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Definition. Let $E$ be a topological space. Suppose that $E$ contains a sequence $\{x_n\}$ such that for every $x\in E$, there exists a subsequence $\{x_{n_k}\}$ of $\{x_n\}$ with $x=\lim x_{n_k}$. Then we say $E$ enjoys sequential separability.

For a given topological space $X$, let $C_p(X)$ be the space of continuous functions on $X$ endowed with the point-wise topology. It is proved that $C_p(X)$ is separable iff $X$ is separably submetrizable (see this paper).

Question.
$$C_p(X) ~\textrm{is sequentially separable} \Leftrightarrow X=? $$

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N. V. Velichko, “On sequential separability,” Mat. Zametki 78 (5), 652–657 (2005) [Math. Notes 78 (5), 610–614 (2005)].

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  • $\begingroup$ A. V. Osipov, “The Separability and Sequential Separability of the Space C(X)”, Math. Notes, 104:1 (2018), 86–95 $\endgroup$
    – Alexander Osipov
    Mar 27, 2022 at 8:08

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