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- Antoine's_necklace abstract "In mathematics, Antoine's necklace, discovered by Louis Antoine (1921), is a topological embedding of the Cantor set in 3-dimensional Euclidean space, whose complement is not simply connected.It is constructed by starting with a solid torus (Stage 0), constructing a "necklace" inside it formed of four linked solid tori (Stage 1), then constructing inside each of these solid tori another necklace of four inside it, and repeating this a countably infinite number of times. Stage n will consist of 4n solid tori, n = 0, 1, 2, 3, . . .. Antoine's necklace A is defined as the intersection of all the stages. Since the solid tori are chosen to become arbitrarily small as the stage 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 A is homeomorphic to the Cantor set.It was used by Alexander (1924) to construct Antoine's horned sphere (similar to but not the same as Alexander's horned sphere).".
- Antoine's_necklace thumbnail Collier_Antoine.jpg?width=300.
- Antoine's_necklace wikiPageID "15067130".
- Antoine's_necklace wikiPageRevisionID "585429398".
- Antoine's_necklace alt "Antoine's necklace".
- Antoine's_necklace authorlink "Louis Antoine".
- Antoine's_necklace caption "Second iteration".
- Antoine's_necklace caption "Third iteration".
- Antoine's_necklace direction "vertical".
- Antoine's_necklace first "Louis".
- Antoine's_necklace footer "Renderings of Antoine's necklace".
- Antoine's_necklace hasPhotoCollection Antoine's_necklace.
- Antoine's_necklace image "Antoine's Necklace.PNG".
- Antoine's_necklace image "Collier Antoine.jpg".
- Antoine's_necklace last "Antoine".
- Antoine's_necklace txt "yes".
- Antoine's_necklace width "200".
- Antoine's_necklace year "1921".
- Antoine's_necklace subject Category:Topology.
- Antoine's_necklace comment "In mathematics, Antoine's necklace, discovered by Louis Antoine (1921), is a topological embedding of the Cantor set in 3-dimensional Euclidean space, whose complement is not simply connected.It is constructed by starting with a solid torus (Stage 0), constructing a "necklace" inside it formed of four linked solid tori (Stage 1), then constructing inside each of these solid tori another necklace of four inside it, and repeating this a countably infinite number of times.".
- Antoine's_necklace label "Antoine's necklace".
- Antoine's_necklace label "Collier d'Antoine".
- Antoine's_necklace label "عقد أنطوان".
- Antoine's_necklace sameAs Collier_d'Antoine.
- Antoine's_necklace sameAs m.03hgf1n.
- Antoine's_necklace sameAs Q2983382.
- Antoine's_necklace sameAs Q2983382.
- Antoine's_necklace wasDerivedFrom Antoine's_necklace?oldid=585429398.
- Antoine's_necklace depiction Collier_Antoine.jpg.
- Antoine's_necklace isPrimaryTopicOf Antoine's_necklace.