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

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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 $(\kappa)$.
Original languageEnglish GB
StatePublished - 2018

Publication series

NameArxiv preprint


  • math.GN
  • math.FA
  • 46A03, 46A08, 54C35


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