---
tag: MM0QPT0GT0
kind: concept
name: Linear continuum
msc: [54F05, 06A05]
specializes: [0FRXBVVR03]
sources:
  - url: https://en.wikipedia.org/wiki/Linear_continuum
same_as:
  wikidata: Q3688953
  wikipedia: Linear continuum
---

## Summary

A linear continuum is a [[totally ordered set|0FRXBVVR03]] with more than one element that has no gaps and no jumps.
Every nonempty subset bounded above has a [[supremum|0R3H5PANRQ]], and between any two elements lies a third.
The real line is the model example.
In the order topology, every linear continuum is [[connected|0V032VWTHD]], and so are its intervals.

## Explanations

### L1 human

A linear continuum is an ordered line without holes.
The rational numbers have holes, for example at $\sqrt{2}$, and the integers have jumps, from $1$ straight to $2$.
The real numbers have neither, so they form a linear continuum.
This is exactly what makes the real line connected and makes the intermediate value theorem true.