Definition:Excluded Set Topology

Definition
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 $\mathcal P \left({S}\right)$ as:
 * $\tau_{\bar H} = \left\{{A \subseteq S: A \cap H = \varnothing}\right\} \cup \left\{{S}\right\}$

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 $T = \left({S, \tau_{\bar H}}\right)$ is called the excluded set space on $S$ by $H$, or just an excluded set space.