# The $\kappa$-Fr\'{e}chet--Urysohn property for locally convex spaces

## TL;DR

This paper investigates the $ppa$-Fre9chet--Urysohn property in topological spaces, showing it implies the Ascoli property and characterizing this in locally convex spaces, especially for spaces of continuous functions.

## Contribution

It establishes that $ppa$-Fre9chet--Urysohn spaces are Ascoli and characterizes this property in various locally convex spaces, answering a specific open question.

## Key findings

- Every $ppa$-Fre9chet--Urysohn Tychonoff space is Ascoli.
- Characterization of the $ppa$-Fre9chet--Urysohn property in locally convex spaces.
- $C_p(X)$ is Ascoli iff $X$ has property $(ppa)$.

## Abstract

A topological space $X$ is $\kappa$-Fr\'{e}chet--Urysohn if for every open subset $U$ of $X$ and every $x\in \overline{U}$ there exists a sequence in $ U$ converging to $x$. We prove that every $\kappa$-Fr\'{e}chet--Urysohn Tychonoff space $X$ is Ascoli. We apply this statement and some of known results to characterize the $\kappa$-Fr\'echet--Urysohn property in various important classes of locally convex spaces. In particular, answering a question posed in [7] we obtain that $C_p(X)$ is Ascoli iff $X$ has the property $(\kappa)$.