We show that the fact that X has a compact resolution swallowing the
compact sets characterizes those Cc(X) spaces which have the so-called 𝔊-base.
So, if X has a compact resolution which swallows all
compact sets, then Cc(X) belongs to the class 𝔊 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 Cc(X) belongs to the class 𝔊 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 Cc(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 Cc(X) is metrizable.