![](/files/happy5.png)
Sticky Cantor sets
![$ C $](/files/tex/05d3558ef52267cc1af40e658352d98883668ee3.png)
![$ \mathbb{R}^n $](/files/tex/2010c953180b3521ec2f66d10e1f40ec71d44574.png)
![$ f $](/files/tex/43374150a8a220f67049937b9790b7d28eb17fb9.png)
![$ \mathbb{R}^n $](/files/tex/2010c953180b3521ec2f66d10e1f40ec71d44574.png)
![$ \epsilon $](/files/tex/816b3cebf962fcc001285ab8e9adce8656388718.png)
![$ 0 $](/files/tex/1d8c59cb34a2a35471b98d11ba99311b971a3879.png)
![$ f $](/files/tex/43374150a8a220f67049937b9790b7d28eb17fb9.png)
![$ \epsilon $](/files/tex/816b3cebf962fcc001285ab8e9adce8656388718.png)
![$ f(C) $](/files/tex/a2c230accce082b7a2be6c4a5181efb12daf6266.png)
![$ C $](/files/tex/05d3558ef52267cc1af40e658352d98883668ee3.png)
![$ f $](/files/tex/43374150a8a220f67049937b9790b7d28eb17fb9.png)
![$ \epsilon $](/files/tex/816b3cebf962fcc001285ab8e9adce8656388718.png)
![$ n $](/files/tex/ec63d7020a64c039d5f6703b8fa3ab7393358b5b.png)
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.
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.
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.