Compactly Generated Spaces

This document is a summary of basic facts about the “standard” category of compactly generated spaces introduced by McCord [McC69] (often referred to in the literature as “compactly generated weak Hausdorff spaces”, or “weak Hausdorff k-spaces”). The main references I’ve used for this material are Gaunce Lewis’s thesis [Lew, App. A], and Neil Strickland’s note [Str09] (especially for material about colimits in compactly generated spaces). Many of the proofs given here stem from ones given in those works, though they may have mutated significantly. There is one innovation here, namely the notion of a “k-Hausdorff” space, which is (perhaps) a bit more convenient than “weak Hausdorff”. Every weak Hausdorff space is k-Hausdorff (11.2), and for a k-space, weak Hausdorff and k-Hausdorff are equivalent (11.4), so our category of “k-Hausdorff k-spaces” is identical to McCord’s category. I have also tried, as much as possible, to rely directly on these definitions and not on inessential intermediate constructions. For instance, I give a direct construction of the topology on mapping spaces, rather than producing it as the k-ification of the compact-open topology. Also notable is the characterization of compactly generated spaces given as (9.10).