The functional characterizations of the Rothberger and Menger properties

Abstract

For a Tychonoff space X, we denote by Cp(X) the space of all real-valued continuous functions on X with the topology of pointwise convergence. In this paper we continue to study different selectors for sequences of dense sets of Cp(X) started to study in the paper [14].A set A⊆Cp(X) will be called 1-dense in Cp(X), if for each x∈X and an open set W in R there is f∈A such that f(x)∈W.We give the characterizations of selection principles S1(A,A), Sfin(A,A) and S1(S,A) where•A — the family of 1-dense subsets of Cp(X);•S — the family of sequentially dense subsets of Cp(X), through the selection principles of a space X. In particular, we give the functional characterizations of the Rothberger and Menger properties.