# 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. It also serves as a counterexample to the claim that all Cantor spaces are ambiently homeomorphic to each other.

## Construction

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

This process can be repeated a countably infinite number of times to create an *A*^{n} 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 as an abstract metric space *A* is homeomorphic to the Cantor set.

However, as a subset of Euclidean space *A* is not ambiently homeomorphic to the standard cantor set *C*. That is, there is no bi-continuous map from **R**^{3} → **R**^{3} that carries *C* onto *A*. To show this, suppose there was such a map *h* : **R**^{3} → **R**, and consider a loop *k* that is interlocked with the necklace. *k* cannot be continuously shrunk to a point without touching *A* because two loops cannot be continuously unlinked. Now consider any loop *j* disjoint from *C*. *j* can be shrunk to a point without touching *C* because we can simply move it through the gap intervals. However, the loop *g* = *h*^{−1}(*k*) is a loop that *cannot* be shrunk to a point without touching *C*, which contradicts the previous statement. Therefore, *h* cannot exist.

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

- 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 - Pugh, Charles Chapman (2002).
*Real Mathematical Analysis*. Springer New York. pp. 106–108. doi:10.1007/978-0-387-21684-3. ISBN 9781441929419.