That's the "paradox" (which I know is just my limited perception) : 1. Since single points are not tori, it seems that the necklace is a set of disjoint points, so the necklace is no longer linked. 2. But the description clearly says that the set's complement is not simply connected, which implies that the necklace at stage Aleph_0 [sic, see below] is still linked.
I like to refer to these as 'infinitely complex dust clouds'. I have some more familiar examples. The rationals are 'disjoint points', but they can't be separated; it's impossible to continuously deform the real line to introduce a nice empty space between the negative rationals and positive rationals. I guess that's the most familiar example, so hopefully the most intuitive.
Transitions from a finite property to an infinite property appear everywhere. For example, every term in the sequence 1, 2, 3, 4, ... (the positive integers) is finite, but the limit of the series (Aleph_0) is infinite.
The limit of the series is the *ordinal* omega. You're only allowed to use aleph-null to refer to the sizes of sets. http://en.wikipedia.org/wiki/Ordinal_number http://en.wikipedia.org/wiki/Cardinal_number Sorry to appear pedantic, but it's an important distinction. The naturals and the fusible numbers are both equally large in terms of cardinal size (you can biject between them) -- they have cardinality aleph-null. But the naturals have order type omega (they can be bijected in a nice, order-preserving way with the set of ordinals below omega, namely the naturals themselves!), whereas the fusible numbers have order type epsilon-nought (they can be bijected beautifully to the ordinals below epsilon-nought such that order is preserved).
I can grasp that. Similarly, the normal (1-dimensional middle-thirds) Cantor set transitions from being a set of intervals with nonzero measure to a set of points. That doesn't seem to bother me either.
But the necklace one bothers me. That transition from linked set of tori to unlinked set of points seems impossible.
Yes, this is one of those things which are obviously false but actually true. Then there are things which are obviously either true or false, but turn out to be equivalent to the continuum hypothesis instead. Sincerely, Adam P. Goucher http://cp4space.wordpress.com