# How do I show that any finite-dimensional (absolute) CW-complex $X$ is locally contractible?

I know that it holds even if $$X$$ has infinite dimension, but I am looking for a specific argument in the finite-dimensional case.

• Many such questions boil down to the following trick. Problem: Given an open subset of the $n$-skeleton. Can we construct an open subset of the $n+1$-skeleton such that the intersection with the $n$-skeleton is the given set? So we choose attaching pushouts and we leave out the midpoints of the $n+1$- cells. Then we have a retraction of the $n+1$-skeleton without the midpoints to the $n$-skeleton. Now we can take the preimage of the given set. Nov 10 '20 at 19:34

see Proposition A.4 p. 523. I do not think there is any significant simplification taking $$X$$ finite-dimensional.