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).

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


1 Answer 1


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

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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.