Here on Wikipedia is the example of how to construct a bijection between $T$ - set of infinite binary strings and $\Bbb{R}$. For me the method of building of such bijection is interesting but I want more details on the part of building the bijection between $[0, 1]$ and $(0, 1)$. Specifically:
How Cantor removed countably infinite sets from $[0,1]$ and $(0,1)$? While I know that every infinite set contains infinite countable subset - it seems to me that if I remove a countable subset from infinite set I will still have infinite set from where I can again remove countable subset. Infinitely repeating it with no "damage" to the power of the infinite set.
1.1. Why Cantor even bothered removing countably infinite subsets from each $[0,1]$ and $(0,1)$?
- Why do we need a family of functions $f_b(t)$?
Constructing a bijection between $T$ and $\Bbb{R}$ is slightly more complicated. Instead of mapping $0111...$ to the decimal $0.0111...$, it can be mapped to the base b number: $0.0111..._b$. This leads to the family of functions: $f_b(t) = 0.t_b$. The functions $f_b(t)$ are injections, except for $f_2(t)$. This function will be modified to produce a bijection between $T$ and $\Bbb{R}$.
I am looking for the advice on where can I find this topic described in details?