Antoine's Necklace
AntoinesNecklace
Construct a chain C of 2n components in a solid torus V. Now thicken each component of C slightly to form a chain C_1 of 2n solid tori in V, where
| pi_1(V-C_1)=pi_1(V-C) |
via inclusion. In each component of C_1, construct a smaller chain of solid tori embedded in that component. Denote the union of these smaller solid tori C_2. Continue this process a countable number of times, then the intersection
| A= intersection _(i=1)^inftyC_i |
which is a nonempty compact subset of R^3 is called Antoine's necklace. Antoine's necklace is homeomorphic with the Cantor set.
See also
Alexander's Horned Sphere, NecklaceExplore with Wolfram|Alpha
WolframAlpha
More things to try:
References
Rolfsen, D. Knots and Links. Wilmington, DE: Publish or Perish Press, pp. 73-74, 1976.Referenced on Wolfram|Alpha
Antoine's NecklaceCite this as:
Weisstein, Eric W. "Antoine's Necklace." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/AntoinesNecklace.html