The $\kappa$-Fr\'{e}chet-Urysohn property for $C_p(X)$ is equivalent to Baireness of $B_1(X)$

TL;DR

This paper establishes an equivalence between the $ abla$-Fréchet-Urysohn property of $C_p(X)$ and the Baire property of $B_1(X)$, linking topological properties of function spaces to those of the underlying space.

Contribution

It proves that the $ abla$-Fréchet-Urysohn property for $C_p(X)$ is equivalent to the Baire property of $B_1(X)$, providing a new characterization of the Banakh property.

Findings

The property $( abla)$ for $X$ is equivalent to Baireness of $B_1(X)$.

Banakh property for $C_p(X)$ is equivalent to the meagerness of $B_1(X)$.

A new topological characterization of the Banakh property for $C_p(X)$.

Abstract

A topological space $X$ is Baire if the intersection of any sequence of open dense subsets of $X$ is dense in $X$. We establish that the property $(\kappa)$ for a Tychonoff space $X$ is equivalent to Baireness of $B_1(X)$ and, hence, the Banakh property for $C_p(X)$ is equivalent to meagerness of $B_1(X)$. Thus, we obtain one characteristic of the Banakh property for $C_p(X)$ through the property of space $X$.