# Is every locally compactly generated space compactly generated?

[Parse it as (locally compact)ly generated.]

I stumbled across this one whilst supervising an undergraduate thesis. Convenient categories for homotopy theory (e.g. CGWH) have been discussed here before. As an alternative to CGWH, Rainer Vogt proposed the category of locally compactly generated spaces; see also the recent German-language point-set topology textbook Grundkurs Topologie by Gerd Laures and Markus Szymik.

If (as is now usual) one means (compact Hausdorff)ly generated when one says compactly generated, then the category of compactly generated spaces is a full subcategory of the category of locally compactly generated spaces. Back in 1971, Vogt asked whether this inclusion is strict or not. Do we know the answer yet?

• In case anyone has the same confusion I had, let me point out that (locally compact Hausdorff)ly generated spaces are the same as (compact Hausdorff)ly generated spaces, as observed by Vogt. And it seems to me that similarly, (locally compact)ly generated spaces should be the same as (compact)ly generated spaces, so the question is really about dropping the Hausdorff assumption rather than "going local". – Tim Campion Dec 6 '15 at 18:42

The paper "A distinguishing example in k-spaces" by John Isbell constructs an example of a locally compact space $X$ which is not compact-Hausdorffly generated.