On the $\sigma$-countable compactness of spaces of continuous functions with the set-open topology
For a Tychonoff space $X$, we obtain a criterion of the $\sigma$-countable compactness of the space of continuous real-valued functions $C(X)$ with the set-open topology. In particular, for extremally disconnected space $X$, we prove that the space $C_{\lambda}(X)$ is a $\sigma$-countably compact space if and only if $X$ is a pseudocompact space, the set $X(P)$ of all $P$-points of $X$ is dense in $X$, and the family $\lambda$ consists of finite subsets of $X(P)$.