It's not correct to say "the necklace" remains linked through the stages of its construction; it's the *stages* that are linked. But it is true that the necklace N is totally disconnected, i.e., its connected components are all single points. And that N is indeed linked with a circle in its complement. (So there is nothing at all that "becomes unlinked".) Here's a rough sketch of a proof that the necklace is linked with a closed curve in its complement that is linked with the first stage in its construction: a single solid torus. First we state what this linking actually means: It means there is a simple closed curve C -- in the complement of the (first stage) solid torus T -- that cannot be shrunk to a point in the complement of T, such that C can also not be shrunk to a point in the complement of Antoine's necklace. Choose C to be a slight enlargement of any meridional circle of bd(T), so that C lies in R^3 - T, and simply links the core circle of T. That is, for any continuous map f: D -> R^3 such that D is a closed 2-disk for which f maps bd(D) homeomorphically onto C, then f(D) must intersect Antoine's necklace. To prove this: It can be shown that for each stage A_n of Antoine's necklace (with the necklace being the intersection of all the stages), any such map f has the property that f(D) intersects A_n in some nonempty compact set, let's say K_n. And such that K_(n+1) is a subset of K_n for all n. This shows that the intersection of all the K_n is a nonempty compact set lying in the intersection of all the A_n -- i.e., lying in Antoine's necklace. QED. Of course, the paradoxical nature of Antoine's necklace is exactly what makes it interesting! Who'd've thought a (topologically) 0-dimensional set like the Cantor set could link a circle? --Dan On 2012-12-06, at 12:44 PM, Robert Munafo wrote:
Adam's mention of Antoine's necklace reminded me of a topology question I've had for a while.
According to the Wikipedia description (which is admittedly basic, but agrees with my own perception of the situation), the necklace remains linked through all finite stages of the construction, yet in the limit (of the countably infinite number of steps) what remains is a set of single points.
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 is still linked.
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. 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.
So the question is, does anyone have another explanation of how the necklace becomes unlinked, which might make it seem less of a paradox?
The answer is probably here, but my mind's a little too foggy to get it:
http://en.wikipedia.org/wiki/Alexander_horned_sphere http://en.wikipedia.org/wiki/Schoenflies_problem http://en.wikipedia.org/wiki/Knot_theory#Knotting_spheres_of_higher_dimensio...
Other things I referred to:
http://en.wikipedia.org/wiki/Simply_connected_space http://en.wikipedia.org/wiki/Cantor_set http://en.wikipedia.org/wiki/Lebesgue_measure http://en.wikipedia.org/wiki/Aleph_number#Aleph-naught
On 12/6/12, Adam P. Goucher <apgoucher@gmx.com> wrote:
[...] Theoretically, you could assemble these magic roundabouts recursively to produce an even larger clockwise one.
-- Robert Munafo -- mrob.com Follow me at: gplus.to/mrob - fb.com/mrob27 - twitter.com/mrob_27 - mrob27.wordpress.com - youtube.com/user/mrob143 - rilybot.blogspot.com
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