On uniformly dense Lindelöf subspaces of function spaces

Abstract

A set Y⊂Cp(X) is uniformly dense in Cp(X) if it is dense in the uniform topology on C(X). We construct a zero-dimensional σ-compact space X such that Cp(X) has a uniformly dense Lindelöf subspace while Cp(X) is not normal. This example answers several published open questions. Additionally, we obtain a version of a theorem of Reznichenko on ω-monolithicity, under MA+¬CH, of a compact space X if Cp(X) has a uniformly dense Lindelöf subspace. We also prove that if X is a dyadic compact then Cp(X) has a uniformly dense subspace of countable pseudocharacter.