Definition:Discrete Topology/Uncountable

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 $S$:
 * $\tau := \set {H: H \subseteq S}$

Let $S$ be an uncountably infinite set.

Then $\tau = \powerset S$ is an uncountable discrete topology, and $\struct {S, \tau} = \struct {S, \powerset S}$ is an uncountable discrete space.

Also see

 * Definition:Finite Discrete Topology
 * Definition:Countable Discrete Topology


 * Properties of Discrete Topology