A characterization of the uniform convergence points set of some convergent sequence of functions

TL;DR

This paper characterizes the sets where a sequence of real-valued functions converges uniformly on a perfectly normal space, generalizing a previous theorem by Ján Borsík.

Contribution

It provides a necessary and sufficient condition for a set to be the uniform convergence set of a convergent function sequence on perfectly normal spaces.

Findings

The uniform convergence set must be a $G_\delta$-set containing all isolated points.

The characterization applies to spaces covered by disjoint dense subsets.

Generalizes Borsík's 2019 theorem.

Abstract

We characterize the uniform convergence points set of a pointwisely convergent sequence of real-valued functions defined on a perfectly normal space. We prove that if $X$ is a perfectly normal space which can be covered by a disjoint sequence of dense subsets and $A\subseteq X$, then $A$ is the set of points of the uniform convergence for some convergent sequence $(f_n)_{n\in\omega}$ of functions $f_n:X\to \mathbb R$ if and only if $A$ is $G_\delta$-set which contains all isolated points of $X$. This result generalizes a theorem of J\'{a}n Bors\'{i}k published in 2019.