Indecomposable continuum
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  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 non-path-connected continuum,  Warsaw circle is a path-connected continuum that is not locally path-connected. A subcontinuum  of a continuum  is a closed, connected subset of . A space is nondegenerate if it is not equal to a single point. A continuum  is decomposable if there exist two subcontinua  and  of  such that  and  but . It follows that  and  are nondegenerate. A continuum that is not decomposable is an indecomposable continuum. A continuum  in which every subcontinuum is indecomposable is said to be hereditarily indecomposable. A composant of an indecomposable continuum  is a maximal set in which any two points lie within some proper subcontinuum of . A continuum  is irreducible between  and  if   and no proper subcontinuum contains both points. For a nondegenerate indecomposable metric continuum , there exists an uncountable subset  such that  is irreducible between any two points of . History
 In 1910 L. E. J. Brouwer described an indecomposable continuum that disproved a conjecture made by Arthur Moritz Schoenflies that, if  and  are open, connected, disjoint sets in  such that , then  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. Bucket handle example
Indecomposable continua are often constructed as the limit of a sequence of nested intersections, or (more generally) as the inverse limit of a sequence of continua. The buckethandle, or Brouwer–Janiszewski–Knaster continuum, is often considered the simplest example of an indecomposable continuum, and can be so constructed (see upper right). Alternatively, take the Cantor ternary set  projected onto the interval  of the -axis in the plane. Let  be the family of semicircles above the -axis with center  and with endpoints on  (which is symmetric about this point). Let  be the family of semicircles below the -axis with center the midpoint of the interval  and with endpoints in . Let  be the family of semicircles below the -axis with center the midpoint of the interval  and with endpoints in . Then the union of all such  is the bucket handle. The bucket handle admits no Borel transversal, that is there is no Borel set containing exactly one point from each composant. Properties

In a sense, 'most' continua are indecomposable. Let  be an -cell with metric ,  the set of all nonempty closed subsets of , and  the hyperspace of all connected members of  equipped with the Hausdorff metric  defined by . Then the set of nondegenerate indecomposable subcontinua of  is dense in . In dynamical systems

In 1932 George Birkhoff described his "remarkable closed curve", a homeomorphism of the annulus that contained an invariant continuum. Marie Charpentier showed that this continuum was indecomposable, the first link from indecomposable continua to dynamical systems. The invariant set of a certain Smale horseshoe map is the bucket handle. Marcy Barge and others have extensively studied indecomposable continua in dynamical systems. See also 
Indecomposability (constructive mathematics)
Lakes of Wada, three open subsets of the plane whose boundary is an indecomposable continuum
Solenoid
Sierpinski carpet

References

External links

 
 explains Brouwer's picture of his indecomposable continuum that appears on the front cover of the journal. Continuum theory