Let $X$ be a topological vector space. Assume that there exists a sequence $\phi_n:X\to X$ of finite range measurable functions with $\lim\phi_n(x)=x$ for every $x\in X$. Can we concluded there exists a sequence $\{X_n\}$ of ***subsets*** of $X$ with $X=\cup X_n$ such that $X_n$'s are all relatively  second-countable? 

Note that the answer will be negative if [$X$ is only assumed to be a **topological space**][1].


  [1]: https://mathoverflow.net/questions/300544/approximation-of-the-identity-by-simple-functions