Let $\mathcal{C}_1$ and $\mathcal{C}_2$ be any two Cantor sets (constructed in Exercise 3.)Show that there exists a function $F:[0,1] \to [0,1]$ with the following properties:1. $F$ is continuous and bijective,2. $F$ is monotonically increasing,3. $F$ maps $\mathcal{C}_1$ surjectively onto $\mathcal{C}_2$.
The hint says to consider the "construction of the standard Cantor-Lebesgue function" but I don't see how imitating the construction will help. I believe that the part actually requiring ideas from the construction of the cantor function is when I have constructed a map from $\mathcal{C}_1$ to $\mathcal{C}_2$ and I want to extend it, but I am not sure how to proceed to that point.
Exercise 3 constructed the Cantor set as follows
Consider the unit interval [0, 1], andlet ξ be a fixed real number with 0 < ξ< 1 (the case ξ = 1/3 corresponds to theCantor set C in the text).In stage 1 of the construction, remove the centrally situated open interval in[0, 1] of length ξ. In stage 2, remove two central intervals each of relative length ξ,one in each of the remaining intervals after stage 1, and so on.Let $\mathcal{C}_ξ$ denote the set which remains after applying the above procedure indefinitely.