Let E(3) be 3-dimensional Euclidean space with its standard topology. Brunnian Rings are subsets of E(3) which are simple closed curves. It is known that an arbitrarily large set S(3) of these Brunnian Rings can be linked together in such a way, that if any one of them is removed, each distinct pair of the remaining ones becomes unlinked. Is this still possible, if the set S(3) is infinite or even uncountable?

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    $\begingroup$ Are you asking if there are Brunnian links with infintely (or uncountably) many components? I can't tell from the question as it is written. $\endgroup$ – Mark Grant Jun 14 '17 at 15:04
  • $\begingroup$ One should make more precise what is meant by "infinitely many rings linked together". Does this mean that for each of them there is no ambient isotopy that brings this ring and the rest into complementary open half-spaces? Then in any collection of uncountably many rings at least one is linked to the rest (because it intersects the closure of their union). $\endgroup$ – Ivan Izmestiev Jun 14 '17 at 15:27
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    $\begingroup$ If we adopt the definition from my previous comment, then an infinite sequence of circles like on the third image here will count as an infinite Brunnian link. $\endgroup$ – Ivan Izmestiev Jun 14 '17 at 15:34
  • $\begingroup$ I am asking whether there Brunniian links wth infinitely (or uncountablyMany $\endgroup$ – Garabed Gulbenkian Jun 15 '17 at 17:15
  • $\begingroup$ I am asking whether there are Brunnian links with infinitely-or even uncountably-many components. If so, is it very difficult to describe how to construct them? $\endgroup$ – Garabed Gulbenkian Jun 15 '17 at 17:27

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