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It looks like CW complexes come up a lot when studying spaces in algebraic topology. However, something that doesn't seem to be mentioned a lot is whether a given topological space as a CW complex. Is this because it is possible for most spaces that are considered to be "useful"? Are there any specific conditions for when a topological space is a CW complex?

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And one should say that every space is weakly equivalent to a CW complex, this is sufficient for many purposes. – Justin Young Mar 3 '14 at 7:52
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Certainly there's no test for whether a topological space is equal to a CW complex. It's no more plausible to test whether it's homeomorphic to one, which may be what you meant by your phrasing "is a CW complex." Another answer has given a condition for a space to be homotopy equivalent to a CW complex, which is of much more significance. Another interesting case is that many spaces of continuous functions into CW complexes are homotopy equivalent to CW complexes: for instance, the loop space $\Omega X$ of maps from $S^1$ to $X$ for $X$ a pointed CW complex is again of the homotopy type of a CW complex. This is quite hard to see in an elementary way when you consider that $\Omega X$ is an infinite-dimensional thing.

However, not all function spaces $X^Y$ where $X$ and $Y$ are CW complexes are homotopy equivalent to CW complexes, which is one of the main reasons most algebraic topologists prefer to work with what are called compactly generated spaces. One the other hand, every space is weakly homotopy equivalent to a CW complex. This means there's a continuous map $X\to Y$ where $Y$ is a CW complex inducing isomorphisms on all homotopy groups. So if you think of just a weak homotopy equivalence as making two spaces "the same" then the answer to your question is that every space is a CW complex-and for homotopy theory this is a pretty reasonable position to take.

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There are plenty of tests for seeing if a space is NOT a CW-complex, like checking to see if it fails to be normal, Hausdorff, locally contractible, etc.

Usually, one only cares about a space being homotopy equivalent to a CW-complex. There is a statement in Hatcher's book (Proposition A.11) that says if $Y$ is a space, $X$ is a CW-complex, and there are maps $i \colon Y \to X$ and $r\colon X \to Y$ so that $ri \simeq \mathrm{id}_Y$, then $Y$ is homotopy equivalent to a CW-complex.

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I like to put what is in some ways the opposite point of view, and ask: how does one specify a space? One way is to give constructions from other spaces, e.g. functions spaces. But how does one give the other spaces?

The history of CW-complexes is that Whitehead generalised them from other constructions in his highly original papers published 1939-1941 which gave the idea of a "membrane complex", a simplicial complex in which some of the simplices were "amalgamated" into larger groups, forming a "membrane". During the war he worked at Bletchley Park. After that, he rewrote these ideas into our familiar CW-complexes, where CW means "closure finite with the weak topology". So the idea is that the space is given constructively by attaching cells in order of increasing dimensions, i.e. complicated spaces built out of simple ones, and this enables inductive proofs of theorems.

Thus CW-complexes give an excellent example of spaces with an additional structure, in this case their skeletal filtration.

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