Let $X$ be a topological vector space (or, perhaps, more generally uniform space). Let $A\subset X$ be a subset. Let $A^s$ denote the set of limits of all convergent sequences (I guess $A^s$ is called sequential closure of $A$, but I am not sure). If $X$ is not metrizable, then $A^s$ may not be closed or even sequentially closed. Thus one may repeat the procedure of taking "sequential closure".

By transfinite induction, for any ordinal $\omega$ one defines $A^\omega$ as follows:

1) if $\omega=1$ then $A^\omega:=A^s$;

2) if $\omega=\omega'+1$ then $A^\omega=(A^{\omega'})^s$;

3) if $\omega$ is a limit ordinal then $A^\omega=\cup_{\omega'<\omega}A^{\omega'}$.

It is easy to see that there exists sufficiently large ordinal $\eta_0$ such that the process stabilizes: for any $\eta\geq \eta_0$ one has $A^{\eta}=A^{\eta_0}$. In particular $A^{\eta_0}$ is sequentially closed.

**QUESTIONS.** (1) Is there a standard name and notation of $A^{\eta_0}$?

(2) Does $A^{\eta_0}$ have to be topologically closed? Particularly I am interested in the situation when $X$ is a locally convex vector space, and $A$ is a linear subspace.