On the pseudouniform topology on C(X)

Abstract

We denote by Cs(X) the set C(X) of real-valued continuous functions defined on X endowed with the topology of the uniform convergence on the closed separable subspaces of X. In this paper we continue the study of Cs(X) initiated in Pseudouniform topologies onC(X)given by ideals (Pichardo-Mendoza et al. (2013) [6]). We prove that Cs(X) is a k-space if and only if Cs(X) is metrizable, and that compactness, sequential compactness and countable compactness coincide in subspaces of Cs(X). In addition, we study the cellularity, density, weight and character of Cs(X). We prove that (1) d((R2λ)s)=λ if λ=λω, and (2) the Continuum Hypothesis is equivalent to the statement: Every non-separable space X satisfies χ(Cs(X))=w(Cs(X)).