TITLE: Why the "W" in CGWH (compactly generated weakly Hausdorff spaces)?
QUESTION [44 upvotes]: 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 CGWH 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?

REPLY [6 votes]: Here is perhaps the simplest example of a CGWH space which fails to be Hausdorff.
Start with a countable metric space X so that with one exception x, each point is open, but so that at the exceptional point, X is not locally compact at x. 
It is easy to find such a subspace of the real line. 
( start with 0 and (1/n)+(1/(m+n))
Now delete each 1/n).
Let Y be the one point compactification, adding to X  a new point y, whose neighborhood complements are compact in X. In the new space Y, compact subsets are closed (and in particular Y is WH), but x and y are inseparable.
See for example, Example 99 from Counterexamples in Topology by Steen and Seebach.