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. 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 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 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 R³ → R³ that carries C onto A. To show this, suppose there was such a map h : R³ → 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

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