Nice argument. Is it possible to find a situation where the dense subset has a cardinality of 2^(aleph_0) where one can still find a smooth closed curve that misses the dense subset? And, yes, you can just remove the points of the dense subset that intersect a particular circle(s) and make a new dense subset that works, but are there constraints on given dense subsets or constraints on given conditions for the curve(s) for it to work? -----Original Message----- From: math-fun-bounces@mailman.xmission.com [mailto:math-fun-bounces@mailman.xmission.com] On Behalf Of Dan Asimov Sent: Monday, September 16, 2013 9:53 AM To: math-fun Subject: [EXTERNAL] Re: [math-fun] curves winding their way around dense subsets Just by cardinality alone, there are 2^aleph_0 circles in the plane with any fixed center and any real radius. But there are only aleph_0 rational points. So there are lots of circles that miss the set of rational points, or for that matter any countable subset, dense or not. (Also, every countable dense subset of the plane is homeomorphic to any other, and every complement of such a subset is homeomorphic to any other as well.) --Dan On 2013-09-16, at 8:39 AM, Cordwell, William R wrote:
Hello,
There was some interesting discussion about rational points being dense in the unit circle (corresponding to all the Pythagorean triples). It is straightforward to show that there are many circles that contain no rational points.
So, is there any theory about (smooth, closed, whatever modifier you like) curves in the plane that completely miss a dense subset?
Thanks, Bill
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