On stable pairs of Hahn and extremal sections of separately continuous functions on the products with a scattered multiplier
Oleksandr Maslyuchenko, Anastasiia Lianha
TL;DR
This work studies stable pairs of Hahn and the extremal sections of separately continuous functions on product spaces, focusing on cases where the second factor is a scattered compact. It characterizes stable pairs via first stable Baire class notions and stable convergence, and proves that for separable $X$ and scattered compact $Y$ the extremal pair $(\wedge_f,\vee_f)$ is stable; it also shows that every stable Hahn pair on $X\times Y$ arises from some separately continuous $f$. A central contribution is the constructive framework that, using disjoint-support techniques and Hilbert cube embeddings, yields explicit $f$ realizing a given stable pair, advancing the understanding of how extremal sections can be realized in broad topological settings.
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 $\vee\!_f=h$.