---
tag: K4D3S1MBZH
kind: concept
name: Dense order
msc: [06A05, 06A06]
sources:
  - url: https://en.wikipedia.org/wiki/Dense_order
same_as:
  wikidata: Q194699
  wikipedia: Dense order
---

## Summary

An order is dense when between any two elements with $x < y$ there is a third element $z$ with $x < z < y$.
The rational numbers and the real numbers are dense, while the integers are not.
By [[Cantor's isomorphism theorem|ABN5PTXEHY]], every countable dense total order without endpoints, and with at least one element, looks exactly like the rationals.

## Explanations

### L1 human

In a dense order there are no gaps of the "next element" kind: any two distinct elements have something strictly between them.
Between the fractions $1/3$ and $1/2$ lies their average $5/12$.
Between the integers $1$ and $2$ there is no integer, so the integers are not dense.
Repeating the argument, between two rationals there are infinitely many rationals.