\begin{definition}[Definition:Closed Extension Topology]
Let $T = \struct {S, \tau}$ be a topological space.
Let $p$ be a new element for $S$ such that $S^*_p := S \cup \set p$.
Let $\tau^*_p$ be the set defined as:
:$\tau^*_p := \set {U \cup \set p: U \in \tau} \cup \set \O$
That is, $\tau^*_p$ is the set of all sets formed by adding $p$ to all the open sets of $\tau$ and including the empty set.
Then:
:$\tau^*_p$ is the closed extension topology of $\tau$
and:
:$T^*_p := \struct {S^*_p, \tau^*_p}$ is the closed extension space of $T = \struct {S, \tau}$.
\end{definition}