Antoine's Necklace

Construct a chain  of  components in a solid torus . Now thicken each component of  slightly to form a chain  of  solid tori in , where

via inclusion. In each component of , construct a smaller chain of solid tori embedded in that component. Denote the union of these smaller solid tori . Continue this process a countable number of times, then the intersection

which is a nonempty compact subset of  is called Antoine's necklace. Antoine's necklace is homeomorphic with the Cantor set.

REFERENCES:

Rolfsen, D. Knots and Links. Wilmington, DE: Publish or Perish Press, pp. 73-74, 1976.