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.