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

TL;DR

This paper explores variants of sequential separability in function spaces, establishing that sigma-separability aligns with sequential separability and linking hereditary variants to the Fréchet-Urysohn property in cosmic spaces.

Contribution

It demonstrates the equivalence of sigma-separability and sequential separability and characterizes hereditary variants via the Fréchet-Urysohn property in cosmic spaces.

Findings

Sigma-separability coincides with sequential separability.

Hereditarily sigma-separable and hereditarily F-separable spaces are characterized by the Fréchet-Urysohn property.

Results are specific to the space of continuous functions with pointwise convergence topology.

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: $\sigma$-separability and $F$-separability. The aim of this paper is to study $\sigma$-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 $\sigma$-separability coincides with sequential separability. Hereditary variants (hereditarily $\sigma$-separablity and hereditarily $F$-separablity) coincides with Frechet-Urysohn property in the class of cosmic spaces.