\begin{definition}[Definition:Discrete Topology]
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'''.
\end{definition}