Definition:Discrete Topology/Countable

Definition

Let $S \ne \O$ be an infinite 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 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.

Also see

• Definition:Finite Discrete Topology
• Definition:Infinite Discrete Topology

• Definition:Uncountable 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: $2$. Countable Discrete Topology