Alexandrov one-point compactification

The Alexandrov one-point compactification of a non-compact topological space $X$ is obtained by adjoining a new point $\infty$ and defining the topology on $X\cup\{\infty\}$ to consist of the open sets of $X$ together with the sets of the form $U\cup\{\infty\}$ , where $U$ is an open subset of $X$ with compact complement.

With this topology, $X\cup\{\infty\}$ is always compact. Furthermore, it is Hausdorff if and only if $X$ is Hausdorff and locally compact.