The $\kappa$-Fréchet--Urysohn property for locally convex spaces

Abstract

A topological space $X$ is $\kappa$-Fréchet--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 $\kappa$-Fréchet--Urysohn Tychonoff space $X$ is Ascoli. We apply this statement and some of known results to characterize the $\kappa$-Fréchet--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 $(\kappa )$.