Kerodon

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$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Remark 3.5.6.9. A continuous function $q: X \rightarrow S$ is a Hurewicz fibration if, for every topological space $Y$, every lifting problem

\[ \xymatrix@R =50pt@C=50pt{ \{ 0\} \times Y \ar [r] \ar [d] & X \ar [d]^{q} \\ \empty [0,1] \times Y \ar [r] \ar@ {-->}[ur] & S } \]

admits a solution. Equivalently, $q$ is a Hurewicz fibration if the evaluation map

\[ \operatorname{Hom}_{\operatorname{Top}}( [0,1], X) \rightarrow \operatorname{Hom}_{\operatorname{Top}}( \{ 0\} , X) \times _{ \operatorname{Hom}_{\operatorname{Top}}( \{ 0\} , S) } \operatorname{Hom}_{\operatorname{Top}}( [0,1], S) \]

admits a continuous section, where we endow $\operatorname{Hom}_{\operatorname{Top}}( [0,1], X)$ and $\operatorname{Hom}_{\operatorname{Top}}( [0,1], S)$ with their compact-open topologies. Every Hurewicz fibration is a Serre fibration. However, the converse is false.