Milnor proved that any paracompact Hausdorff space which is equi-locally convex (and hence in particular locally contractible) is homotopy equivalent to a CW complex. However, unlike being paracompact and Hausdorff, the property of being equi-locally convex seems slightly arbitrary here, while the weaker property of being locally contractible is more conceptual. Does anyone know of a reference for this possible strengthening of Milnor's result, or, possibly, a counterexample?
Homotopy Equivalence of Paracompact, Hausdorff, Locally Contractible Spaces to CW Complexes
at.algebraic-topology
Best Answer
After some more digging I found a (somewhat non-explicit) counterexample to the original question. In his paper "un espace metrique lineaire qui n’est pas un retracte absolu" Cauty constructs a metric vector space $V$ which is not an absolute neighborhood retract. According to the characterization established in "une caracterisation des retractes absolus de voisinage" (also Cauty) a metric space $X$ is an absolute neighborhood retract if and only if each open subset of $X$ is homotopy equivalent to a CW complex. It follows that the metric vector space $V$ above contains an open subset $U \subset V$ which is not homotopy equivalent to a CW. Since $U$ is metrizable it is paracompact and Hausdorff, and since it $V$ is locally contractible so is $U$. Hence $U$ is a counterexample to the original question.