The $κ$-Fréchet--Urysohn property for locally convex spaces

S. Gabriyelyan

Published 2018-12-25Version 1

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 $x$. For a Tychonoff space $X$, denote by $C_p(X)$ and $C_k(X)$ the space $C(X)$ of all real-valued continuous functions on $X$ endowed with the pointwise topology and the compact-open topology, respectively. The space $X$ is Ascoli if every compact subset of $C_k(X)$ is evenly continuous. We prove that every $\kappa$-Fr\'{e}chet--Urysohn space $X$ is Ascoli. We apply this statement and some of the main results from [1,4-8,12] 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 show that $C_p(X)$ is Ascoli iff $X$ has the property $(\kappa)$.