Definition:Discrete Topology

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

Let $\tau = \powerset S$ be the power set of $S$.

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

Then $\tau$ is called the discrete topology on $S$ and $\struct {S, \tau} = \struct {S, \powerset S}$ the discrete space on $S$, or just a discrete space.

Also see

 * Properties of Discrete Topology
 * Definition:Indiscrete Topology