Image of Cantor set is Cantor set I would like to understand what kind of functions map Cantor sets (nowhere dense, no isolated points, compact) in $\mathbb{R}$ onto other Cantor sets of $\mathbb{R}$. 
Is this true for homeomorphisms? (sounds natural, as they should preserve the topological structure).
Otherwise, what are sufficient conditions for a function $f$ to have this property?
Many thanks.
 A: If $X$ is compact and metrisable, then there is a continuous function from the Cantor set that is onto $X$. 
This $X$ can be made a homeomorphism iff the only connected subsets of $X$ are singletons and $X$ has no isolated points.
But the projections $C \times C \to C$ show that such $f$ need not be 1-1, even though it is a map between Cantor sets.
A: An excellent question. One that's honestly better answered by the professional analysts on the site, but I'll give it a shot. 
Homeomorphisms would indeed map the Cantor set into other subsets of $\mathbb R$ that are topologically identical to it. If by "a Cantor set", you mean any subset of  $\mathbb R$ that is homomorphic to the original ternary constructed Cantor set, then absolutely this is true. In fact, topologists have classified any topological space (regardless of whether or not it's a subset of $\mathbb R$) that is homomorphic to the original ternary constructed Cantor set is called a Cantor space. It turns out there are many subspaces of the real line that are Cantor spaces. 
Trivally,of course, the Cantor set is a Cantor space. (Duh!) But you might be very surprised to learn that the countably infinite topological product of {0, 1} with the discrete topology is also a Cantor space. This topological product of this 2 point subset of $\mathbb R$ is usually written as $2^{\mathbb {N}}$ or $2^ω$ where 2 denotes the 2-element set {0,1} with the discrete topology. A point in $2^ω$ is an infinite binary sequence, that is a sequence which assumes only the values 0 or 1. As bizarre as it sounds, a homeomorphism f does exist onto the Cantor set and it's constructed as follows: 
$$\sum_{n=0}^\infty \frac{2 a_n}{3^{n+1}}$$
It turns out that every nonempty totally disconnected perfect compact metric space is homeomorphic to the Cantor set.
Much much more can be found on the Cantor set and its topological counterparts in Stephen Willard's classic General Topology. If you're fascinated with such questions, that's the natural place to begin and it's available in inexpensive Dover paperback now. 
Hope that helped,not sure if I did. 
