The buckethandle, or B-J-K 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 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.
The buckethandle admits no Borel transversal: there is no Borel set containing exactly one point from each composant.
Read more about this topic: Indecomposable Continuum