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.