Well, my intention is basically to understand what is done in the answer here, which I will summarize in the following... and discuss a little more (hence I made a new question).
Problem: Given a family $C$ of subsets of the real numbers, how can we characterize all elements from the $\sigma$-algebra generated by the family $C$?
In this link the answer is almost clear to me, but there are some facts I'd like to discuss.
Construction: For any family $\mathcal{B}\subseteq\mathcal{P}(\Bbb{R})$, we define $\mathcal{B}_c :=\{\bigcup_{k\in \Bbb{N}}A_k: A_k \in \mathcal{B}\}\cup \{A^c: A\in \mathcal{B}\}$ the set which elements are all possible countable unions of elements of $\mathcal{B}$ and also all complements of sets in $\mathcal{B}$.
Now consider $\mathcal{B}_0=C$. And then, for sucessor ordinals $\lambda+1$ we define $$\mathcal{B}_{\lambda+1}=(\mathcal{B}_{\lambda})_c$$ and for limit ordinals $\eta$ we define $$\mathcal{B}_{\eta}=\bigcup_{\alpha<\eta}\mathcal{B}_{\alpha}$$
Claim: Let $\Omega$ be the first uncountable ordinal. Then $\sigma(C)=\mathcal{B}_{\Omega}$
The demonstration of the claim is here.
First doubt: my teacher said that this construction just hold if $C$ is the family of Lebesgue-measurable sets, but I can't see why who is $C$ matters at all... here $C$ is the family of open sets, but does that matter? So if $C$ doesn't matter can we say that this give us a method for characterizing any sigma algebra that is generated by a set?
Second doubt: I don't know much of ordinals. Is it $\Omega$ the only limit ordinal such that every countable subset of $[0,\Omega)$ has an upper bound in $[0,\Omega)$? Because it seems like if yes then we can prove that $\sigma(C)=\mathcal{B}_{\kappa}$ for any ordinal $\kappa$ that satisfies this assertion.
Third doubt: At step 2 of the answer (here $C$ is the family of of open subsets of $\Bbb{R}$), we could say $E=\Bbb{R}\in\mathcal{B}_0$, right? Of course it is in $\mathcal{B}_2$ also... but am I missing anything?
Fourth doubt: How can we prove that $(\mathcal{B}_{\lambda})_{\lambda}$ is strictly increasing??
I'm sorry if I am being too redundant in my question... But thank's in advance for any help!