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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.

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  • $\begingroup$ Since you refer to May's Concise Course, I infer that "topological space" means compactly-generated weak-Hausdorff space, right? And "cofibration" means CGWH-Hurewicz cofibration? $\endgroup$
    – Tim Campion
    Commented Feb 12, 2021 at 18:01
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    $\begingroup$ Probably, whichever category you use, Hurewicz cofibrations (as opposed to Serre cofibrations) are closed under pullback along open embeddings, but not under pullback along all maps. Note that in the universal diagram May uses, there's a mapping cylinder. Mapping cylinders involve both a product and a pushout to construct. Pushouts do not necessarily commute with pullbacks in any category of topological spaces. OTOH, pushouts do commute with pullbacks in simplicial sets. Indeed, cofibrations of simplicial sets are just monomorphisms, which are closed under pullback along an arbitrary map. $\endgroup$
    – Tim Campion
    Commented Feb 12, 2021 at 18:06
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    $\begingroup$ @Tim Thanks, I would be interested in knowing the answer for both Serre and Hurewicz cofibrations, especially if the answer is negative for one and positive for the other. Perhaps more broadly my question is: what is the largest class of topological spaces and cofibrations for which this property holds? $\endgroup$
    – user1092847
    Commented Feb 12, 2021 at 18:16

2 Answers 2

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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.

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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).
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  • $\begingroup$ It seems that what's needed for the existence of $\tau$ is a suitably strong sense of perfect normality, and, eg, exists whenever $X$ is a CW complex (metrisable or not). $\endgroup$
    – Tyrone
    Commented Feb 15, 2021 at 19:26

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