Explicit homemorphism between code space

Question

Given a finite set we can construct the string (or code)space $S^\omega$ with the metric $\rho(w,w')=r^m$ where $w,w'\in S^\omega$ and $m$ is the maximum number such that the two strings $w,w'$ coincide. This is an ultrametric.

I want to know an explicit homeomorphism between $S_1^\omega$ and $S_2^\omega$ where $S_1$ and $S_2$ are different set with different finite cardinality.

It is known that they are homeomorphic because they both are Cantor sets.

Are these maps in some sense unique?

Answer 1

Take $S_1=\{0,1\}$ and $S_2=\{00,01,1\}$. There exists a canonical function $\phi:S_2^{\omega}\rightarrow S_1^{\omega}$ which i easily seen to be a homeomorphism. You can do this with every finite set.