Ascoli’s theorem for pseudocompact spaces

Abstract

A Tychonoff space X is called (sequentially) Ascoli if every compact subset (resp. convergent sequence) of $$C_k(X)$$ is equicontinuous, where $$C_k(X)$$ denotes the space of all real-valued continuous functions on X endowed with the compact-open topology. The classical Ascoli theorem states that each compact space is Ascoli. We show that a pseudocompact space X is Asoli iff it is sequentially Ascoli iff it is selectively $$\omega $$ -bounded. The class of selectively $$\omega $$ -bounded spaces is studied.