# 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

- 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