---
tag: HG4S5HNWXF
kind: definition
name: Continuum
msc: [54F15]
defines: NJQ196M1WR
uses: [0VKD1TP7F3, 0TVJZYB6SH, 0V032VWTHD, 0S1FJ8VEX9]
sources:
  - url: https://en.wikipedia.org/wiki/Continuum_(topology)
---

## Definition

A *continuum* is a nonempty [[metric space|0VKD1TP7F3]] that is [[compact|0TVJZYB6SH]] and [[connected|0V032VWTHD]].
A continuum is *nondegenerate* if it has more than one point.
A *subcontinuum* of a continuum $X$ is a subset of $X$ that is a continuum in the induced metric.

## Remarks

The closed interval $[0, 1]$ is a continuum: it is compact by the Heine–Borel theorem and connected because it is an [[interval|0Q1D85B8G6]].
The set $[0, 1] \cup [2, 3]$ is compact but not connected, so it is not a continuum.
Some authors define a continuum as a nonempty compact connected [[Hausdorff space|0S1FJ8VEX9]], and call the metric ones metric continua.
The intersection of a decreasing sequence of continua is a continuum.

## Explanations

### L1 human

A continuum is a compact space in one piece.
Arcs, circles and filled squares are continua.