mappings, if it is defined on the product of a strongly zero-dimensional metrizable and a topological space and acts into a topological space.