On precompact sets in spaces Cc (X)

We show that the fact that X has a compact resolution swallowing the compact sets characterizes those C-c (X) spaces which have the so-called C-base. So, if X has a compact resolution which swallows all compact sets, then C-c (X) belongs to the class C of Cascales and Orihuela (a large class of locally convex spaces which includes the (LM) and (DF)-spaces) for which all precompact sets are metrizable and, conversely, if C, (X) belongs to the class C and X satisfies an additional mild condition, then X has a compact resolution which swallows all compact sets. This fully applicable result extends the classification of locally convex properties (due to Nachbin, Shirota, Warner and others) of the space C-c (X) in terms of topological properties of X and leads to a nice theorem of Cascales and Orihuela stating that for X containing a dense subspace with a compact resolution, every compact set in C-c (X) is metrizable.