---
tag: 0D198GPMTJ
kind: definition
name: Dense set
aliases: [dense-subset]
msc: [54A05]
defines: 0BSJSX7ME2
uses: [0DRKT1QJR7]
sources:
  - url: https://en.wikipedia.org/wiki/Dense_set
    locator: Definition
---

## Definition

Let $X$ be a [[topological space|0CNQXVBHCX]] and let $D \subseteq X$.
The set $D$ is dense in $X$ when its [[closure|0DRKT1QJR7]] is $X$, that is, $\overline{D} = X$.

## Remarks

Equivalently, $D$ is dense if and only if $D$ meets each nonempty open set of $X$.
In a metric space, $D$ is dense if and only if for each $x \in X$ and each $\varepsilon > 0$ there is $y \in D$ with $d(x, y) < \varepsilon$.
The rationals $\mathbb{Q}$ are dense in $\mathbb{R}$: this is the [[density of the rationals|0STX0WRDNG]]. The irrationals are also dense in $\mathbb{R}$.

## Explanations

### L1 human

A dense set has points everywhere in the space: near each point of the space there are points of the dense set.
Each real number has rational numbers as close to it as you want.