The $κ$-Fréchet-Urysohn property for $C_p(X)$ is equivalent to Baireness of $B_1(X)$

TL;DR

The paper proves that for any Tychonoff space $X$, the $\kappa$-Fréchet-Urysohn property of $C_p(X)$ is equivalent to the Baireness of $B_1(X)$, thereby equating the Banakh property of $C_p(X)$ with the meagerness of $B_1(X)$. It develops a chain of equivalences among $(\kappa)$ on $X$, $C_p(X)$ being $\kappa$-Fréchet-Urysohn and Ascoli, and the existence of strongly $Coz_{\delta}$-disjoint subsequences for disjoint finite families of $X$, linking function-space properties to base-space structure. The results yield several corollaries for special classes (almost $K$-analytic, $k$-scattered, scattered, pseudocompact, first-countable) and provide constructions showing how Baire or meager behavior of $B_1(X)$ translates into Banakh or non-Banakh behavior of $C_p(X)$. These findings offer a concrete criterion to assess the Banakh property of $C_p(X)$ via the Baire status of $B_1(X)$ and related topological invariants of $X$.

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 $(κ)$ 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 $X$.

Theorems & Definitions (12)

• Theorem 3.1
• proof
• Corollary 3.2
• Remark 3.3
• Corollary 3.4
• Proposition 3.5
• Proposition 3.6
• Proposition 3.7
• Remark 3.8
• Proposition 3.9