Antoine's necklace
Encyclopedia
In mathematics, Antoine's necklace, discovered by , is an embedding of the Cantor set
in 3-dimensional Euclidean space whose complement is not simply connected.
It is constructed by starting with a solid torus
, replacing it by a "necklace" inside it formed of linked tori, then replacing each of these tori by another necklace inside it, and repeating this an infinite number of times.
It was used by to construct Antoine's horned sphere (similar to but not the same as Alexander's horned sphere).
Cantor set
In mathematics, the Cantor set is a set of points lying on a single line segment that has a number of remarkable and deep properties. It was discovered in 1875 by Henry John Stephen Smith and introduced by German mathematician Georg Cantor in 1883....
in 3-dimensional Euclidean space whose complement is not simply connected.
It is constructed by starting with a solid torus
Solid torus
In mathematics, a solid torus is a topological space homeomorphic to S^1 \times D^2, i.e. the cartesian product of the circle with a two dimensional disc endowed with the product topology. The solid torus is a connected, compact, orientable 3-dimensional manifold with boundary...
, replacing it by a "necklace" inside it formed of linked tori, then replacing each of these tori by another necklace inside it, and repeating this an infinite number of times.
It was used by to construct Antoine's horned sphere (similar to but not the same as Alexander's horned sphere).