Finite CW complex

A finite CW complex has finitely many cells. Its cellular chain complex consists of finitely generated free abelian groups and vanishes outside finitely many degrees, so its integral homology groups are finitely generated.

Suppose a compact skeleton $Y$ embeds as $e(Y)\subset {\mathbb{R}}^m$. Attach a disk by $a:S^{k-1}\to Y$ and extend its boundary coordinate map radially by $G(ru)=re(a(u))$. Map the disk to $(G(x),(1-|x|)x,1-|x|)$ and $Y$ to $(e(y),0,0)$. The boundary identifications agree; a positive last coordinate recovers each interior point. This defines a continuous injection of the compact quotient into Euclidean space, hence an embedding. Repeating proves Hausdorffness and Euclidean embeddability for every finite CW complex, without assuming triangulability of arbitrary attaching maps.