Antoine's necklace

First iteration

Second iteration

Renderings of Antoine's necklace

In mathematics, Antoine's necklace 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. It was discovered by Louis Antoine (1921).^[1]

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.

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, embedded in R³ on a line segment. 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 → R³, 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.

In fact, there is no homeomorphism of R³ sending A to a set of Hausdorff dimension < 1, since the complement of such a set must be simply-connected.

Antoine's necklace was used by James Waddell Alexander (1924) to construct Antoine's horned sphere (similar to but not the same as Alexander's horned sphere).^[2] This construction can be used to show the existence of uncountably many embeddings of a disk or sphere into three-dimensional space, all inequivalent in terms of ambient isotopy.^[3]

• Cantor dust – Set of points on a line segment
• Knaster–Kuratowski fan – Topological space that becomes totally disconnected with the removal of a single point
• List of topologies – List of concrete topologies and topological spaces
• Sierpinski carpet – Plane fractal built from squaresPages displaying short descriptions of redirect targets
• Whitehead manifold – open 3-manifold that is contractible, but not homeomorphic to R^3Pages displaying wikidata descriptions as a fallback
• Wild knot
• Superhelix
• Hawaiian earring

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

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