Antoine's necklace

Antoine's necklace
First iteration
Antoine's necklace
Second iteration
Renderings of Antoine's necklace

In mathematics, Antoine's necklace, discovered by Louis Antoine (1921), is a topological embedding of the Cantor set in 3-dimensional Euclidean space, whose complement is not simply connected.

Construction

Antoine's necklace is constructed iteratively like so: Begin with a solid torus A0 (iteration 0). Next, construct a "necklace" of smaller, linked tori that lie inside A0. This necklace is A1 (iteration 1). Each torus composing A1 can be replaced with another smaller necklace as was done for A0. Doing this yields A2 (iteration 2).

This process can be repeated a countably infinite number of times to create an An for all n. Antoine's necklace A is defined as the intersection of all the iterations.

Properties

Since the solid tori are chosen to become arbitrarily small as the iteration number increases, the connected components of A must be single points. It is then easy to verify that A is closed, dense-in-itself, and totally disconnected, having the cardinality of the continuum. This is sufficient to conclude that A is homeomorphic to the Cantor set.

Antoine's necklace was used by Alexander (1924) to construct Antoine's horned sphere (similar to but not the same as Alexander's horned sphere).

See also

References

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