# Sticky Cantor sets

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

I borrowed this conjecture from this forum thread.

Certainly I understand this conjecture wrongly: is a subset of a line segment. Consider a homeomorphism which moves all points of orthogonally to this line segment by . 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.

## Bibliography

* indicates original appearance(s) of problem.

### Misunderstanding

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.

### M

"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.