Arbitrary Intersection of Compact Sets in Non-Hausdorff Spaces

Question

I know that the arbitrary intersection of compact sets in Hausdorff spaces is always compact, but is this true in general? I suspect not, but struggle to think of a counterexample.

Answer 1

Here's a counterexample. Let $X=\mathbb{N}\cup\{\infty,\infty'\}$, where a subset $U\subseteq X$ is open iff either $U\subseteq\mathbb{N}$ or $X\setminus U$ is finite. Then $A=\mathbb{N}\cup\{\infty\}$ and $B=\mathbb{N}\cup\{\infty'\}$ are both compact (they are one-point compactifications of $\mathbb{N}$), but $A\cap B=\mathbb{N}$ has the discrete topology and is not compact.