On dense subspaces of countable pseudocharacter in function spaces

A set A⊆Cp(X) is uniformly dense in Cp(X) if, for any f∈Cp(X) and ε>0, there is g∈A such that |g(x)−f(x)|<ε for all x∈X. We prove that Cp(X) has a uniformly dense subspace that condenses onto a second countable space if and only if Cp(X) has cardinality of the continuum. This answers a question of Tkachuk published in 2003. It is also proved that Cp(X) has a dense Fσ-metrizable subspace if X is metrizable or Eberlein compact. If X is Corson compact or Cp(X) is a Lindelöf Σ-space, then it is possible to find a dense set Y⊆Cp(X) with countable pseudocharacter.