TITLE: Point-wise limit of finite valued functions
QUESTION [6 upvotes]: 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 $X$?

REPLY [7 votes]: 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.  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.