Take the 2-minute tour ×  I'm sorry to be asking a (possibly)
elementary question, but I've run into a problem in point-set
topology; I've just read that there exists paracompact Hausdoff
spaces which are not compactly generated. I ask the following:
Question: If $X$ is paracompact Hausdorff, is its compactly
generated replacement, $k\left(X\right),$ paracompact Hausdorff?
Recall: The inclusion $i:CGH \to Haus$ of compactly generated
Hausdorff spaces into Hausdorff spaces has a right adjoint $k,$
which replaces the topology of $X$ with the following topology:
$U \subset X$ is open in $k\left(X\right)$ if and only if for
all compact subsets $K \subset X,$ $U \cap K$ is open in $K$.
Another way of describing this topology is that it is the final
topology with respect to all maps into $X$ with compact
Hausdorff domain. (For the experts, $CGH$ is the mono-
coreflective Hull of the category of compact Hausdorff spaces in
the category of Hausdorff spaces)  share|improve this question
It seems that it's certainly