Topological groups and cardinality of their commutator group (example request)

Question

What would be examples of non abelian, non discrete, second countable compact topological groups with a finite or at most countable commutator (derived) subgroup ?

Answer 1

Proposition. Let $G$ be a compact group with in which $[G,G]$ is countable. Then $[G,G]$ is finite, and the center of $G$ is open of finite index.

[So $G$ is extension of a finite group (as normal subgroup) by an abelian compact group.]

Note that a priori, it is not clear that the closure of $[G,G]$ is countable, so this is not immediate.

Here's how to prove this. Suppose that $[G,G]$ is countable. For fixed $y$, consider $x\mapsto xyx^{-1}y^{-1}$. It has a countable image. Right-translating, we deduce that the continuous $x\mapsto xyx^{-1}$ has countable image. But this is an orbit map (for $G$ acting on itself by conjugacy), so the image is both compact and homogeneous, and countable. So it is finite. That is, $y$ has a finite conjugacy class. Since this holds for every $y$, each conjugacy class in $G$ is finite. A group with this property is called FC-group.

So the main result of Hofmann and Russo, which describe FC-groups that admit a compact group topology, applies. (Ref: K. Hofmann, F. Russo. The probability that x and y commute in a compact group. Math. Proc. Cambridge Philos. Soc. 153 (2012), no. 3, 557–571. ArXiv link.