Math 2.0

General topology Definition

Indecomposable continuum

Builds on ContinuumInterior

Definition

Let XX be a continuum. The continuum XX is decomposable if there are subcontinua AA and BB of XX with A≠XA \neq X, B≠XB \neq X and X=A∪BX = A \cup B. The continuum XX is indecomposable if it is not decomposable. It is hereditarily indecomposable if every subcontinuum of XX is indecomposable.

Remarks

The interval [0,1][0, 1] is decomposable, since [0,1]=[0,1/2]∪[1/2,1][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 are nondegenerate indecomposable continua. A continuum is indecomposable if and only if every proper subcontinuum has empty interior.

Reading path

To understand this page from the ground up, read these 3 pages in order. The foundations come first.

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Continuum, concept, no evidence levelContinuumconceptInterior, concept, no evidence levelInteriorconceptIndecomposable continuum, definition, no evidence levelIndecomposable contin…definition
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  1. Continuum concept
  2. Interior concept
  3. Indecomposable continuum definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Indecomposable_continuum
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Changed2 November 2026

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