Compact subspace of products of linearly ordered spaces and co-Namioka spaces

TL;DR

The paper proves that for certain compact spaces and Baire spaces, separately continuous functions are jointly continuous on a dense G_delta subset, extending the understanding of continuity in product spaces.

Contribution

It introduces a new class of compact subspaces of product spaces of linearly ordered spaces where separate continuity implies joint continuity on a dense G_delta set.

Findings

Existence of dense G_delta sets where functions are jointly continuous

Extension of Namioka property to specific compact subspaces

Characterization of co-Namioka spaces in this context

Abstract

It is shown that for any Baire space $X$, linearly ordered compact spaces $Y_1,\dots, Y_n$, compact space $Y\subseteq Y_1\times\cdots \times Y_n$ such that for every parallelepiped $W\subseteq Y_1\times\cdots \times Y_n$ the set $Y\cap W$ is connected, and separately continuous mapping $f:X\times Y\to\mathbb R$ there exists a dense in $X$ $G_\delta$-set $A\subseteq X$ such that $f$ is jointly continuous at every point of $A\times Y$.