For a number field $K$ let $I_K$ be the idele class group and let $$U := \prod _ {\mathfrak q \nmid p} U_\mathfrak q$$ where $\mathfrak q$ is prime in $K$ and $p$ is a prime in $\mathbb Q$. I want to prove that $I_K/ \overline{K^*U}$ is totally disconnected.
It looks like something that should be proved using some result as I don't see how it could be proven from definitions. From the claim 2 of the highest rated answer here https://mathoverflow.net/questions/290075/subgroups-of-a-topological-group-such-that-quotient-space-is-totally-disconnecte it seems that I should prove that $\overline {K^*U}$ contains the connected component of identity. But I cannot figure out how to do that and also the proof of the claim 2 is not short, so I will prefer to see a proof without using that claim.