It is well known that if $X$ and $Y$ are topological spaces with $X$ locally compact Hausdorff and $Y$ compactly generated, then $X \times Y$ (with the ordinary product topology) is compactly generated. Does this fail if $X$ is compact, compactly generated, but not Hausdorff?
1 Answer
I think the space $Z=\mathbb{Q}^* \times \mathbb{Q}$ works for your Question. (Here $\mathbb{Q}^*$ is the one point compactification of the rational numbers)
At first let me recall some properties of $\mathbb{Q}^*$ which you could find Them in the Monthly Article (Between $T_1$ and $T_2$) http://www.jstor.org/discover/10.2307/2316017?uid=3738280&uid=2&uid=4&sid=47699073521117.
 $\mathbb{Q}^*$ is a $KC$ space.(i.e. every compact subset of this space is closed)
 It's easy to show that $\mathbb{Q}^*$ and $\mathbb{Q}$ are compactly generated.
Now I bring Two theorems from the Article "On KC and Kspaces" which are important in the sequel to show that $Z$ is the space which you needed. you could find this Article from:
Theorem 1: Let $X , Y$ be topological spaces.If $X$ is $KC$ and $Y$ is hausdorff; Then $X\times Y$ is $KC$.
Theorem 2: In the topological space $X$ the following are equivalent:
 $X$ is locally compact and Hausdorff.
 $X^*\times X$ is $KC$ and compactly generated.
For the sake of Theorem 1, you could see that $\mathbb{Q}^* \times \mathbb{Q}$ is $KC$. Also from part one of theorem 2 and because $\mathbb{Q}$ is not locally compact, We can conclude that $\mathbb{Q}^* \times \mathbb{Q}$ is not compactly generated.

$\begingroup$ What is $KC$? Do you have a counterexample for product of compactly generated spaces being compactly generated? $\endgroup$– SigurCommented Mar 21, 2014 at 18:09

$\begingroup$ @Sigur what Ali_Reza has done is exactly what you asked for isn't it ? He explained what KC meant and gave you the requested counter example $Q\times Q^*$ $\endgroup$– brunohCommented May 24, 2023 at 19:17