On the classification of second-countable Stone spaces

Question

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

1. $X$ is second countable
2. $X$ is metrizable
3. $X$ has countably many clopen subsets
4. $X$ 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?

Answer 1

The paper linked by YCor in the comments,

• R. S. Pierce. Existence and uniqueness theorems for extensions of zero-dimensional compact metric spaces, Trans. Amer. Math. Soc., 148:1–21, 1970, doi:10.1090/S0002-9947-1970-0254804-4

gives a complete answer. To see this, let's use the term scattered space for a countable totally disconnected Hausdorff space. In Prop 3.2, Pierce shows that every scattered space $S$ is a topologically-disjoint union $S = T \amalg U$ where $U$ is a compact scattered space and $T$ is of the form $\omega^\alpha$. Now if $X$ is a second-countable Stone space with perfect part $P$ and scattered part $S = T \amalg U$, then note that $P \cup T$ is closed in $X$: since $P$ is closed, we need only note that a sequence in $T$ has a limit in $X$, and that limit can't be in $U$ because it is topologically disjoint from $T$. Therefore $P \cup T$ is compact Hausdorff. There is a continuous bijection $(P \cup T) \amalg U \to X$, which must be a homeomorphism because these are compact Hausdorff spaces.

Next, if $X$ is a second-countable Stone space, then write $X^\alpha$ for the $\alpha$th Cantor-Bendixson derivative of $X$, and $P= X^\infty$ for the perfect core of $X$ and $S = X - P$ for the scattered part. Write $X_\alpha = P \cap \overline{X^\alpha - P} = P \cap \overline{S^\alpha}$ for the limit points in $P$ of the $\alpha$th derivative of $S$. Then $(X_\alpha)_{\alpha < \mu}$ is a decreasing filtration of $P$ by nonempty closed subsets indexed by a countable ordinal.

In Thm 1.2,Pierce shows that if $\mathcal C \supseteq \mathcal C_0 \supseteq \cdots$ is any descending filtration of nonempty closed subspaces $(\mathcal C_\alpha)_{\alpha < \mu}$ of Cantor space indexed by a countable ordinal $\mu$, then there is a second-countable Stone space $X$ with perfect part $P$, and scattered part $\omega^\mu$, and $X_\alpha = \mathcal C_\alpha$. By taking a topologically-disjoint union with a scattered Stone space, we obtain that for any such filtration of Cantor space and any scattered space $S$ which is a topologically-disjoint union of $\omega^\mu$ with a scattered Stone space, there is a second-countable Stone space $X$ with perfect part $\mathcal C$, scattered part $S$, and $X_\alpha = \mathcal C_\alpha$.

Finally, Pierce shows in Thm 1.1 that a second-countable Stone space $X$ is determined up to homeomorphism by its scattered part along with the filtration $(X_\alpha)_{\alpha < \mu}$ of its perfect core. More precisely, Given $X,Y$ second-countable Stone spaces, a homeomorphism $f_\infty : P \to Q$ between perfect cores extends to a homeomorphism $f : X \to Y$ if and only of the scattered parts are homeomorphic and $f_\infty(X_\alpha)= Y_\alpha$. Of course, the perfect core is homeomorphic to $\mathcal C$ iff it is nonempty.

Summary:

To sum up, we can construct an uncountable second countable Stone space from the following data:

1. A scattered space which is of the form $\omega^\mu \amalg (n \times (\omega^\lambda + 1))$ for countable ordinals $\mu \leq \lambda$ and $n \in \mathbb N$, and

2. A decreasing filtration of Cantor space indexed by $\mu$ and comprising nonempty closed sets.

Every uncountable second-countable Stone space arises this way up to homeomorphism, and two such are homeomorphic iff there is a homeomorphism between the scattered parts in (1) and a self- homeomorphism of Cantor space relating the filtrations of (2). The invariants (1) and (2) can be read off from a general $X$ as indicated above.