Proof about aspherical spaces are determined up to homotopy equivalence by its fundamental group

Question

It is said that Hurewicz proved that an aspherical is determined up to homotopy equivalence by its fundamental group. But I can only find the proofs to show that two aspherical CW complexes are homotopy equivalent if and only if their fundamental groups are isomorphic. I wonder if the statement is only true for the CW case, or it's also true for the general case?

Answer 1

It's not true for arbitrary spaces. For instance, let $X$ be a Warsaw circle, obtained by "closing up" a topologist's sine curve so that it is path-connected. Then $X$ has trivial (singular) homology and homotopy groups, and in particular is an aspherical space with the same fundamental group as a point, but $X$ is not homotopy equivalent to a point.