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It is a standard fact that the inclusion of a sub-CW-complex into a CW-complex is a cofibration, it follows from the fact that the inclusions $S^k\to D^{k+1}$ are, and that they are preserved by pushouts.

My question is about a general closed embedding of a CW-complex into another one, say $f:Y\to X$; but it's not necessarily cellular, and even if it were, it doesn't necessarily witness $Y$ as a sub-CW-complex of $X$.

Is it still necessarily a cofibration ?

If it helps/changes the answer, we may assume that $Y$ or both $X,Y$ are finite dimensional, or even finite (though if the answer is "yes" for one of these cases with more hypotheses and "no" with fewer hypothese, I would still be interested in counterexamples for fewer hypotheses)

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    $\begingroup$ Surely the Alexander horned sphere is not an NDR, but a proof escapes me. $\endgroup$ – user98602 Feb 13 at 22:31
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    $\begingroup$ @MikeMiller I had the same idea and tried to prove it - but I discovered that is an NDR. $\endgroup$ – Paul Frost Feb 14 at 10:16
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This is only a partial answer:

If $X$ is a locally finite CW-complex and $A \subset X$ is a closed subspace which is also a CW-complex (but not necessarily a subcomplex), then $i : A \hookrightarrow X$ is a cofibration.

This is based on three well-known facts.

(1) Locally finite CW-complexes are metrizable.

See e.g. Proposition 1.5.17 of [1].

(2) Metrizable CW-complexes are ANRs.

This is due to the fact that CW-complexes are absolute neighborhood extensors for metrizable spaces.

(3) If $X$ is an ANR and $A$ a closed subset of $X$, then the following are equivalent:

a) the inclusion $i : A \to X$ a cofibration.

b) $A$ is an ANR.

See for example Proposition A.6.7 of [1].

[1] Fritsch, Rudolf, and Renzo Piccinini. Cellular structures in topology. Vol. 19. Cambridge University press, 1990.

https://epub.ub.uni-muenchen.de/4493/1/4493.pdf

See also https://sites.google.com/site/ksakaiidtopology/home/homepage-of-katsuro-sakai/anr.

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  • $\begingroup$ This is already great, thank you ! I'll wait and see of someone can bring more information (e.g. the general case, or replacing "locally finite" with "finite dimensional" ) $\endgroup$ – Max Feb 14 at 10:51
  • $\begingroup$ Wow, (3) is much more flexible than I would have ever guessed. $\endgroup$ – user98602 Feb 19 at 19:39
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If you don‘t insist on finiteness of the CW-complex, then you may take for $X$ an infinite CW-complex and $Y$ its one-point-compactification. For example, if $X=R^n$, then $Y=S^n$ is a CW-complex. But the inclusion $X\to Y$ is not a cofibration because its image is not closed. (Or because there exist homotopies which are not proper and thus do not extend.)

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  • $\begingroup$ Thank you; but I actually meant a closed embedding, I'm sorry about that; I'll edit my question accordingly $\endgroup$ – Max Feb 13 at 22:29

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