A note on $\kappa$-Fréchet--Urysohn property in function spaces

Abstract

We establish that a Tychonoff space $X$ has the property $(\kappa)$ if all countable subsets of $X$ are scattered. Therefore a countably compact space $X$ does not need to be scattered if $C_p(X)$ is $\kappa$-Fréchet-Urysohn. However, if a hereditarily Baire space is sequential and has the property $(\kappa)$, then it must be scattered. We also prove, for any space $X$, that $C_p(X)$ is $\kappa$-Fréchet-Urysohn if and only if so is $C_p(X, [0,1])$. A pseudocompact first countable space must be scattered if it has the property $(\kappa)$.