19.69 (2018)

(P. Wesolek). The acronym tdlc stands for totally disconnected and locally compact, and tdlcsc for tdlc and second countable. Let $\text{Res}(G)$ denote the intersection of all open normal subgroups of a topological group $G$. The class of elementary tdlcsc groups is defined as the smallest class $\mathcal{E}$ of tdlcsc groups such that (1) $\mathcal{E}$ contains all second countable profinite groups and countable discrete groups; (2) $\mathcal{E}$ is closed under taking closed subgroups, Hausdorff quotients, directed unions of open subgroups, and group extensions.

This class admits a well-behaved, ordinal-valued decomposition rank $\xi$ defined recursively as follows: $\xi(\{1\}) = 1$, and if $G \in \mathcal{E}$ is a union of an increasing sequence $(O_i)$ of compactly generated open subgroups, then $\xi(G) = \sup_i \{\xi(\text{Res}(O_i))\} + 1$.

What is the supremum of the decomposition ranks of elementary tdlcsc groups? In particular, is it countable?