Prove that any Cantor-like set is uncountable.

Question

I have known that there exists some Cantor-like sets with positive measure, and wondering if I can prove that any any Cantor-like set is uncountable by construting or just proving the existence of a bijection of any two Cantor-like sets.
Maybe there are several different proofs of this property, but I want to know will this method work.
A Cantor-like set is a set construting in the following way:
Removing $2^{k-1}$ centrally situated open intervals of length $l_k$ at each $k^{th}$ stage, with $$l_1+2l_2+...+2^{k-1}l_k<1$$

Answer 1

If you take two sets which are constructed by removing centrally situated open intervals you have a for each iteration a homeomorphism between the two constructions, e.g. a piecewise linear map defined on the partition of [0,1]. These homeomorphism have a limit function which provides a bijection on the two Cantor-like sets.

Answer 2

If you're considering fat Cantor sets, then they can't be countable, because they have positive Lebesgue measure.