Compact subset of a Hausdorff space is closed

Compact subset of a Hausdorff space is closed: Let $X$ be a Hausdorff space . If $K\subseteq X$ is compact , then $K$ is a closed set in $X$.

This is one of the basic structural features of Hausdorff spaces and is used repeatedly in compactness arguments, including the compact-to-Hausdorff homeomorphism criterion and uniqueness phenomena such as uniqueness of limits .