An example of Hausdorff space that is not a k-space

Question

A space $X$ is a $k$-space if a subset $U$ of $X$ is open if and only if $U \cap K$ is open in $K$, for every compact $K \subseteq X$.

Is there any simple examples of Hausdorff spaces that are not k-spaces?

Answer 1

Observation: If $X$ is a $T_2$ k-space in which every compact subset is finite, then $X$ is discrete. $\;\square$

The answer given by bof here contains an example of a nondiscrete $T_2$ space in which each compact subset is finite. Of course, such a space cannot be a k-space.

Other, more striking examples can be obtained. Let $\beta\mathbb{N}$ be the Stone-Čech compactification of the natural numbers and let $p\in\beta\mathbb{N}\setminus\mathbb{N}$. Then $\mathbb{N}\cup\{p\}$ in the subspace topology is a nondiscrete Hausdorf space (which is paracompact and perfectly normal) in which every compact subspace is finite. Thus $\mathbb{N}\cup\{p\}$ is not a k-space.

Here is another approach. Arhangelskii has shown that a $T_2$ space $X$ is Fréchet-Urysohn if and only if every subspace of $X$ is a k-space. There are plentiful examples of $T_2$ spaces which are not Fréchet-Urysohn, and each such will contain a subspace which is not a k-space. For example, the compact space $\beta\mathbb{N}$ is a k-space, but it is not Fréchet-Urysohn.