TITLE: homeomorphism of cantor set extends to the plane?
QUESTION [7 upvotes]: 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.

REPLY [8 votes]: 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.