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" formula_1 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 and Warsaw circle are examples of non -path-connected continua. A "subcontinuum" formula_2 of a continuum formula_1 is a closed, connected subset of formula_1. A space is "nondegenerate" if it is not equal to a single point. A continuum formula_1 is "decomposable" if there exist two nondegenerate subcontinua formula_6 and formula_7 of formula_1 such that formula_9 and formula_10 but formula_11. A continuum that is not decomposable is an "indecomposable continuum". A continuum formula_1 in which every subcontinuum is indecomposable is said to be "hereditarily indecomposable". A "composant" of an indecomposable continuum formula_1 is a maximal set in which any two points lie within some proper subcontinuum of  formula_1. A continuum formula_1 is "irreducible between formula_16 and formula_17" if formula_18 and no proper subcontinuum contains both points. An indecomposable continuum is irreducible between any two of its points. History. In 1910 L. E. J. Brouwer described an indecomposable continuum that disproved a conjecture made by Arthur Moritz Schoenflies that, if formula_19 and formula_20 are open, connected, disjoint sets in formula_21 such that formula_22, then formula_22 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 formula_24 projected onto the interval formula_25 of the formula_26-axis in the plane. Let formula_27 be the family of semicircles above the formula_26-axis with center formula_29 and with endpoints on formula_24 (which is symmetric about this point). Let formula_31 be the family of semicircles below the formula_26-axis with center the midpoint of the interval formula_33 and with endpoints in formula_34. Let formula_35 be the family  of semicircles below the formula_26-axis with center the midpoint of the interval formula_37 and with endpoints in formula_38. Then the union of all such formula_35 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 formula_40 be an formula_41-cell with metric formula_42, formula_43 the set of all nonempty closed subsets of formula_40, and formula_45 the "hyperspace" of all connected members of formula_43 equipped with the Hausdorff metric formula_47 defined by formula_48. Then the set  of nondegenerate indecomposable subcontinua of formula_40 is dense in formula_45. 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.