mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, Antoine's necklace is a topological embedding of the

Cantor set In mathematics, the Cantor set is a set of points lying on a single line segment that has a number of unintuitive properties. It was discovered in 1874 by Henry John Stephen Smith and mentioned by German mathematician Georg Cantor in 1883. Throu ...

in 3-dimensional

Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...

, whose complement is not

simply connected In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every Path (topology), path between two points can be continuously transformed into any other such path while preserving ...

. It also serves as a counterexample to the claim that all

Cantor space In mathematics, a Cantor space, named for Georg Cantor, is a topological abstraction of the classical Cantor set: a topological space is a Cantor space if it is homeomorphic to the Cantor set. In set theory, the topological space 2ω is called "the ...

s are ambiently homeomorphic to each other. It was discovered by .

Construction

Antoine's necklace is constructed iteratively like so: Begin with a

solid torus In mathematics, a solid torus is the topological space formed by sweeping a disk around a circle. It is homeomorphic to the Cartesian product S^1 \times D^2 of the disk and the circle, endowed with the product topology. A standard way to visual ...

''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 In general topology, a subset A of a topological space is said to be dense-in-itself or crowded if A has no isolated point. Equivalently, A is dense-in-itself if every point of A is a limit point of A. Thus A is dense-in-itself if and only if A\su ...

, and

totally disconnected In topology and related branches of mathematics, a totally disconnected space is a topological space that has only singletons as connected subsets. In every topological space, the singletons (and, when it is considered connected, the empty set) ...

, having the

cardinality of the continuum In set theory, the cardinality of the continuum is the cardinality or "size" of the set of real numbers \mathbb R, sometimes called the continuum. It is an infinite cardinal number and is denoted by \bold\mathfrak c (lowercase Fraktur "c") or \ ...

. 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 In geometry, a line segment is a part of a line (mathematics), straight line that is bounded by two distinct endpoints (its extreme points), and contains every Point (geometry), point on the line that is between its endpoints. It is a special 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 → 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 In mathematics, Hausdorff dimension is a measure of ''roughness'', or more specifically, fractal dimension, that was introduced in 1918 by mathematician Felix Hausdorff. For instance, the Hausdorff dimension of a single point is zero, of a line ...

< 1, since the complement of such a set must be simply-connected. Antoine's necklace was used by to construct Antoine's horned sphere (similar to but not the same as

Alexander's horned sphere The Alexander horned sphere is a Pathological (mathematics), pathological object in topology discovered by . It is a particular topological embedding of a two-dimensional sphere in three-dimensional space. Together with its inside, it is a topolo ...

). 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 In the mathematical subject of topology, an ambient isotopy, also called an ''h-isotopy'', is a kind of continuous distortion of an ambient space, for example a manifold, taking a submanifold to another submanifold. For example in knot theory, o ...

References

External links

{{Sister project links, auto=y, wikt= Topology Eponyms in geometry 1921 introductions

Construction

Properties

See also

References

Further reading

External links