Countably tight, zero-dimensional and totally disconnected dense subspaces of Cp(X)

Abstract

In this paper we prove that under CH there exists a space X such that $C_p\left(X\right)$ does not admit dense subspaces of countable tightness, partially answering Problem 1 of [13]. We prove that if X has cardinality at most continuum, then $C_p\left(X\right)$ contains a dense zero-dimensional subspace and a dense totally disconnected non zero-dimensional subspace. We also provide an example of a compact space X such that $C_p(X,2)$ is exponentially separable but X is not Corson compact.