In mathematics, Antoine's necklace is a topological embedding of the Cantor set in 3-dimensional Euclidean space, whose complement is not simply connected.
This process can be repeated a countably infinite number of times to create an An for all n. Antoine's necklace A is defined as the intersection of all the iterations.
Since the solid tori are chosen to become arbitrarily small as the iteration number increases, the connected components of A must be single points.
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.
[2] 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.