Functorial CW-approximation via the singular complex

Remark 2.3.5. An explicit construction of such a functorial CW-approximation may be given as follows. Recall from Definition 1.6.9 that the singular complex functor \(\Sing \colon \Top \to \sSet \) admits the geometric realization functor \(\abs {-}\colon \sSet \to \Top \) as a left adjoint. One can prove that the geometric realization \(\abs {K}\) is a CW-complex for every simplicial set \(K\). Furthermore, for any topological space \(X\) the counit \(\abs {\Sing (X)} \to X\) of the adjunction is a weak homotopy equivalence.