Definition:Discrete Topology/Infinite

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 infinite set.

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

Countable Discrete Topology

Let $S$ be a countably infinite set.

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

Uncountable Discrete Topology

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

• Properties of Discrete Topology

• 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.

Sources

• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $2 \text { - } 3$. Infinite Discrete Topology