\begin{definition}[Definition:Alexandroff Extension]
Let $T = \struct {S, \tau}$ be a non-empty topological space.
Let $p$ be a new element not in $S$.
Let $S^* := S \cup \set p$.
Let $\tau^*$ be the topology on $S^*$ defined such that $U \subseteq S^*$ is open {{iff}}:
:$U$ is an open set of $T$
or
:$U$ is the complement in $T^*$ of a closed and compact subset of $T$.
This topology is called the '''Alexandroff extension''' on $S$.
\end{definition}