Alexandrov one-point compactification AlexandrovOnePointCompactification 2003-07-27 10:29:03 2005-02-06 04:32:33 Definition compactification \PMlinkescapeword{open} The \emph{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.