This discussion from the main chatroom seems to be related to descriptive set theory:

I searched a bit for some reference for the claim about Bore sets.

This is stated in Kechris (Descriptive Set Theory) as Theorem 13.6 (The Perfect Set Theorem for Borel Sets; Alexandrov, Hausdorff).

Let $X$ be Polish and $A\subseteq X$ be Borel. Then either $A$ is countable or else it contains a Cantor set. In particular, every uncountable standard Borel space has cardinality $2^{\aleph_0}$.

The convention used there is: For convenience we will say that a subset $C$ of a topological space is \emph{a Cantor set} if it is homeomorphic to the Cantor space $\mathcal C$.

And this is related to descriptive set theory, too:

And also a topic of chains in $\mathcal P(X)$ resurfaced again.

Thanks for your answer from a couple of days ago @Martin, I haven't had time to read it in detail, but it already helped me with those doubts I had

I am glad to hear it.

To be honest, I did not have time to check all the details at the time. But I mentioned explicitly that there might be some gaps or inaccuracies.

But since you write that you have been able to figure it out, it probably does not matter that much.

Some time ago with some friends we decided to go through Oxtoby's book. But we did not finish it. I believe we finished with the chapter about Banach category theorem. (And I am not sure whether there are some parts before this chapter that we skipped.) So we did not get to the chapter about Sierpinski-Erdos duality theorem.