# Are second-countable subsets of topological vector spaces metrizable?

Let $X$ be a topological vector space of size $\mathfrak{c}$. Assume that there exists a countable union $X=\cup X_n$ such that all subsets $X_n$'s are relatively second countable.

Q. Does there exists a a countable union $X=\cup Y_n$ such that all $Y_n$'s are relatively second countable metrisable?

## 1 Answer

This follows from general facts.