For any space X, either \(C_p(X)\) or \(C_pC_p(X)\) is \(\psi \)-separable

Abstract

We establish that, for any Tychonoff space X, at least one of the spaces \(C_p(X)\) and \(C_pC_p(X)\) has a dense subspace of countable pseudocharacter. Under MA, we give an example of a space X such that \(C_p(X)\) does not have a dense subspace of countable functional tightness. We also show that there exists a compact zero-dimensional space K such that \(C_p(K,\{0,1\})\) is exponentially separable while K is not a Sokolov space. For compact scattered spaces K of countable dispersion index, we show that \(C_p(K)\) has a dense exponentially separable subspace if and only if K is \(\omega \)-monolithic. Our results provide answers to several published open questions.