Sorting through old papers after the move to our new house, I came across a communication to Categories by Bill Lawvere, dated 21 Nov 2001, entitled "categories:K-spaces and Hurewich", and concerned with the history of k-spaces and related concepts. I thought the following bit of history worth contributing. Before studying monoidal closed categories in his well-known doctoral thesis, Brian Day wrote a very pleasant Masters thesis on monoidal closed structures on variants of topological spaces. For some reason this never got published - perhaps it was not thought original enough at the time - but it contained the perfect way of introducing k-spaces; and not just hausdorff ones - restricting to those is an error. One starts with the category Top of topological spaces, and the category Comp of compact hausdorff spaces. Based on Comp, one forms Steenrod's category of quasi-spaces: a quasi-space is a set X along with, for each A in Comp, a subset of Set(A,X) whose elements may be called the "allowable" maps - one imposes a few evident axioms on these. The quasi-spaces form a category Qu, whose morphisms from X to Y are those set-maps whose composites with allowables are allowable. This is of course classical; but what Brian had is the following. There is an evident functor f: Top --> Qu; just call A --> X allowable if it is continuous. There is an equally evident functor g: Qu --> Top; call a subset open if its characteristic function into the Sierpinski space 2 lies in Qu. We have the adjunction g --| f. As with any adjunction, we have an equivalence between the full subcategory of Top where the counit is invertible and the full subcategory of Qu where the unit is invertible. The subcategory of Top here, of course reflective in Top, is the category of k-spaces, better called the "compactly-generated" spaces; it is also a coreflective full subcategory of Qu. Others have noticed this since and published it; but certainly subsequent to Brian's 1968 (I think) Master's thesis. Of course one is not obliged to use Comp in defining one's quasi-spaces; write Qu' for the quasi-spaces based instead on Top. Now Top --> Qu' is fully faithful, and reflective: we know the reflexion explicitly. Again Qu' is cartesian-closed, although Top is not. This is how Brian and I proved those results in [On topological quotient maps preserved by pullbacks or products, Proc. Cambridge Phil. Soc.67, 1970, 553 - 558]. We did the pulling back in the cartesian closed Qu', applied the reflexion, and wrote down the condition for preservation. We feared, however, that topologists would be frightened off by these "abstract categorical notions"; so we went through all that we had done, translating it into the usual language of topology, before we submitted it for publication. The readers, with our motives and techniques so concealed, must have thought it black magic. Of course, as Bill Lawvere said, the whole "quasi" business should be done abstractly, and turns out to involve subcategories of presheaf categories, with associated toposes like that of Peter Johnstone. I see that this letter has become very long. I must apologize: but so much of our history is getting lost forever. Max Kelly.