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I am looking at the paper

Covering homotopy properties of maps between CW complexes or ANRs by Mark Steinberger and James West

and a claim is made in the proof of their first main theorem that (slightly rephrased)

since $U$ is a contractible subspace of the CW complex $B$, $U$ "is" a CW complex

Question: Is it generally true that an open subspace of a CW complex can be given the structure of a CW complex? Is it true in general only for contractible subspaces? Why? Is there a reference?

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  • $\begingroup$ As John mentions, there are counter-examples to open subsets of CW-complexes being CW-complexes, but I believe under reasonable "finiteness" assumptions the result should be true. I have not looked at the counter-examples John cites but I imagine they are due to the CW-complex having fairly "bad" attaching maps. In my mind I'm imagining an adaptation of the proof that open subsets of $\mathbb R^n$ admit CW-structures. $\endgroup$
    – Ryan Budney
    Commented Aug 26, 2019 at 20:12
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    $\begingroup$ One of Cauty's counterexamples has only three cells, so the problem is not about finiteness, but rather about the attaching maps. Regular CW complexes can be triangulated, and open subsets of simplicial complexes can again be triangulated, so it is true that open subsets of regular CW complexes are CW spaces. $\endgroup$
    – John Rognes
    Commented Aug 26, 2019 at 20:18

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It is not generally true that each open subset of a CW complex admits the structure of a CW complex. Counterexamples were given in

\bib{MR1157891}{article}{
   author={Cauty, Robert},
   title={Sur les ouverts des CW-complexes et les fibr\'{e}s de Serre},
   language={French},
   journal={Colloq. Math.},
   volume={63},
   date={1992},
   number={1},
   pages={1--7},
   issn={0010-1354},
   review={\MR{1157891}},
   doi={10.4064/cm-63-1-1-7},
}

in response to the paper you mention by Steinberger and West. Here is the abstract:

M. Steinberger et J. West ont prouvé dans [7] qu'un fibré de Serre p:E → B entre CW-complexes a la propriété de relèvement des homotopies par rapport aux k-espaces. Malheureusement, leur démonstration contient une légère erreur. Ils affirment que certains ensembles (notés U et p−1U×U) sont des CW-complexes car ce sont des ouverts de CW-complexes. Ceci est généralement faux, et notre premier objectif dans cette note est de donner des exemples d'ouverts de CW-complexes n'admettant aucune décomposition CW. Malgré cela, le théorème de Steinberger et West est vrai, et notre deuxième objectif est de montrer comment leur démonstration peut être rectifiée.

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