\begin{definition}[Definition:Open Extension Topology]
Let $T = \struct {S, \tau}$ be a topological space.
Let $p$ be a new element which is not in $S$.
Let $S^*_p = S \cup \set p$.
Let $\tau^*_p$ be the set defined as:
:$\tau^*_{\bar p} = \set {U: U \in \tau} \cup \set {S^*_p}$
That is, $\tau^*_{\bar p}$ is the set of all sets formed by taking all the open sets of $\tau$ and adding to them the set $S^*_p$.
Then:
:$\tau^*_{\bar p}$ is the open extension topology of $\tau$
and:
:$T^*_{\bar p} = \struct {S^*_p, \tau^*_{\bar p} }$ is the open extension space of $T = \struct {S, \tau}$.
The open sets of $T^*_{\bar p}$ can be seen to be the same as the open sets of $T$, but with $S^*_p$ added.
\end{definition}