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

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 ${\wedge }_f(x)=\underset{y\in Y}{\inf }f(x,y)$ and $\vee {}_f(x)=\underset{y\in Y}{\sup }f(x,y)$ 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 $h(x)=\underset{n\in \mathbb{N}}{\min }{u}_n(x)$ and $g(x)=\underset{n\in \mathbb{N}}{\max }{u}_n(x)$ 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 $\vee {}_f=h$.

19 pages