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Why is every compact subset of a Hausdorff space closed?

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topology Because both a doughnut and a coffee cup have one hole (handle), they can be mathematically, or topologically, transformed into one another without cutting them in any way. For this reason, it has often been joked that topologists cannot tell the difference between a coffee cup and a doughnut.

In topology, a Hausdorff space is defined by the separation property where any two distinct points can be enclosed in disjoint open sets.21

The reason every compact subset of a Hausdorff space is closed relies on this ability to separate points. If you take a point $y$ that is not in a compact subset $K$ , you can use the Hausdorff property to separate $y$ from every point $x$ in $K$ using disjoint open neighborhoods. Because $K$ is compact, you can extract a finite subcollection of these neighborhoods to cover $K$ while keeping them disjoint from a neighborhood of $y$ . This construction ensures that $y$ has an open neighborhood that does not intersect $K$ , meaning the complement of $K$ is open and $K$ itself must be closed.21

This property is a fundamental result in general topology, distinguishing Hausdorff spaces from more general topological spaces where compact sets are not necessarily closed.2