Simplicial Approximation Theorem for maps roughly states:
If $X$ and $Y$ are two finite simplicial complexes and $f:|X|→|Y|$ is a continuous map between their geometric realizations, then there exists a subdivision $X'$ of $X$ and a simplitial map $g: X' \to Y$ such that $|g|$ is homotopic to $f$.
Can someone help me how to show (by using this theorem) that the set $[X,Y]$ of homotopy classes $f: X \to Y$ is at most countably infinite.
The point here is that one can choose $X'$ to be an iterated barycentric subdivision $X^{(k)}$ of $X$. There are finitely many simplicial maps from $X^{(k)}$ to $Y$ for a given $k$, so countably many overall.