Is it possible to draw in the plane an uncountable set of the digit 8 such that the lines of two different digits won't cut? I think it's wrong, I guess I need to find rational coordinates in every digit, but I don't know how to prove it.
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How are you defining the digit $8$? – Parcly Taxel Apr 18 '20 at 12:17
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1Standard trick: Given an "$8$" you can pick a pair of rational points, one in each of the two circles. There are only countably many such pairs. – lulu Apr 18 '20 at 12:18
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1Every "8" contains a "Y" and it is already impossible to place uncountably many "Y"'s in the plane - though the Y-problem is harder to prove than the 8-problem – Hagen von Eitzen Apr 18 '20 at 12:18
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@ParclyTaxel for the intended proof to work: The connected boundary of a non-connected bounded open set :) – Hagen von Eitzen Apr 18 '20 at 12:22