Antoine's necklace

First iteration

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

• Cantor dust
• Sierpinski carpet

References

• Antoine, Louis (1921), "Sur l'homeomorphisme de deux figures et leurs voisinages", Journal Math Pures et appl. 4: 221–325
• Alexander, J. W. (1924), "Remarks on a Point Set Constructed by Antoine", Proceedings of the National Academy of Sciences of the United States of America 10 (1): 10–12, doi:10.1073/pnas.10.1.10, JSTOR 84203, PMC 1085501, PMID 16576769
• Brechner, Beverly L.; Mayer, John C. (1988), "Antoine's Necklace or How to Keep a Necklace from Falling Apart", The College Mathematics Journal 19 (4): 306–320, doi:10.2307/2686463, JSTOR 2686463