On stable pairs of Hahn and extremal sections of separately continuous functions on the products with a scattered multiplier

TL;DR

This paper investigates the properties of minimal and maximal sections of separately continuous functions on product spaces, establishing conditions under which these sections form stable pairs of Hahn and constructing functions with prescribed sections.

Contribution

It characterizes when the pair of minimal and maximal sections of separately continuous functions form stable pairs of Hahn, and constructs such functions for given stable pairs.

Findings

Pairs of sections are stable under certain topological conditions.

Existence of separately continuous functions with prescribed sections.

Characterization of stable pairs of Hahn in product spaces.

Abstract

The minimal and the maximal sections $\wedge_f,\vee\!_f:X\to\overline{\mathbb R}$ of a function $f:X\times Y\to\overline{\mathbb R}$ are defined by $\wedge_f(x)=\inf\limits_{y\in Y}f(x,y)$ and $\vee\!\!_f(x)=\sup\limits_{y\in Y}f(x,y)$ for any $x\in X$. A pair $(g,h)$ of functions on $X$ is called a stable pair of Hahn if there exists a sequence of continuous functions $u_n$ on $X$ such that $h(x)=\min\limits_{n\in\mathbb{N}}u_n(x)$ and $g(x)=\max\limits_{n\in\mathbb{N}}u_n(x)$ for any $x\in X$. Evidently, every stable pair of Hahn is a countable pair of Hahn, and hence a pair of Hahn. We prove that for any separately continuous function $f$ on the product of compact spaces $X$ and $Y$ such that $Y$ is scattered and at least one of them has the countable chain property, the pair $(\wedge_f,\vee\!_f)$ is a stable pair of Hahn. We prove that for any stable pair of Hahn $(g,h)$ on the product of a topological space $X$ and an infinity completely regular space $Y$ there exists a separately continuous function $f$ on $X\times Y$ such that $\wedge_f=g$ and $\vee\!_f=h$.