The "right" topological spaces

The convenient category CGH of compactly generated Hausdorff spaces has some poor colimits, since Hausdorffification may change the underlying point sets. The category CGWH of compactly generated weak Hausdorff spaces is even better behaved. The advantages are discussed in Chris McCord's paper "Classifying Spaces and Infinite Symmetric Products", Trans. A.M.S. (1969), which credits John Moore for these ideas. Lewis-May-Steinberger and Elmendorf-Kriz-Mandell-May do their serious work on spectra and S-modules in CGWH. This is probably the category of topological spaces (as opposed to simplicial sets) in which most algebraic topologists really work. A search for "CGWH" leads to several related questions on this site, including a link to Neil Strickland's note http://neil-strickland.staff.shef.ac.uk/courses/homotopy/cgwh.pdf . (I think his reference to Jim McClure's thesis might really be to Gaunce Lewis' thesis.)

I'm a little late to the party. I agree with Dennis Nardin's comment that really any convenient category of spaces should be good for homotopy theory. By this, I mean any complete and cocomplete cartesian closed full subcategory of $\mathsf{Top}$ which includes the CW complexes and whose limits and colimits are not too far from those in $\mathsf{Top}$; in particular, homotopy groups of CW complexes should not change from those in $\mathsf{Top}$. $k$-spaces and its variants are rather large categories as these convenient categories go.[1]

But sometimes you actually want to work with a smaller category of topological spaces -- usually asking for there to be a universal way to turn any topological space into one in your category without changing the weak homotopy type. For example, sequential spaces or delta-generated spaces, or variants with separation conditions, are good options for point-set level categories. These smaller categories have the advantage, unlike $k$-spaces and variants, of being locally presentable, which gives even better categorical control, and in particular allows all the techniques of combinatorial model categories to be brought to bear. An additional advantage of delta-generated spaces is that connected components are the same as path components, and there is a connected component functor, which is left-adjoint to the discrete space functor.

There is a general recipe which allows one to cook up convenient categories of topological spaces. Start with a subcategory $\mathcal{C} \subseteq \mathsf{Top}$, and consider the category $\mathsf{Top}_\mathcal{C}$ which is the closure of $\mathcal{C}$ under colimits in $\mathsf{Top}$ [2]. If $\mathcal{C}$ consists of locally compact Hausdorff spaces [3] and is closed under finite products in $\mathsf{Top}$ [4], then $\mathsf{Top}_\mathcal{C}$ will be cartesian closed, and coreflective in $\mathsf{Top}$, so its colimits are computed as in $\mathsf{Top}$, and limits are computed by taking the ordinary limit and then making the topology finer to land back in $\mathsf{Top}_\mathcal{C}$. So as long as $\mathsf{Top}_\mathcal{C}$ includes the unit interval, it will also contain all CW complexes and have the right homotopy groups thereof, and hence be a convenient category of topological spaces. If in addition $\mathcal{C}$ is small, then $\mathsf{Top}_\mathcal{C}$ will have the added benefit of being locally presentable. If desired, additional separation conditions can also be added without much fuss.

Delta-generated spaces are $\mathsf{Top}_\mathcal{C}$ when $\mathcal{C}$ is the category of topological simplices [5]. Sequential spaces are $\mathsf{Top}_\mathcal{C}$ when $\mathcal{C}$ consists of all metric spaces [6]. And $k$-spaces are $\mathsf{Top}_\mathcal{C}$ when $\mathcal{C}$ is all compact Hausdorff spaces.

[1] To be fair, it's maybe an unnatural restriction to require the category to be a subcategory of $\mathsf{Top}$ - e.g. a supercategory like quasitopological spaces might do as well. But I'll let others discuss these options.

[2] Another way to describe $\mathsf{Top}_\mathcal{C}$, familiar in the case of $k$-spaces, is the following. If $X$ is a space, say that $U\subseteq X$ is $\mathcal{C}$-open iff for every $C \in \mathcal{C}$ and map $f: C \to X$, the set $f^{-1}(U)$ is open. Then $X \in \mathsf{Top}_\mathcal{C}$ iff every $\mathcal{C}$-open subset of $X$ is open. An equivalent description (again familiar in the case of $k$-spaces) says that $X \in \mathsf{Top}_\mathcal{C}$ iff for every $Y \in \mathsf{Top}$ and every function $f: X \to Y$, if the induced map $\mathsf{Top}(C,X) \to \mathsf{Top}(C,Y)$ is continuous for every $C \in \mathcal{C}$, then $f$ is continuous. Finally, $\mathsf{Top}_\mathcal{C}$ is also equivalently described as the closure of $\mathcal C$ in $\mathsf{Top}$ under coproducts and quotients.

[3] Or more generally, exponentiable spaces.

[4] This condition can be weakened to say that finite products in $\mathsf{Top}$ of objects of $\mathcal{C}$ are at least in $\mathsf{Top}_\mathcal{C}$.

[5] Or equivalently, you could take $\mathcal{C}$ to be the category of CW complexes, or the category of manifolds, or you even just the one-object category consisting of the unit interval, or the real line; in all these cases $\mathsf{Top}_\mathcal{C}$ will still be exactly the delta-generated spaces.

[6] Or equivalently, you could take $\mathcal{C}$ to be the category of first-countable spaces, or the category of second-countable spaces, or just the one-object category consisting of the one-point compactification of the natural numbers.

You might look at On a topological topos, n-category café.