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 of a continuum that is not path-connected. The Warsaw circle is a path-connected continuum that is not locally path-connected. A subcontinuum is a closed. A space is nondegenerate is decomposable are nondegenerate. A continuum that is not decomposable is an indecomposable continuum in which every subcontinuum is indecomposable is said to be hereditarily indecomposable. A composant is .

History

In 1910 L. E. J. Brouwer described an indecomposable continuum that disproved a conjecture made by Arthur Moritz Schoenflies 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 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

be an n-cell with metric the hyperspace equipped with the Hausdorff metric is dense.

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.