When working in the compactly generated spaces (weak Hausdorff $k$-spaces) it is true that any space is weakly equivalent to a CW complex. I'm interested in the converse: let $X$ be a CW complex and $Y$ be a topological space such that there is a weak equivalence $X \to Y$, then is it necessarily true that $Y$ is compactly generated? If that is not true in general, does it hold for homotopy equivalence? I figure the proof would be an abysmal exercise in point-set topology if either were true.
I wondered about this while trying to show that a certain space does not have the homotopy type${}^*$ of a CW complex, and that if one wanted to look for such a space one shouldn't waste any time looking in the category of compactly generated spaces. If the above were true I could prove no such weak equivalence exists if the space isn't compactly generated.
${}^*$It seems some authors take homotopy type to mean weak equivalence, which makes sense in the category of CW spaces, but in general is homotopy type taken to mean homotopy equivalence?
Every space is weak equivalent to a CW-complex; in fact, I'm not familiar with any proof that only compactly generated spaces (as opposed to all spaces) are weak equivalent to CW-complexes. For instance, the proof in Hatcher (Proposition 4.13) works for all spaces.
In any case, if you know the result for $k$-spaces, you can immediately deduce it for arbitrary spaces, since the map $kX\to X$ to any space from its $k$-ification is a weak equivalence (this is immediate from the fact that "weak equivalence" is defined in terms of maps from certain compact Hausdorff spaces to $X$).
For strong homotopy type, however, compact generation is neither necessary nor sufficient to be homotopy equivalent to a CW-complex. For instance, if $X$ is any non-compactly generated space, the cone on $X$ is contractible and hence homotopy equivalent to a CW-complex, but not compactly generated. On the other hand, plenty of compact Hausdorff spaces such as $\{0\}\cup\{1,1/2,1/3,\dots\}$ are not homotopy equivalent to CW-complexes.