# Kerodon

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Corollary 3.5.6.11. Let $q: X \rightarrow S$ be a continuous function between topological spaces. Suppose that $q$ is a fiber bundle: that is, for every point $s \in S$, there exists an open set $U \subseteq S$ containing $s$ and a homeomorphism $U \times _{S} X \simeq U \times Y$ for some topological space $Y$ (compatible with the projection to $U$). Then $q$ is a Serre fibration.

Proof. By virtue of Proposition 3.5.6.10, it suffices to check this locally on $S$ and we may therefore assume that there exists a pullback diagram

$\xymatrix@R =50pt@C=50pt{ X \ar [d]^{q} \ar [r] & Y \ar [d] \\ S \ar [r] & \{ \ast \} }$

for some topological space $Y$. Using Remark 3.5.6.3, we are reduced to showing that the projection map $Y \rightarrow \{ \ast \}$ is a Serre fibration, which follows from Example 3.5.6.2. $\square$