# Is a closed embedding of CW-complexes a cofibration?

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)

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

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

• 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" ) – Max Feb 14 at 10:51
• Wow, (3) is much more flexible than I would have ever guessed. – user98602 Feb 19 at 19:39

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.)

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