Separately continuous functions on products and its dependence on $\aleph$ coordinates

TL;DR

This paper explores the conditions under which separately continuous functions on product spaces depend on a limited number of coordinates, focusing on the influence of the cardinality  on this dependence.

Contribution

It provides necessary and sufficient conditions for the dependence of separately continuous functions on at most  coordinates in product spaces.

Findings

Characterization of topological spaces where functions depend on  coordinates

Conditions linking the structure of product spaces to coordinate dependence

Extension of classical results to spaces with arbitrary  cardinality

Abstract

It is investigated necessary and sufficient conditions on topological spaces $X=\prod\limits _{s\in S}X_s$ and $Y=\prod\limits _{t\in T}Y_t$ for the dependence of every separately continuous functions $f:X\times Y\to \mathbb R$ on at most $\aleph$ coordinates with respect to the first or the second variable.