Let $X$ be a second countable topological vector space. Does there exist any sequence of finite valued functions $f\_n\colon X\to X$ converging point-wise to the identity mapping on $X$?



Point-wise limit of finite valued functions
To answer a question raised in comments: if $X$ is merely separable then this can fail. Let $X = \mathbb{R}^\mathbb{R}$ with the product topology, which is a Hausdorff topological vector space. Then $X$ is separable. ([Any product of continuum many separable spaces is separable.](https://dantopology.wordpress.com/2009/11/06/product-of-separable-spaces/) Alternatively, viewing $X$ as the space of all functions from $\mathbb{R}$ to $\mathbb{R}$, one can show that the set of polynomials with rational coefficients is dense.)


Now let $f\_n : X \to X$ be a sequence of finite-range functions converging pointwise to some $f$. Let $A = \bigcup\_n f\_n(X)$ be the union of the images of the $f\_n$, so that $A$ is countable. Then the number of sequences of elements of $A$ has cardinality at most $(\aleph\_0)^{\aleph\_0} = \mathfrak{c}$, so that the set $B$ of possible limits of sequences from $A$ likewise has cardinality at most $\mathfrak{c}$. But the image of $f$ is contained in $B$. Since $X$ has cardinality $\mathfrak{c}^\mathfrak{c} > \mathfrak{c}$, the function $f$ cannot be surjective.


