Product of a $k$-space and a compact space

Question

I am beginning to learn about compactly generated spaces. I would like to know whether the following is true: if $X$ is a compact Hausdorff space and $Y$ is Hausdorff compactly generated space, then the usual product topology $X\times Y$ is compactly generated as well; I think it is true but I am not able to prove this. My definition of $k$-space is one in which $A\subseteq X$ is closed iff $A\cap K$ is closed in $K$ for each compact $K\subseteq X$.

Answer 1

It’s even true if $X$ is just locally compact, but the proof that I know is quite indirect, using the following results.

• $Y$ is a Hausdorff $k$-space iff it is a quotient of some locally compact Hausdorff space.
• For any locally compact Hausdorff space $X$ and any quotient map $q:Z\to Y$, the map $f:X\times Z\to X\times Y:\langle x,y\rangle\mapsto\langle x,q(y)\rangle$ is a quotient map.

Now let $X$ be a locally compact Hausdorff space and $Y$ a Hausdorff $k$-space. There is a locally compact Hausdorff space $Z$ such that $Y$ is the image of $Z$ under a quotient map $q$, so the map

$$X\times Z\to X\times Y:\langle x,z\rangle\mapsto\langle x,q(z)\rangle$$

is a quotient map. Finally, products of finitely many locally compact spaces are locally compact, so $X\times Z$ is locally compact, and its quotient $X\times Y$ is therefore a $k$-space.