Baireness of the space of pointwise stabilizing functions of the first Baire class

TL;DR

This paper investigates the Baire property of the space of pointwise stabilizing Baire-one functions, showing it inherits the Baire property from the larger space of Baire-one functions, thus answering a recent open question.

Contribution

It proves that the space of pointwise stabilizing Baire-one functions is Baire whenever the space of Baire-one functions is Baire, resolving a recent open problem.

Findings

The space of pointwise stabilizing Baire-one functions is Baire if the space of Baire-one functions is Baire.

This result extends the understanding of Baire properties in function spaces.

Answers a recent open question by T. Banakh and S. Gabriyelyan.

Abstract

A topological space $X$ is Baire if the Baire Category Theorem holds for $X$, i.e., the intersection of any sequence of open dense subsets of $X$ is dense in $X$. In this paper, we have obtained that the space $B^{st}_1(X)$ of pointwise stabilizing Baire-one functions is Baire if the space $B_1(X)$ of Baire-one functions is so. This answers a question posed recently by T. Banakh and S. Gabriyelyan.