Definition:Discrete Topology

Definition
Let $A \ne \varnothing$ be a set.

Let $\vartheta = \mathcal P \left({A}\right)$ be the power set of $A$.

Then $\vartheta$ is called the discrete topology on $A$ and $\left({A, \vartheta}\right) = \left({A, \mathcal P \left({A}\right)}\right)$ the discrete space on $A$, or just a discrete space.

Finite Discrete Topology
If $A$ is finite, $\vartheta = \mathcal P \left({A}\right)$ is a finite discrete topology, and $\left({A, \vartheta}\right) = \left({A, \mathcal P \left({A}\right)}\right)$ is a finite discrete space.

Infinite Discrete Topology
If $A$ is infinite, $\vartheta = \mathcal P \left({A}\right)$ is an infinite discrete topology, and $\left({A, \vartheta}\right) = \left({A, \mathcal P \left({A}\right)}\right)$ is an infinite discrete space.

Countable Discrete Topology
If $A$ is countably infinite, $\vartheta = \mathcal P \left({A}\right)$ is a countable discrete topology, and $\left({A, \vartheta}\right) = \left({A, \mathcal P \left({A}\right)}\right)$ is a countable discrete space.

Uncountable Discrete Topology
If $A$ is uncountable, $\vartheta = \mathcal P \left({A}\right)$ is an uncountable discrete topology, and $\left({A, \vartheta}\right) = \left({A, \mathcal P \left({A}\right)}\right)$ is an uncountable discrete space.