Definition:Discrete Topology/Uncountable

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 $S$:
 * $\tau := \left\{ {H: H \subseteq S}\right\}$

Let $S$ be an uncountably infinite set.

Then $\tau = \mathcal P \left({S}\right)$ is an uncountable discrete topology, and $\left({S, \tau}\right) = \left({S, \mathcal P \left({S}\right)}\right)$ is an uncountable discrete space.

Also see

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


 * Properties of Discrete Topology