# Definition:Discrete Topology

*This page is about Discrete Topology in the context of topology. For other uses, see discrete.*

## 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 $S$:

- $\tau := \set {H: H \subseteq S}$

Then $\tau$ is called **the discrete topology on $S$** and $\struct {S, \tau} = \struct {S, \powerset S}$ **the discrete space on $S$**, or just **a discrete space**.

### Finite Discrete Topology

Let $S$ be a finite set.

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

### Infinite Discrete Topology

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**.

## 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

- 1964: Steven A. Gaal:
*Point Set Topology*... (previous) ... (next): Chapter $\text {I}$: Topological Spaces: $1$. Open Sets and Closed Sets - 1975: Bert Mendelson:
*Introduction to Topology*(3rd ed.) ... (previous) ... (next): Chapter $3$: Topological Spaces: $\S 2$: Topological Spaces: Example $5$ - 1975: W.A. Sutherland:
*Introduction to Metric and Topological Spaces*... (previous) ... (next): $3$: Continuity generalized: topological spaces: $3.1$: Topological Spaces: Example $3.1.4$ - 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*(2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $1 \text { - } 3$. Discrete Topology