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?
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.