Sticky Cantor sets

Importance: Medium ✭✭
Subject: Topology
Keywords: Cantor set
Recomm. for undergrads: no
Posted by: porton
on: February 6th, 2011
Conjecture   Let $ C $ be a Cantor set embedded in $ \mathbb{R}^n $. Is there a self-homeomorphism $ f $ of $ \mathbb{R}^n $ for every $ \epsilon $ greater than $ 0 $ so that $ f $ moves every point by less than $ \epsilon $ and $ f(C) $ does not intersect $ C $? Such an embedded Cantor set for which no such $ f $ exists (for some $ \epsilon $) is called "sticky". For what dimensions $ n $ do sticky Cantor sets exist?

I borrowed this conjecture from this forum thread.

Certainly I understand this conjecture wrongly: $ C $ is a subset of a line segment. Consider a homeomorphism which moves all points of $ \mathbb{R}^n $ orthogonally to this line segment by $ \epsilon/2 $. This would be a solution of this problem. Obviously it is not what is meant.

Indeed I submit the problem to OPG as is in the hope that somebody will correct my wrong understanding and adjust the formulation to not be misunderstood as by me.


* indicates original appearance(s) of problem.


Your misunderstanding comes from the definition of a Cantor set. A Cantor set is a set homeomorphic to the usual middle-thirds Cantor set. In general it does not have to lie on a line segment.


"embedded" does not imply that it is still a subset of the line. It just says that it's one-to-one and a homeomorphism with the image. The conjecture requires to prove that there exists a Cantor which cannot be separated from itself, so showing an example where it can be separated is not relevant.

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