Suppose $A \to X$ is a cofibration in topological spaces, and $U \subseteq X$ is an open subset. Is $U \cap A \to U$ a cofibration?

Sorry if this is rather simple, but I don't have much experience with this sort of algebraic topology. Naively, it looks as though the universal example for the homotopy extension property (see May's Concise course 6.2) implies that cofibrations are stable under all pullbacks. However, if this were true, I would have expected to see it stated somewhere.

Concise Course, I infer that "topological space" means compactly-generated weak-Hausdorff space, right? And "cofibration" means CGWH-Hurewicz cofibration? $\endgroup$docommute with pullbacks in simplicial sets. Indeed, cofibrations of simplicial sets are just monomorphisms, which are closed under pullback along an arbitrary map. $\endgroup$