On uniform spaces with a small base and K -analytic Cc(X)Cc(X) spaces

A base {Uα:α∈NN}{Uα:α∈NN} of a uniformity is a GG-base if Uβ⊆UαUβ⊆Uα whenever α≤βα≤β. If X is a completely regular space we show that there exists an admissible uniformity on X   with a GG-base that contains the Nachbin uniformity if and only if there exists a resolution of the space Cc(X)Cc(X) of real-valued continuous functions on X equipped with the compact-open topology consisting of equicontinuous sets. This result is applied to show, among other things, that if G   is a kRkR-space topological group such that Cc(G)Cc(G) is K-analytic then G   has a GG-base. In the opposite direction, if G   is a topological group with a GG-base and enjoys the so-called property U  , then Cc(G)Cc(G) has a resolution consisting of equicontinuous sets.