# Simplicial Approximation Theorem

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.
• There is a procedure called simplicial subdivision. If one does this $k$ times to $X$ we get $X^{(k)}$. The proof of simplicial approximation shows that one can take $X'$ to be any sufficiently fine subdivision of $X$. – Angina Seng Apr 24 '17 at 17:09