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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?

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This follows from general facts.

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