= Euclidean embedding by finite cell attachment
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 $G(ru)=r e(a(u))$. 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.
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