Does noncompact manifold or orbifold have the homotopy type,of CW complex?

Question

I forget for a while, we don't need the compactness condition here right?

Answer 1

According to The Topology of CW-Complexes by Lundell and Weingram (Van Nostrand Reinhold, 1969) the answer is yes for (separable) manifolds.

Answer 2

For smooth manifolds the following holds. By the existence of Morse functions one can deduce a handle-body decomposition of a manifold. This decomposition then yields a CW structure on a space homotopy equivalent to the manifold. I don't think that compactness is needed in any of the above arguments.