Let $K$ be an algebraic number field. Let $\mathbb{I}$ be the group of ideles and let $\mathbb{I}_f$ be the group of finite ideles. We embed $K^\times$ diagonally in both.
It is know that $K^\times$ is a discrete subgroup of $\mathbb{I}$. Hence it is also closed in $\mathbb{I}$. However, if the ring of integers of $K$ has an infinite unit group, then $K^\times$ is not discrete in $\mathbb{I}_f$.
Question: Is $K^\times$ at least closed in $\mathbb{I}_f$?