On the σ-countable compactness of spaces of continuous functions with the set-open topology

For a Tychonoff space X , we obtain a criterion of the σ -countable compactness of the space of continuous functions C ( X ) with the set-open topology. In particular, for the class of extremally disconnected spaces X , we prove that the space C λ ( X ) is σ -countably compact if and only if X is a pseudocompact space, the set X ( P ) of all P -points of the space X is dense in X , and the family λ consists of finite subsets of the set X ( P ).