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?