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I try to understand the significance of CW-complexes in homotopy theory, in particular with respect to the classical models structure on $\mathbf{Top}$. Why do we chose Serre cofibrations for the classical model structure instead of Hurewitz cofibration?

About every source on homotopy theory mentions that the category of (spaces with homotopy type of) CW-complexes is the place to do homotopy theory, but I haven't found any source that explains why (except for comments like: if you read the last 200 pages, it should be clear).

Two nice properties I've singled out thus far are that CW-pairs satisfy the HEP, and the Whitehead theorem. But I doubt that this is the whole story.

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There are three important model structures on spaces: Quillen, with Serre fibrations and weak equivalences; Strom, with Hurewicz fibrations and closed Hurewicz cofibrations and homotopy equivalences; mixed, with Hurewicz fibrations and weak equivalences. The last two are Quillen equivalent (which means they present the same notion of homotopy theory,) but they're different from the first. The homotopy theory of the last two has proved to be more fundamental; it's the free homotopy theory in various axiomatizations, most precisely for quasicategories and derivators.

More practically, it's just easier to prove things up to weak equivalence, when you can replace everything with a CW complex. These permit inductive proofs by climbing up skeleta, level by level. Obstruction theory is a hugely important application of this possibility; this is the kind of thing your books are presumably hinting at. You can't do this with general spaces, and I don't think you even can if you work with the Strom version of simplicial sets, because conceptually the possibility of these inductive arguments depends on the fact that the "good" model structures are cofibrantly generated, which is an immensely useful technical property without which model categories are horribly unwieldy.

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