Definition:Excluded Point Topology

Definition
Let $S$ be a set which is non-null.

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.

Finite Excluded Point Topology
If $S$ is finite, $\tau_{\bar p}$ is a finite excluded point topology, and $\left({S, \tau_{\bar p}}\right)$ is a finite excluded point space.

Infinite Excluded Point Topology
If $S$ is infinite, $\tau_{\bar p}$ is an infinite excluded point topology, and $\left({S, \tau_{\bar p}}\right)$ is a infinite excluded point space.

Countable Excluded Point Topology
If $S$ is countably infinite, $\tau_{\bar p}$ is a countable excluded point topology, and $\left({S, \tau_{\bar p}}\right)$ is a countable excluded point space.

Uncountable Excluded Point Topology
If $S$ is uncountable, $\tau_{\bar p}$ is an uncountable excluded point topology, and $\left({S, \tau_{\bar p}}\right)$ is an uncountable excluded point space.