In his 1967 paper *A convenient category of topological spaces*,
Norman Steenrod introduced the category *CGH* of **compactly generated Hausdorff spaces**
as a good replacement of the category *Top* topological spaces, in order to do homotopy theory.


The most important difference between *CGH* and *Top* is that in *CGH* there is a functorial homeomorphism $$\mathrm{map}(X,\mathrm{map}(Y,Z))\cong \mathrm{map}(X\times Y,Z),$$
a fact that is only true in *Top* under the extra assumption that $Y$ is locally compact.




---


But in more recent papers, I see that people use *CG**W**H* spaces instead of *CGH* spaces...
Why?  



Could someone explain to me what goes wrong in *CGH* spaces
(please illustrate with an example),  
 and explain how the *"w"* fixes everything?


Also (following Jeff's comment), to whom should the *"w"* be attributed?


One more wish: can someone give me an example of a *CGWH* space that isn't *CGH*?



Why the "W" in CGWH (compactly generated weakly Hausdorff spaces)?
I *believe* that CGWH spaces were first used in a *systematic* way in the work of Lewis-May-Steinberger on spectra. It is certainly the case that Gaunce Lewis's (unpublished) thesis contains the best reference on CGWH spaces that I'm aware of. (I haven't looked at the McCord paper Andrey mentions. **Update:** Having looked at McCord's paper, it does indeed seem to be the one to introduce CGWH (the idea of which he attributes to J.C. Moore.))


As to why one might prefer to use CGWH spaces, I'm not precisely sure. But here is one possibility.


A key property of the category of CG spaces is that the product of a quotient map with a space is still a quotient map. In CGWH spaces, something even nicer is true: any *pullback* of a quotient map (along any map) is still a quotient map. (I don't know whether this nicer fact fails in CGH, but I suspect it does.)


Another nice fact about CGWH: regular monomorphisms are precisely the closed inclusions.("Regular monomorphism" means the monomorphism is an equalizer of some pair.) (I originally said here that regular epis in CGWH are precisely quotient maps, but on reflection I'm not sure this is true.) 


