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? 



Are second-countable subsets of topological vector spaces metrizable?
This follows from general facts.


* [Topological vector spaces are (completely) regular](https://math.stackexchange.com/questions/2332760/every-topological-vector-space-is-regular).
* Subspaces of regular spaces are regular.
* A second-countable space is metrisable if and only if it is regular (see Theorems 32.1, 32.2, and 34.1 in Munkres' *Topology*).


