Question

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)

Answer 1

(This should be a comment, but my rep is too low.)

It seems that it's certainly Hausdorff, as the topology of $k(X)$ is finer (if $U$ is open in $X$ then $U\cap K$ is open in $K$ for all compacta $K$, by definition of the subspace topology.) So the two separating sets that worked for $X$ still work for $k(X)$.