Definition:Discrete Topology/Uncountable

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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

  • Results about discrete topologies can be found here.

Linguistic Note

Be careful with the word discrete.

A common homophone horror is to use the word discreet instead.

However, discreet means cautious or tactful, and describes somebody who is able to keep silent for political or delicate social reasons.