In point-set topology, an indecomposable continuum is a continuum that is indecomposable, i.e. that cannot be expressed as the union of any two of its proper subcontinua. In 1910, L. E. J. Brouwer was the first to describe an indecomposable continuum. Indecomposable continua have been used by topologists as a source of counterexamples. They also occur in dynamical systems.
Definitions A continuum C {\displaystyle C} is a nonempty compact connected metric space. The arc, the n-sphere, and the Hilbert cube are examples of path-connected continua; the topologist's sine curve is an example of a continuum that is not path-connected. The Warsaw circle is a path-connected continuum that is not locally path-connected. A subcontinuum C ′ {\displaystyle C'} of a continuum C {\displaystyle C} is a closed, connected subset of C {\displaystyle C} . A space is nondegenerate if it is not equal to a single point. A continuum C {\displaystyle C} is decomposable if there exist two subcontinua A {\displaystyle A} and B {\displaystyle B} of C {\displaystyle C} such that A ≠ C {\displaystyle A\neq C} and B ≠ C {\displaystyle B\neq C} but A ∪ B = C {\displaystyle A\cup B=C} . It follows that A {\displaystyle A} and B {\displaystyle B} are nondegenerate. A continuum that is not decomposable is an indecomposable continuum. A continuum C {\displaystyle C} in which every subcontinuum is indecomposable is said to be hereditarily indecomposable. A composant of an indecomposable continuum C {\displaystyle C} is a maximal set in which any two points lie within some proper subcontinuum of C {\displaystyle C} . A continuum C {\displaystyle C} is irreducible between c {\displaystyle c} and c ′ {\displaystyle c'} if c , c ′ ∈ C {\displaystyle c,c'\in C} and no proper subcontinuum contains both points. For a nondegenerate indecomposable metric continuum X {\displaystyle X} , there exists an uncountable subset J {\displaystyle J} such that X {\displaystyle X} is irreducible between any two points of J {\displaystyle J} .
History In 1910 L. E. J. Brouwer described an indecomposable continuum that disproved a conjecture made by Arthur Moritz Schoenflies that, if X 1 {\displaystyle X_{1}} and X 2 {\displaystyle X_{2}} are open, connected, disjoint sets in R 2 {\displaystyle \mathbb {R} ^{2}} such that ∂ X 1 = ∂ X 2 {\displaystyle \partial X_{1}=\partial X_{2}} , then ∂ X 1 = ∂ X 2 {\displaystyle \partial X_{1}=\partial X_{2}} must be the union of two closed, connected proper subsets. Zygmunt Janiszewski described more such indecomposable continua, including a version of the bucket handle. Janiszewski, however, focused on the irreducibility of these continua. In 1917 Kunizo Yoneyama described the Lakes of Wada (named after Takeo Wada) whose common boundary is indecomposable. In the 1920s indecomposable continua began to be studied by the Warsaw School of Mathematics in Fundamenta Mathematicae for their own sake, rather than as pathological counterexamples. Stefan Mazurkiewicz was the first to give the definition of indecomposability. In 1922 Bronisław Knaster described the pseudo-arc, the first example found of a hereditarily indecomposable continuum.
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