# Intersection of nested compact subspaces in non-Hausdorff space

Why is it that if a space is not Hausdorff and you take the intersection of nested compact subspaces the intersection could be empty? Could you give me an example?

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Take $X$ to be any set with the trivial topology (only the empty set and $X$ are open). Every subset of $X$ is compact. Now take any chain of subsets whose intersection is empty.
For example, $\mathbb N$ with the trivial topology and intersect $A_k = \{n\in\mathbb N\mid n>k\}$.