On a theorem of Grothendieck in Cp-theory

A theorem of Grothendieck is extended in several directions. In particular, it is proved that if X is a Lindelöf ∑-space, then the closure of every relatively countably compact subset of Cp(X) is compact. We also investigate when the closure of every pseudocompact subspace of Cp(X) is compact, and when every pseudocompact subset of Cp(X) is itself compact, or even belongs to a given class of compacta. New open questions are formulated.