---
tag: 0S2TEHX4E1
kind: example
name: Discrete topology
aliases: [discrete-space]
msc: [54A05]
instance_of: [0CQHFPGJ45]
sources:
  - url: https://en.wikipedia.org/wiki/Discrete_space
    locator: Definitions
same_as:
  wikidata: Q175116
  wikipedia: Discrete space
---

## Example

Let $X$ be a set and let $\tau = \mathcal{P}(X)$ be the set of all subsets of $X$.
Then $\tau$ is a [[topology|0CNQXVBHCX]] on $X$, the discrete topology. The space $(X, \tau)$ is a discrete space.

## Remarks

Each subset of $X$ is both open and closed. Each point $x$ gives an open set $\{x\}$.
It is the finest topology on $X$. It is the [[metric topology|0A6YRNZB2R]] of the [[discrete metric|0NA4FK7B8S]].
Each function from a discrete space to any topological space is [[continuous|0SY9ECQ19Q]].
A discrete space is [[compact|0QPHW703PV]] if and only if it is finite.

## Explanations

### L1 human

In the discrete topology, every set is open, so each point is isolated from all other points.
The integers $\mathbb{Z}$, as a subspace of $\mathbb{R}$, carry the discrete topology.