Definition:Discrete Topology

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

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

That is, let $\tau$ be the set of all subsets of $S$:
 * $\tau := \left\{ {H: H \subseteq S}\right\}$

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

Also see

 * Properties of Discrete Topology
 * Definition:Indiscrete Topology