Warning In the formulation of Proposition, the hypothesis that $X$ and $Y$ have the homotopy type of a CW complex cannot be omitted. For any topological space $Y$, the counit map $v: | \operatorname{Sing}_{\bullet }(Y) | \rightarrow Y$ is a weak homotopy equivalence (Corollary, whose domain is a CW complex (Remark If $Y$ satisfies the conclusion of Proposition, then $v$ is a homotopy equivalence, so $Y$ has the homotopy type of a CW complex.