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.

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.

Linguistic Note
Be careful how you spell this. A common homophone horror is to refer to this as the discreet topology.

However, discreet means cautious or tactful, and describes somebody who is able to keep silent for political or delicate social reasons.