---
tag: 0KSM23SN3H
kind: definition
name: Nowhere dense set
msc: ["54E52"]
defines: 0VQYCCQVJ5
uses: [0DRKT1QJR7, 0C38S0KMJK]
sources:
  - url: https://en.wikipedia.org/wiki/Nowhere_dense_set
    locator: Definition by closure
---

## Definition

Let $X$ be a [[topological space|0CNQXVBHCX]] and let $A \subseteq X$.
The set $A$ is nowhere dense in $X$ when the [[interior|0C38S0KMJK]] of its [[closure|0DRKT1QJR7]] is empty, that is, $\operatorname{int} \overline{A} = \emptyset$.

## Remarks

The set $\mathbb{Z}$ is nowhere dense in $\mathbb{R}$. So is each finite subset of $\mathbb{R}$.
The [[Cantor set|0BXNDZFQME]] is nowhere dense in $\mathbb{R}$, though it is uncountable.
The set $\mathbb{Q}$ is not nowhere dense in $\mathbb{R}$, because its closure is $\mathbb{R}$.
A closed set is nowhere dense if and only if its complement is open and [[dense|0D198GPMTJ]].

## Explanations

### L1 human

A nowhere dense set is thin everywhere: inside each open set, you can find a smaller open set that misses it completely.
The integers on the real line are an example, since there are gaps between them.