Definition 3.6.3.1. Let $X$ and $Y$ be topological spaces. We say that a continuous function $f: X \rightarrow Y$ is a weak homotopy equivalence if the induced map of singular simplicial sets $\operatorname{Sing}_{\bullet }(f): \operatorname{Sing}_{\bullet }(X) \rightarrow \operatorname{Sing}_{\bullet }(Y)$ is a homotopy equivalence (Definition 3.1.6.1).
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