Two new properties of the space Cp(X)

If X is a completely regular Hausdorff topological space we show that: (i) if H is a compact subset of Cp(X) and X contains a quasi-Souslin subspace which separates the functions of H, then H is Talagrand compact, and (ii) Cp(X) is quasi-Souslin if and only if it is K-analytic.