Geodesic
The Separability and Sequential Separability of the Space~$C(X)$
Matematičeskie zametki, Tome 104 (2018) no. 1, pp. 87-98

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The following results are obtained: (1) a criterion for the separability of the space of continuous functions $C(X)$ with the set-open topology; (2) a criterion for the sequential separability of the space $C_{p}(A|X)$, where $A\subseteq X$; (3) an answer to Velichko's question of whether a set-theoretic condition on a metric space $X$ in the criterion for the sequential separability of $C_{p}(X)$ is essential.
Keywords: space of continuous functions, separability, sequential separability, set-open topology. 
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     author = {A. V. Osipov},
     title = {The {Separability} and {Sequential} {Separability} of the {Space~}$C(X)$},
     journal = {Matemati\v{c}eskie zametki},
     pages = {87--98},
     publisher = {mathdoc},
     volume = {104},
     number = {1},
     year = {2018},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2018_104_1_a8/}
}
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A. V. Osipov. The Separability and Sequential Separability of the Space~$C(X)$. Matematičeskie zametki, Tome 104 (2018) no. 1, pp. 87-98. http://geodesic.mathdoc.fr/item/MZM_2018_104_1_a8/