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http://mathworld.wolfram.com/AntoinesNecklace.html
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Start with a torus V. C 1 is a chain of tori linked together. In each component of C 1, construct a chain of smaller tori (generally with the same number of links as in C 1 ). C 2 denotes the union of the tori at this level. Continue this process ad infinitum, and Antoine’s necklace is the intersection of the C i. At each subsequent stage in the linking process we have a finite approximation to Antoine’s necklace, with increasing complexity because the total number of links grows exponentially. The topology of the necklace seems to grow more complicated.
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In fact, Antoine’s necklace is topologically equivalent to the middle thirds Cantor set, which is basically just scattered dust. The Cantor set is compact (closed and bounded), uncountable, completely disconnected, and self- similar. The finite approximations for Antoine’s necklace are multiply-connected yet Antoine’s necklace isn’t even connected.
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