mrob> 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. ------ rwg>A similar thing happens with increasingly fine polygonal approximations to spacefilling functions. In http://www.tweedledum.com/rwg/rad2fill.htm the upper figure defines the iteration. For any finite order, six of these equipartition the area near the center of the lower figure. But at the limit, the areas disappear and everything is boundary. I don't get the many-torus illustrations in http://en.wikipedia.org/wiki/Antoine%27s_necklace . Doesn't the text say four tori? Shouldn't three work? (Bogus Borromean rings.) --rwg Mrob> 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. http://en.wikipedia.org/wiki/Antoine%27s_necklace