# Are there any disadvantages to working in the category of k-spaces as opposed to Top?

Unlike the category Top of topological spaces with continuous maps as the arrows, the full subcategory of compactly generated spaces (k-spaces) is Cartesian closed.

It seems like a very nice category of spaces, since every manifold, CW complex, first countable space, compact space, and locally compact space is a k-space.

Is there any compelling reason to work in the larger category of topological spaces rather than restricting to k-spaces? For instance, are there any spaces of interest to "topology-users" such as analysts, physicists, engineers, applied mathematicians, etc. that are not k-spaces?

• Its worth pointing out that you can also fix $\mathbf{Top}$ by generalizing; e.g. you can replace topological spaces with convergence spaces. This is, in my opinion, is the most natural definition of the word "space," if by "space" we mean "thing with an underlying set such that functions between such things are either continuous or not." – goblin Feb 19 '16 at 9:45