Spaces not distinguishing ideal pointwise and $\sigma$-uniform convergence

Abstract

We examine topological spaces not distinguishing ideal pointwise and ideal $\sigma$-uniform convergence of sequences of real-valued continuous functions defined on them. For instance, we introduce a purely combinatorial cardinal characteristic (a sort of the bounding number $\mathfrak{b}$) and prove that it describes the minimal cardinality of topological spaces which distinguish ideal pointwise and ideal $\sigma$-uniform convergence. Moreover, we provide examples of topological spaces (focusing on subsets of reals) that do or do not distinguish the considered convergences. Since similar investigations for ideal quasi-normal convergence instead of ideal $\sigma$-uniform convergence have been performed in literature, we also study spaces not distinguishing ideal quasi-normal and ideal $\sigma$-uniform convergence of sequences of real-valued continuous functions defined on them.