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In mathematics, Antoine's necklace, discovered by Louis Antoine (1921), 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 Alexander (1924) to construct Antoine's horned sphere (similar to but not the same as Alexander's horned sphere).
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

Mathematics Encyclopedia

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