The following quote is found in the (~1969) book of Saunders MacLane,
"Categories for the working mathematician"
"All told, this suggests that in Top we have been studying
the wrong mathematical objects.The right ones are the spaces in CGHaus."
CGHaus is the category of compactly generated Hausdorff spaces.
It is advocated that it is a better category than Top
because it is cartesian closed.
Almost 50 years later, what is the general feeling on that point?
Is CGHaus the "right" topology to study algebraic topology
(which is what MacLane is interested in)?
Is there a better choice? Or no consensus on the question?
Is CGHaus used in other fields of mathematics?
Best Answer
The convenient category CGH of compactly generated Hausdorff spaces has some poor colimits, since Hausdorffification may change the underlying point sets. The category CGWH of compactly generated weak Hausdorff spaces is even better behaved. The advantages are discussed in Chris McCord's paper "Classifying Spaces and Infinite Symmetric Products", Trans. A.M.S. (1969), which credits John Moore for these ideas. Lewis-May-Steinberger and Elmendorf-Kriz-Mandell-May do their serious work on spectra and S-modules in CGWH. This is probably the category of topological spaces (as opposed to simplicial sets) in which most algebraic topologists really work. A search for "CGWH" leads to several related questions on this site, including a link to Neil Strickland's note http://neil-strickland.staff.shef.ac.uk/courses/homotopy/cgwh.pdf . (I think his reference to Jim McClure's thesis might really be to Gaunce Lewis' thesis.)