The Namioka property of $KC$-functions and Kempisty spaces

TL;DR

This paper investigates the properties of Kempisty spaces, showing that Valdivia compact spaces are Kempisty spaces and that their finite or infinite products also retain this property.

Contribution

It establishes that Valdivia compact spaces are Kempisty spaces and that the product of any family of compact Kempisty spaces is also Kempisty, expanding understanding of these spaces.

Findings

Valdivia compact spaces are Kempisty spaces

Product of any family of compact Kempisty spaces is Kempisty

Properties of Kempisty spaces are characterized in the context of $KC$-functions

Abstract

A topological space $Y$ is called a Kempisty space if for any Baire space $X$ every function $f:X\times Y\to\mathbb R$, which is quasi-continuous in the first variable and continuous in the second variable has the Namioka property. Properties of compact Kempisty spaces are studied in this paper. In particular, it is shown that any Valdivia compact is a Kempisty space and the cartesian product of an arbitrary family of compact Kempisty spaces is a Kempisty space.