---
tag: 0GKJ78948V
kind: definition
name: Separable space
msc: [54D65]
defines: 0Z2P41SVDP
uses: [0D198GPMTJ, 0ZG61TFBES]
sources:
  - url: https://en.wikipedia.org/wiki/Separable_space
    locator: lead section
---

## Definition

A [[topological space|0CNQXVBHCX]] $X$ is separable when there is a [[countable|0ZG61TFBES]] subset $D \subseteq X$ that is [[dense|0D198GPMTJ]] in $X$.

## Remarks

The real line is separable, because $\mathbb{Q}$ is countable and dense. In the same way $\mathbb{R}^n$ is separable, with the dense set $\mathbb{Q}^n$.
Each [[second-countable space|00RE1R4W7N]] is separable. For metric spaces, the converse is also true.
An uncountable set with the [[discrete metric|0NA4FK7B8S]] is not separable, because each dense subset is the whole set.

## Explanations

### L1 human

A list of points can approximate a separable space. Each point of the space has points of the list as close as you want.
For $\mathbb{R}$, the list of all fractions does the job.