[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](https://mathoverflow.net/questions/47702/why-the-w-in-cgwh-compactly-generated-weakly-hausdorff-spaces). As an alternative to CGWH, Rainer Vogt [proposed](http://www.ams.org/mathscinet-getitem?mr=300277) 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?



Is every locally compactly generated space compactly generated?
The paper "[A distinguishing example in k-spaces](http://www.ams.org/journals/proc/1987-100-03/S0002-9939-1987-0891170-3/)" by John Isbell constructs an example of a locally compact space $X$ which is not compact-Hausdorffly generated. 


