Let $K$ be a number field. Why is the intersections of all open subgroups of finite index in the idele group $\mathbb I_K$ equal to $\overline{K^{\times}(K^{\times}_{\infty})^0}$? Also, I'm having a hard time thinking about $\overline{K^{\times}(K^{\times}_{\infty})^0}$: how much larger is the closure?
(EDIT: As David Loeffler writes, we need to consider subgroups of finite index containing $K^{\times}$.)
Here, $$ (K_\infty^{\times})^0:=\prod_{v\text{ real}}\mathbb R_{>0}\times \prod_{v\text{ complex}}\mathbb C\times \prod_{v\in V_K^0} 1. $$ where $V_K^0$ is the set of finite places of $K$.
(I'll refer to my notes for the standard definitions and statements, http://web.mit.edu/~holden1/www/math/ant.pdf, 10.5.1 for $\mathbb I_K$, 10.5.4 for the embedding $K^{\times}\hookrightarrow \mathbb I_K$.)
Context: This question comes up when trying to prove [p.263] the topological isomorphism [10.6.5] $ G(K^{\text{ab}}/K)\cong \mathbb I_K/\overline{K^{\times}(K^{\times}_{\infty})^0}$ ("1-dimensional Langlands") from the usual statement of global class field theory [10.6.1].
Given that the map is surjective, it remains to show the kernel is $\overline{K^{\times}(K^{\times}_{\infty})^0}$, which will follow (from the existence theorem) if it is the intersection of all open subsets of finite index in $\mathbb I_K$.