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I keep running across comments and answers to questions that imply that the mapping cylinder for “nice” functions is a CW complex. Why is this necessarily so?

Consider any function $ f \colon X \to Y $. Using Hatcher’s (p2) standard definition, the mapping cylinder will not be a CW complex unless Y, the range of f, is a CW complex.

If f is a nice function, its image may be nice, but the rest of Y can be almost anything.

What am I missing?

PossumP
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  • Maybe they're defining "nice" to mean "whatever I need for the mapping cylinder to be a CW complex." – Qiaochu Yuan Sep 13 '20 at 22:27
  • When people say is a CW complex they normally mean is homeomorphic to a CW complex, or more frequently is homotopy equivalent to a CW complex. For example Hatcher's definition does not give you an actual CW complex unless both $X,Y$ are CW complexes and $f$ is cellular. On the other hand, as you point out, the mapping cylinder is homotopy equivalent to a CW complex if and only if $Y$ is. – Tyrone Sep 13 '20 at 23:16
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    You should make precise what "nice" means. In the present form it could be anything and an answer is impossible. – Paul Frost Sep 14 '20 at 09:04
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    @Tyrone If being a CW-complex means "homotopy equivalent to a CW complex", then as you say everything is trivial because the mapping cylinder is homotopy equivalent to its base. I cannot imagine that this is the intended interpretation. – Paul Frost Sep 14 '20 at 09:09
  • @Paul & @ Qiaochu - It doesn’t really matter how you define nice, because in every case you have to look at at the structure of Y as well. But, for example, f(x)=c , a constant. How can we say anything about the structure of the mapping cylinder even In this case? – PossumP Sep 14 '20 at 16:39
  • The mapping cylinder contains both $X,Y$ as a closed subspaces, so if they are not both at least, say, perfectly normal hereditarily paracompact Hausdorff spaces, then $M_f$ cannot be a CW complex, or even homeomorphic to one, regardless of how 'nice' $f$ might be. If $X,Y$ are CW complexes, then as I said above, 'nice' should mean 'skeletal'. If you want $M_f$ to be an actual CW complex then there aren't any weaker assumptions on $f$ you can make. – Tyrone Sep 14 '20 at 19:58

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