Indecomposable continuum
Encyclopedia
In point-set topology, an indecomposable continuum is a continuum
that is not the union of any two of its proper subcontinua. The pseudo-arc
is an example of a hereditarily indecomposable continuum. L. E. J. Brouwer discovered the first indecomposable continuum in 1910.
Indecomposable continua have historically appeared as counterexamples to various conjectures, and because of this they are often viewed as pathological objects. However, they can occur in applications, such as attractors in dynamical systems.
, Janiszewski
and Knaster) is an indecomposable plane continuum which has a simple construction as the Cantor ternary set C, with semicircle
s linking its points. We can lay C out along the X-axis of the plane from 0 to 1. If x is in C then so is 1-x, and these points are linked by a semicircle in the positive Y direction. If x is in C, and if it lies between 2/3n and 3/3n (inclusive) for a certain n, then the point (5/3n - x) is also in C and in the same range. These two points are linked by a semicircle in the negative Y direction.
Continuum (topology)
In the mathematical field of point-set topology, a continuum is a nonempty compact connected metric space, or less frequently, a compact connected Hausdorff topological space...
that is not the union of any two of its proper subcontinua. The pseudo-arc
Pseudo-arc
In general topology, the pseudo-arc is the simplest nondegenerate hereditarily indecomposable continuum. Pseudo-arc is an arc-like homogeneous continuum. R.H...
is an example of a hereditarily indecomposable continuum. L. E. J. Brouwer discovered the first indecomposable continuum in 1910.
Indecomposable continua have historically appeared as counterexamples to various conjectures, and because of this they are often viewed as pathological objects. However, they can occur in applications, such as attractors in dynamical systems.
Buckethandle
The buckethandle, or B-J-K continuum (for BrouwerLuitzen Egbertus Jan Brouwer
Luitzen Egbertus Jan Brouwer FRS , usually cited as L. E. J. Brouwer but known to his friends as Bertus, was a Dutch mathematician and philosopher, a graduate of the University of Amsterdam, who worked in topology, set theory, measure theory and complex analysis.-Biography:Early in his career,...
, Janiszewski
Zygmunt Janiszewski
Zygmunt Janiszewski was a Polish mathematician.-Life:His mother was Julia Szulc-Chojnicka. His father, Czeslaw Janiszewski, was a graduate of the University of Warsaw and was an important person in finance, being the director of the Société du Crédit Municipal in Warsaw.Janiszewski taught at the...
and Knaster) is an indecomposable plane continuum which has a simple construction as the Cantor ternary set C, with semicircle
Semicircle
In mathematics , a semicircle is a two-dimensional geometric shape that forms half of a circle. Being half of a circle's 360°, the arc of a semicircle always measures 180° or a half turn...
s linking its points. We can lay C out along the X-axis of the plane from 0 to 1. If x is in C then so is 1-x, and these points are linked by a semicircle in the positive Y direction. If x is in C, and if it lies between 2/3n and 3/3n (inclusive) for a certain n, then the point (5/3n - x) is also in C and in the same range. These two points are linked by a semicircle in the negative Y direction.