---
tag: EK0FJBNE1Q
kind: definition
name: Indecomposable continuum
msc: [54F15]
defines: 56J7DSVAZH
uses: [NJQ196M1WR, 0RZH6W98HK]
sources:
  - url: https://en.wikipedia.org/wiki/Indecomposable_continuum
---

## Definition

Let $X$ be a [[continuum|NJQ196M1WR]].
The continuum $X$ is *decomposable* if there are subcontinua $A$ and $B$ of $X$ with $A \neq X$, $B \neq X$ and $X = A \cup B$.
The continuum $X$ is *indecomposable* if it is not decomposable.
It is *hereditarily indecomposable* if every subcontinuum of $X$ is indecomposable.

## Remarks

The interval $[0, 1]$ is decomposable, since $[0, 1] = [0, 1/2] \cup [1/2, 1]$ with both pieces proper subcontinua.
A one-point continuum is indecomposable in a trivial way, because its only subcontinuum is itself.
The Knaster bucket handle and the [[pseudo-arc|VGCTDCGS6V]] are nondegenerate indecomposable continua.
A continuum is indecomposable if and only if every proper subcontinuum has empty [[interior|0RZH6W98HK]].

## Explanations

### L1 human

An indecomposable continuum is a compact space in one piece that cannot be written as the union of two smaller compact pieces in one piece.