In
point-set topology
In mathematics, general topology is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differential topology, geomet ...
, an indecomposable continuum is a
continuum
Continuum may refer to:
* Continuum (measurement), theories or models that explain gradual transitions from one condition to another without abrupt changes
Mathematics
* Continuum (set theory), the real line or the corresponding cardinal number ...
that is indecomposable, i.e. that cannot be expressed as the union of any two of its
proper
Proper may refer to:
Mathematics
* Proper map, in topology, a property of continuous function between topological spaces, if inverse images of compact subsets are compact
* Proper morphism, in algebraic geometry, an analogue of a proper map for ...
subcontinua. In 1910,
L. E. J. Brouwer
Luitzen Egbertus Jan Brouwer (; ; 27 February 1881 – 2 December 1966), usually cited as L. E. J. Brouwer but known to his friends as Bertus, was a Dutch mathematician and philosopher, who worked in topology, set theory, measure theory and compl ...
was the first to describe an indecomposable continuum.
Indecomposable continua have been used by topologists as a source of
counterexample
A counterexample is any exception to a generalization. In logic a counterexample disproves the generalization, and does so rigorously in the fields of mathematics and philosophy. For example, the fact that "John Smith is not a lazy student" is a ...
s. They also occur in
dynamical systems
In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a p ...
.
Definitions
A ''continuum''
is a nonempty
compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in British ...
connected
Connected may refer to:
Film and television
* ''Connected'' (2008 film), a Hong Kong remake of the American movie ''Cellular''
* '' Connected: An Autoblogography About Love, Death & Technology'', a 2011 documentary film
* ''Connected'' (2015 TV ...
metric space
In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general settin ...
. The
arc, the
''n''-sphere, and the
Hilbert cube
In mathematics, the Hilbert cube, named after David Hilbert, is a topological space that provides an instructive example of some ideas in topology. Furthermore, many interesting topological spaces can be embedded in the Hilbert cube; that is, c ...
are examples of
path-connected
In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint non-empty open subsets. Connectedness is one of the principal topological properties that ...
continua; the
topologist's sine curve
In the branch of mathematics known as topology, the topologist's sine curve or Warsaw sine curve is a topological space with several interesting properties that make it an important textbook example.
It can be defined as the graph of the functio ...
and
Warsaw circle
Shape theory is a branch of topology that provides a more global view of the topological spaces than homotopy theory. The two coincide on compacta dominated homotopically by finite polyhedra. Shape theory associates with the Čech homology theory ...
are examples of non-path-connected continua. 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 nondegenerate subcontinua
and
of
such that
and
but
. 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 In point-set topology, the composant of a point ''p'' in a continuum ''A'' is the union of all proper subcontinua of ''A'' that contain ''p''. If a continuum is indecomposable, then its composants are pairwise disjoint. The composants of a cont ...
'' 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. An indecomposable continuum is irreducible between any two of its points.
History
![Lakes of Wada](https://upload.wikimedia.org/wikipedia/commons/b/b9/Lakes_of_Wada.png)
In 1910 L. E. J. Brouwer described an indecomposable continuum that disproved a conjecture made by
Arthur Moritz Schoenflies
Arthur Moritz Schoenflies (; 17 April 1853 – 27 May 1928), sometimes written as Schönflies, was a German mathematician, known for his contributions to the application of group theory to crystallography, and for work in topology.
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
Zygmunt Janiszewski (12 July 1888 – 3 January 1920) was a Polish mathematician.
Early life and education
He was born to mother Julia Szulc-Chojnicka and father, Czeslaw Janiszewski who was a graduate of the University of Warsaw and served as t ...
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
was a Japanese mathematician at Kyoto University working in topology. In 1917, he published the construction of the Lakes of Wada, which he named after his teacher Takeo Wada
was a Japanese mathematician at Kyoto University working in anal ...
described the
Lakes of Wada
In mathematics, the are three disjoint connected open sets of the plane or open unit square with the counterintuitive property that they all have the same boundary. In other words, for any point selected on the boundary of ''one'' of the lakes ...
(named after
Takeo Wada
was a Japanese mathematician at Kyoto University working in analysis and topology
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deform ...
) whose common boundary is indecomposable. In the 1920s indecomposable continua began to be studied by the
Warsaw School of Mathematics
Warsaw School of Mathematics is the name given to a group of mathematicians who worked at Warsaw, Poland, in the two decades between the World Wars, especially in the fields of logic, set theory, point-set topology and real analysis. They publishe ...
in ''
Fundamenta Mathematicae
''Fundamenta Mathematicae'' is a peer-reviewed scientific journal of mathematics with a special focus on the foundations of mathematics, concentrating on set theory, mathematical logic, topology and its interactions with algebra, and dynamical syst ...
'' for their own sake, rather than as pathological counterexamples.
Stefan Mazurkiewicz
Stefan Mazurkiewicz (25 September 1888 – 19 June 1945) was a Polish mathematician who worked in mathematical analysis, topology, and probability. He was a student of Wacław Sierpiński and a member of the Polish Academy of Learning (''PAU''). ...
was the first to give the definition of indecomposability. In 1922
Bronisław Knaster
Bronisław Knaster (22 May 1893 – 3 November 1980) was a Polish mathematician; from 1939 a university professor in Lwów and from 1945 in Wrocław.
He is known for his work in point-set topology and in particular for his discoveries in 1922 of ...
described the
pseudo-arc In general topology, the pseudo-arc is the simplest nondegenerate hereditarily indecomposable continuum. The pseudo-arc is an arc-like homogeneous continuum, and played a central role in the classification of homogeneous planar continua. R. H. Bi ...
, 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
In mathematics, the inverse limit (also called the projective limit) is a construction that allows one to "glue together" several related objects, the precise gluing process being specified by morphisms between the objects. Thus, inverse limits can ...
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
In mathematics, the Cantor set is a set of points lying on a single line segment that has a number of unintuitive properties. It was discovered in 1874 by Henry John Stephen Smith and introduced by German mathematician Georg Cantor in 1883.
Thr ...
projected onto the interval