Indecomposable continuum

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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.

The first four stages of the construction of the bucket handle as the limit of a series of nested intersections

Indecomposable continua have been used by topologists as a source of counterexamples. They also occur in dynamical systems.

Definitions

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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   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  .[1]

History

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Fifth stage of the Lakes of Wada

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.[2] 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.[3]

Bucket handle example

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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.[4]

The bucket handle admits no Borel transversal, that is there is no Borel set containing exactly one point from each composant.

Properties

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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

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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.[5]

See also

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References

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  1. ^ Nadler, Sam (2017). Continuum Theory: An Introduction. CRC Press. ISBN 9781351990530.
  2. ^ Brouwer, L. E. J. (1910), "Zur Analysis Situs", Mathematische Annalen, 68 (3): 422–434, doi:10.1007/BF01475781, S2CID 120836681
  3. ^ Cook, Howard; Ingram, William T.; Kuperberg, Krystyna; Lelek, Andrew; Minc, Piotr (1995). Continua: With the Houston Problem Book. CRC Press. p. 103. ISBN 9780824796501.
  4. ^ Ingram, W. T.; Mahavier, William S. (2011). Inverse Limits: From Continua to Chaos. Springer Science & Business Media. p. 16. ISBN 9781461417972.
  5. ^ Kennedy, Judy (1 December 1993). "How Indecomposable Continua Arise in Dynamical Systems". Annals of the New York Academy of Sciences. 704 (1): 180–201. Bibcode:1993NYASA.704..180K. doi:10.1111/j.1749-6632.1993.tb52522.x. ISSN 1749-6632. S2CID 85143246.
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  • Solecki, S. (2002). "Descriptive set theory in topology". In Hušek, M.; van Mill, J. (eds.). Recent progress in general topology II. Elsevier. pp. 506–508. ISBN 978-0-444-50980-2.
  • Casselman, Bill (2014), "About the cover" (PDF), Notices of the AMS, 61: 610, 676 explains Brouwer's picture of his indecomposable continuum that appears on the front cover of the journal.