In mathematics 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 introduced by German mathematician Georg Cantor in 1883. Thr ...

in 3-dimensional Euclidean space, 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 between two points can be continuously transformed (intuitively for embedded spaces, staying within the spac ...

. 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 visuali ...

''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 Closed may refer to: Mathematics * Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set * Closed set, a set which contains all its limit points * Closed interval, ...

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 \mathfrak c (lowercase fraktur "c") or , \mathb ...

. 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 straight line that is bounded by two distinct end points, and contains every point on the line that is between its endpoints. The length of a line segment is given by the Euclidean distance between ...

. 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 to construct Antoine's horned sphere (similar to but not the same as

Alexander's horned sphere The Alexander horned sphere is a pathological object in topology discovered by . Construction The Alexander horned sphere is the particular embedding of a sphere in 3-dimensional Euclidean space obtained by the following construction, starting ...

References

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Construction

Properties

See also

References

Further reading