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

TL;DR

This paper investigates topological spaces that differentiate between ideal pointwise and ideal σ-uniform convergence of real-valued function sequences, introducing a combinatorial cardinal characteristic and providing examples related to subsets of reals.

Contribution

It introduces a new combinatorial cardinal characteristic that determines the minimal size of spaces distinguishing these convergences and compares different types of convergence.

Findings

The minimal cardinality is characterized by a new combinatorial cardinal.

Examples of spaces that do or do not distinguish the convergences are provided.

The study extends to ideal quasi-normal convergence, comparing it with ideal σ-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.