Definition:Excluded Point Topology

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

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