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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 second-countable topological space.

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.

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 just assume a second countable topological space ( enter link description here ).

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 second-countable topological space.