One example of a non-Hausdorff topological space in which all compact subsets are closed is the co-countable topology on an uncountable set, as demonstrated here.
It was claimed (as a now-deleted answer to the above question) that the compact complement topology on $\mathbb{R}$ was another example, but this was proven incorrect here.
Can someone provide an additional example of a non-Hausdorff space in whihc all compact subsets are closed?
