Does a locally contractible compact space have the homotopy type of a finite CW complex? (I think it probably does, but I need a reference anyway.)
EDIT: My intuition was wrong [to see why, read C.T.C.Wall, Finiteness conditions for CW complexes, Ann. of Math. 81 (1965) 56-69.]. Apparently, the right question is whether a locally contractible compact is dominated by a finite CW complex. This is true if it can be embedded into a manifold, but I don't know what to do with the general case.