Intereting Posts

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Example of two open balls such that the one with the smaller radius contains the one with the larger radius.
Prove geometric sequence question
Find the least number b for divisibility
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Math Wizardry – Formula for selecting the best spell
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Orthogonal and symmetric Matrices
F.g. groups with a finite index abelian subgroup
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Suppose C is a Cantor set in the Euclidean plane, or even in R^3. Suppose h is a homeomorphism of C onto itself. Can h be extended to a homeomorphism of the whole space? What about if h preserves the ´end points´ of the Cantor set? Seems that the latter is a necessary condition but I don´t know how to prove it.

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- The relation between order isomorphism and homeomorphism
- complement of zero set of holomorphic function is connected
- Question(s) about uniform spaces.
- Open Dense Subset of $M_n(\mathbb{R})$

In $\mathbb{R}^2$, Schonflies theorem ensures that any two embeddings of Cantor sets are equivalently embedded, or equivalently that any homeomorphism $h$ between the two Cantor sets (it can be proven that any two Cantor sets have a homeomorphism between them – I think this is proven in Edwin E. Moise’s book *Geometric Topology in Dimensions 2 and 3*) can be extended to a homeomorphism $H \colon \mathbb{R}^2 \to \mathbb{R}^2$.

In $\mathbb{R}^3$, we have many examples of *rigid* Cantor sets; a personal favourite is the following paper which generalises Skora’s construction of a wildly embedded Cantor set in $S^3$ with a simply connected complement, to produce a rigid wildly embedded Cantor set in $\mathbb{R}^3$, with a simply connected complement.

This depends on what you want to call a Cantor set. The usual middle-thirds Cantor set is one embedding of the cantor set into the plane/$3$-space, however there are many other ways of embedding the Cantor set into a space and there appears to be a rich theory behind all the possible ways (I don’t work in this area so I can’t give too many details).

The paper Homogeneity groups of ends of open $3$-manifolds, Garity & Repovs, offers a brief glimpse of some of the questions that can arise, and you can get a feel for how non-trivial some of these embeddings can be by reading the introduction.

The standard middle thirds Cantor set embedding in the plane is known to be *strongly homeogenously embedded* which means any automorphism extends to the whole of the plane. However, on the other end of things there exist embeddings of the Cantor set in $\mathbb{R}^n$ such that no non-trivial automorphism of the Cantor set can be extended – these are called *rigidly embedded*.

There are many open problems related to these terms involving tame and wild embeddings in Eucldiean space.

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