Velichko's notions close to sequentially separability and their hereditary variants in $C_p$-theory

Abstract

A space $X$ is sequentially separable if there is a countable $S\subset X$ such that every point of $X$ is the limit of a sequence of points from $S$. In 2004, N.V. Velichko defined and investigated concepts close to sequentially separability: $σ$-separability and $F$-separability. The aim of this paper is to study $σ$-separability and $F$-separability (and their hereditary variants) of the space $C_p(X)$ of all real-valued continuous functions, defined on a Tychonoff space $X$, endowed with the pointwise convergence topology. In particular, we proved that $σ$-separability coincides with sequential separability. Hereditary variants (hereditarily $σ$-separablity and hereditarily $F$-separablity) coincides with Frechet-Urysohn property in the class of cosmic spaces.