Definition:Particular 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 $\vartheta_p$ of the power set $\mathcal P \left({X}\right)$ as:
 * $\vartheta_p = \left\{{A \subseteq S: p \in A}\right\} \cup \left\{{\varnothing}\right\}$

... that is, all the subsets of $S$ which include $p$, along with the empty set.

Then $\vartheta_p$ is a topology called the particular point topology on $S$ by $p$, or just a particular point topology.

The topological space $T = \left({S, \vartheta_p}\right)$ is called the particular point space on $S$ by $p$, or just a particular point space.

Finite Particular Point Topology
If $S$ is finite, $\vartheta_p$ is a finite particular point topology, and $\left({S, \vartheta_p}\right)$ is a finite particular point space.

Infinite Particular Point Topology
If $S$ is infinite, $\vartheta_p$ is an infinite particular point topology, and $\left({S, \vartheta_p}\right)$ is a infinite particular point space.

Countable Particular Point Topology
If $S$ is countably infinite, $\vartheta_p$ is a countable particular point topology, and $\left({S, \vartheta}\right)$ is a countable particular point space.

Uncountable Particular Point Topology
If $S$ is uncountable, $\vartheta_p$ is an uncountable particular point topology, and $\left({S, \vartheta_p}\right)$ is an uncountable particular point space.