  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.
