Diagonals of separately continuous maps with values in box products

TL;DR

This paper proves that for certain spaces, Baire-one maps can be extended to separately continuous maps on the product space with the box topology, preserving the diagonal values.

Contribution

It establishes the existence of separately continuous maps with prescribed diagonals for maps into box products of equiconnected metrizable spaces.

Findings

Existence of separately continuous extensions for Baire-one maps.

Extension preserves the diagonal map.

Applicable to paracompact connected spaces and box products.

Abstract

We prove that if $X$ is a paracompact connected space and $Z=\prod_{s\in S}Z_s$ is a product of a family of equiconnected metrizable spaces endowed with the box topology, then for every Baire-one map $g:X\to Z$ there exists a separately continuous map $f:X^2\to Z$ such that $f(x,x)=g(x)$ for all $x\in X$.