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307:). 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
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Since the solid tori are chosen to become arbitrarily small as the iteration number increases, the connected components of
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Brechner, Beverly L.; Mayer, John C. (1988), "Antoine's
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Proceedings of the
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This process can be repeated a countably infinite number of times to create an
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because we can simply move it through the gap intervals. However, the loop
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because two loops cannot be continuously unlinked. Now consider any loop
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389:(1921), "Sur l'homeomorphisme de deux figures et leurs voisinages",
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Antoine's necklace is constructed iteratively like so: Begin with a
521:. Undergraduate Texts in Mathematics. Springer New York. pp.
170:. This is sufficient to conclude that as an abstract metric space
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are ambiently homeomorphic to each other. It was discovered by
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can be replaced with another smaller necklace as was done for
76:. It also serves as a counterexample to the claim that all
335: – List of concrete topologies and topological spaces
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cannot be continuously shrunk to a point without touching
411:(1924), "Remarks on a Point Set Constructed by Antoine",
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is not ambiently homeomorphic to the standard Cantor set
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Embedding of Cantor set in 3-dimensional
Euclidean space
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Pages displaying short descriptions of redirect targets
269:, which contradicts the previous statement. Therefore,
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must be single points. It is then easy to verify that
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is defined as the intersection of all the iterations.
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193:. That is, there is no bi-continuous map from
209:. To show this, suppose there was such a map
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391:Journal de Mathématiques Pures et Appliquées
245:can be shrunk to a point without touching
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341: – Plane fractal built from squares
177:However, as a subset of Euclidean space
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323: – Set of points on a line segment
225:that is interlocked with the necklace.
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276:In fact, there is no homeomorphism of
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111:(iteration 1). Each torus composing
174:is homeomorphic to the Cantor set.
64:is a topological embedding of the
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303:(similar to but not the same as
50:Renderings of Antoine's necklace
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477:The College Mathematics Journal
291:Antoine's necklace was used by
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168:cardinality of the continuum
293:James Waddell Alexander
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72:, whose complement is not
531:10.1007/978-0-387-21684-3
305:Alexander's horned sphere
301:Antoine's horned sphere
327:Knaster–Kuratowski fan
221:, and consider a loop
134:. Antoine's necklace
436:10.1073/pnas.10.1.10
164:totally disconnected
119:. Doing this yields
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286:Hausdorff dimension
107:. This necklace is
573:1921 introductions
348:Whitehead manifold
333:List of topologies
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339:Sierpinski carpet
261:) is a loop that
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58:mathematics
557:Categories
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362:Superhelix
142:Properties
66:Cantor set
397:: 221–325
357:Wild knot
563:Topology
463:16576769
315:See also
280:sending
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130:for all
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454:1085501
423:Bibcode
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