View source for Antoine's necklace

{{Short description|Embedding of Cantor set in 3-dimensional Euclidean space}}
{{multiple image
 | direction = vertical
 | width     = 200
 | footer    = Renderings of Antoine's necklace
 | image1    = Antoine's Necklace Iteration 1.png
 | alt1      = Antoine's necklace
 | caption1  = First iteration
 | image2    = Antoine's Necklace Iteration 2.png
 | alt2      = Antoine's necklace
 | caption2  = Second iteration
}}
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 space]]s are ambiently homeomorphic to each other. It was discovered by {{harvs|txt=yes|last=Antoine|first=Louis|year=1921}}.<ref>{{citation|first=Louis|last=Antoine|authorlink=Louis Antoine|title=Sur l'homeomorphisme de deux figures et leurs voisinages
|journal= Journal de Mathématiques Pures et Appliquées|volume=4|year=1921|pages=221–325}}</ref>

==Construction==

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

This process can be repeated a countably infinite number of times to create an ''A''<sup>''n''</sup> 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 [[Closure (mathematics)|closed]], [[Dense-in-itself|dense-in-itself]], and totally disconnected, having the [[Cardinality of the continuum|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 '''R'''<sup>3</sup> on a [[Line segment|line segment]]. That is, there is no bi-continuous map from '''R'''<sup>3</sup> → '''R'''<sup>3</sup> that carries ''C'' onto ''A''. To show this, suppose there was such a map ''h'' : '''R'''<sup>3</sup> → '''R'''<sup>3</sup>, 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''<sup>−1</sup>(''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 '''R'''<sup>3</sup> 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 {{harvs|txt|last=Alexander|first=James Waddell|year=1924}} to construct '''Antoine's horned sphere''' (similar to but not the same as Alexander's horned sphere).<ref>{{citation|title=Remarks on a Point Set Constructed by Antoine
|journal= Proceedings of the National Academy of Sciences of the United States of America
|volume=10|issue= 1|year= 1924|pages= 10–12
|doi=10.1073/pnas.10.1.10|pmid=16576769|pmc=1085501|jstor=84203|bibcode=1924PNAS...10...10A|doi-access=free}}</ref> 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|ambient isotopy]].<ref>{{citation|title=Antoine's Necklace or How to Keep a Necklace from Falling Apart
|first1=    Beverly L. |last1=Brechner|first2= John C.|last2= Mayer 
 |journal=   The College Mathematics Journal|volume= 19|issue= 4|year= 1988|pages= 306–320 
|doi=10.2307/2686463|jstor=2686463}}</ref>

== See also ==

* {{annotated link|Knaster–Kuratowski fan}}
* {{annotated link|List of topologies}}
* {{annotated link|Sierpinski carpet}}
* {{annotated link|Whitehead manifold}} 
* [[Wild knot]]
* [[Biology:Superhelix|Superhelix]]
* [[Hawaiian earring]]

==References==
{{reflist}}

==Further reading==
*{{cite book |last=Pugh |first=Charles Chapman |date=2002 |title=Real Mathematical Analysis |series=Undergraduate Texts in Mathematics |url=https://archive.org/details/realmathematical00char |url-access=registration |doi=10.1007/978-0-387-21684-3 |publisher=Springer New York |pages=[https://archive.org/details/realmathematical00char/page/106 106–108] |isbn=9781441929419}}

[[Category:Topology]]

{{Sourceattribution|Antoine's necklace}}