Definition:Alexandroff Extension

Definition
Let $T = \left({S, \tau}\right)$ be a non-empty topological space.

Let $p$ be a point which is not in $S$.

Let $S^* := S \cup \left\{{p}\right\}$.

Let $\tau^*$ be the topology on $S^*$ defined such that $U \subseteq S^*$ is open :
 * $U$ 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 one point compactification topology on $S$.

Also see

 * One Point Compactification Topology is Topology