Properties of function spaces reflected by uniformly dense subspaces

A set A⊂Cp(X) is uniformly dense in Cp(X) if, for any f∈Cp(X) and any ε>0, there is g∈A such that |g(x)−f(x)|<ε for all x∈X. We prove that, for many properties P, if a uniformly dense subspace of Cp(X) has P then the whole Cp(X) has P. This is true, in particular, for P∈{Lindelöf Σ-property, tightness ⩽κ, network weight ⩽κ, Fréchet–Urysohn property}. If Cp(X) has a uniformly dense σ-compact subspace then X is compact. We give an example of a compact space X such that ψ(Cp(X))>ω while Cp(X) has a uniformly dense subspace of countable pseudocharacter.