Let $X$ be a countable closed set of the 2-dimensional sphere $S^2$. It is easy to show using Baire category theorem that $X$ is the closure of its isolated points, i.e., $X = \overline{\mathrm{Isol}(X)}$.
Let now $X_1 = \mathrm{Isol}(X)$ and $X_{n + 1} = \mathrm{Isol}(X - \bigcup_{i = 1}^n X_i)$ for $n \geq 1$. In other words, this sequence of sets is constructed by the following iterative procedure: we consider the set of isolated points of the original set, remove it, then again consider the set of isolated points of what is left, then remove it from the current set again, etc.
Is it true that $X - \bigcup_{n \geq 1} X_n$ is at most finite set?
I believe if one assumes that $X - \bigcup_{n \geq 1} X_n$ is infinite, then it is possible to construct an uncountable subset of $X$. However, writing this argument in all details can be pretty annoying. But, perhaps, there is a more elegant way, potentially involving Baire category theorem?
