Compact coverings for Baire locally convex spaces

Abstract Very recently Tkachuk has proved that for a completely regular Hausdorff space X the space C p ( X ) of continuous real-valued functions on X with the pointwise topology is metrizable, complete and separable iff C p ( X ) is Baire (i.e. of the second Baire category) and is covered by a family { K α : α ∈ N N } of compact sets such that K α ⊂ K β if α ⩽ β . Our general result, which extends some results of De Wilde, Sunyach and Valdivia, states that a locally convex space E is separable metrizable and complete iff E is Baire and is covered by an ordered family { K α : α ∈ N N } of relatively countably compact sets. Consequently every Baire locally convex space which is quasi-Suslin is separable metrizable and complete.