3. http://mathworld.wolfram.com/AntoinesNecklace.html

4.  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.

5.  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.