 Indecomposable continuum

In pointset topology, an indecomposable continuum is a continuum that is not the union of any two of its proper subcontinua. The pseudoarc is an example of a hereditarily indecomposable continuum. L. E. J. Brouwer discovered the first indecomposable continuum in 1910.^{[1]}
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 BJK continuum (for Brouwer, Janiszewski and Knaster) is an indecomposable plane continuum which has a simple construction as the Cantor ternary set C, with semicircles linking its points. We can lay C out along the Xaxis of the plane from 0 to 1. If x is in C then so is 1x, and these points are linked by a semicircle in the positive Y direction. If x is in C, and if it lies between 2/3^{n} and 3/3^{n} (inclusive) for a certain n, then the point (5/3^{n}  x) is also in C and in the same range. These two points are linked by a semicircle in the negative Y direction.
See also
References
 ^ Charles E. Aull, Robert Lowen (2001). Handbook of the history of general topology. Springer.
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