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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?

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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 $T_2$)

  • $\mathbb{Q}^*$ 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 K-spaces" 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.

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What is $KC$? Do you have a counterexample for product of compactly generated spaces being compactly generated? – Sigur Mar 21 '14 at 18:09

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