Kerodon

Example 3.6.3.7. Let $X$ be a topological space with the property that every continuous path $p: [0,1] \rightarrow X$ is constant (this condition is satisfied, for example, if $X$ is totally disconnected). Let $X'$ denote the topological space whose underlying set coincides with $X$, but endowed with the discrete topology. Then the identity map $f: X' \rightarrow X$ induces an isomorphism of singular simplicial sets $\operatorname{Sing}_{\bullet }(X') \rightarrow \operatorname{Sing}_{\bullet }(X)$, and is therefore a weak homotopy equivalence of topological spaces. However, $f$ is a homotopy equivalence if and only if the topology on $X$ is discrete (since any homotopy inverse of $f$ must coincide with the identity map $f^{-1}: X \rightarrow X'$).