Let $X$ be a Stone space (i.e. totally disconnected compact Hausdorff). Then the following are equivalent:

- $X$ is second countable
- $X$ is metrizable
- $X$ has countably many clopen subsets
- $X$ is an inverse limit of a sequence $\{X_n\}_{n \in \mathbb N}$ of finite (discrete) spaces

In (4), the transition maps can be take to be surjective. Such spaces have received new attention in their role in Clausen and Scholze's light condensed mathematics.

By Stone duality and (3), classifying second-countable Stone spaces $X$ is equivalent to classifying countable Boolean algebras, which sounds pretty hopeless. Nevertheless, the Cantor-Bendixson theorem tells us that $X$ is either countable, or the disjoint union of a countable space and a copy of the Cantor set $\mathcal C$.

In the case where the perfect core is empty, the Sierpinski-Mazurkiewicz theorem tells us that the countable part is homeomorphic to $n \times (\omega^\alpha+1)$ for some finite (discrete) $n$, where $\alpha + 1$ is the Cantor-Bendixson rank of $X$ and $\omega^\alpha + 1$ is the set of ordinals $\leq \omega^\alpha$ in the order topology. ($\alpha$ can be an arbitrary countable ordinal)

Unfortunately, when the perfect core is nonempty, the remaining scattered part is generally not itself compact, so the Sierpinski-Mazurkiewicz theorem doesn't apply to it. This still sounds pretty hopeless, but still I wonder:

**Question:** What more can be said about the classification of second-countable Stone spaces up to homeomorphism?

Is there any analog of the Sierpinski-Mazurkiewicz theorem which can help us here? For instance, is there a classification of second-countable, countable totally disconnected spaces which are not assumed to be compact, which could be applied to the scattered part of $X$ to at least understand that piece?