The Serre-fibration structure on topological spaces

Proof. It is clear that the weak homotopy equivalences satisfy 2-out-of-3, and it is also straightforward to check that Serre fibrations are closed under pullbacks and composition and contain all homeomorphisms. This gives conditions (1) and (2) from Definition 2.2.4.

To see condition (3), consider a pullback square

of topological spaces, where \(p\) is both a Serre fibration and a weak homotopy equivalence. We need to show that also \(p'\) is a weak homotopy equivalence. By Corollary 2.3.15, it suffices to show that for every point \(b' \in B'\) the fiber \(F'_{b'} := {p'}^{-1}(b')\) is weakly contractible. But since \(p'\) is a pullback of \(p\), this fiber is homeomorphic to the fiber \(F_b = p^{-1}(b)\), which in turn is weakly contractible by Corollary 2.3.15.

Finally, for condition (4), note that a continuous map \(f\colon E \to B\) may be factored as \[ E \xrightarrow {i} P(f) \xrightarrow {p} B, \] where \(p\) is the path space fibration from Lemma 2.3.17, and where \(i\) maps \(e\) to \((e,\const _{f(e)})\). The first map is a homotopy equivalence, hence in particular a weak homotopy equivalence, and the second map is a Serre fibration by Lemma 2.3.17. Since the definition of \(P(f)\) is functorial in \(f\), this finishes the proof that \((\Top , W_{\text {whe}}, F_{\text {Serre}})\) is a category with weak equivalences and fibrations, with functorial factorizations. It is also clear that a product of Serre fibrations is again a Serre fibration, finishing the proof. □