Is an open subset of a cofibration a cofibration? 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.
 A: By Satz 1 of Dold's Die Homotopieerweiterungseigenschaft (=HEP) ist eine lokale Eigenschaft, it suffices that there exist a continuous function $\tau \colon X \to [0,1]$ with $(\overline{A} \cap V) \subset \tau^{-1}((0,1]) \subset V$. Interestingly, there is also a converse, Satz 2: if $\{V_i\}_{i \in I}$ is a numerable open cover $X$ such that each $A \cap V_i \hookrightarrow V_i$ is Hurewicz cofibration, then $A \hookrightarrow X$ is Hurewicz cofibration.
A: The answer is yes if $X$ is metrizable.  As noted after Satz 1 in the paper by Dold cited in the answer by skupers
Dold, Albrecht
Die Homotopieerweiterungseigenschaft (=HEP) ist eine lokale Eigenschaft. 
Invent. Math. 6 (1968), 185–189.

such a map $\tau$ exists if $X$ is metrizable, since then $A = \bar A$ is closed in $X$ and you can take $\tau(x)$ to be the distance from $x$ to $X - U$.
Closed cofibrations are closed under pullbacks along Hurewicz fibrations.  This is  Theorem 12 of Strøm's paper
Strøm, Arne
Note on cofibrations. II.
Math. Scand. 22 (1968), 130–142 (1969).

