\begin{definition}[Definition:Excluded Set Topology]
Let $S$ be a set which is non-empty.
Let $H \subseteq S$ be some subset of $S$.
We define a subset $\tau_{\bar H}$ of the power set $\powerset S$ as:
:$\tau_{\bar H} = \set {A \subseteq S: A \cap H = \O} \cup \set S$
that is, all the subsets of $S$ which are disjoint from $H$, along with the set $S$.
Then $\tau_{\bar H}$ is a topology called the '''excluded set topology on $S$ by $H$''', or just '''an excluded set topology'''.
The topological space $T = \struct {S, \tau_{\bar H} }$ is called the '''excluded set space on $S$ by $H$''', or just '''an excluded set space'''.
\end{definition}