This self-contained paper characterises those locally convex spaces whose (weakly) precompact (respectively, compact) subsets are metrisable. Applications and examples are provided. Our approach also applies to get Cascales-Orihuela's, Valdivia's and Robertson's metrisation theorems for (pre)compact sets.

This paper characterizes the K-analyticity-framedness in ℝ^X for C_p(X) (the space of real-valued continuous functions on X with pointwise topology) in terms of C_p(X). This is used to extend Tkachuk’s result about the K-analyticity of spaces C_p(X) and to supplement the Arkhangel ′skiĭ–Calbrix characterization of σ-compact cosmic spaces. A partial answer to an Arkhangel ′skiĭ–Calbrix problem is also provided.