\begin{definition}[Definition:Excluded Point Topology]
Let $S$ be a set which is non-empty.
Let $p \in S$ be some '''particular point''' of $S$.
We define a subset $\tau_{\bar p}$ of the power set $\powerset S$ as:
:$\tau_{\bar p} = \set {A \subseteq S: p \notin A} \cup \set S$
That is, all the subsets of $S$ which do not include $p$, along with the set $S$.
Then $\tau_{\bar p}$ is a topology called the '''excluded point topology on $S$ by $p$''', or just '''an excluded point topology'''.
The topological space $T = \struct {S, \tau_{\bar p} }$ is called the '''excluded point space on $S$ by $p$''', or just '''an excluded point space'''.
\end{definition}