New answers tagged 4-manifolds
Hatcher, Algebraic Topology, Corolary A.9 Every compact manifold, with or without boundary, is an E[uclidean]N[eighborhood]R[etract]. See also the following three numbered conclusions. Or, in simpler terms, CW is a lot more flexible than, say, PL. By "compact manifold", I expect you mean, in particular, an object $X$ with a finite very good cover ...
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