Q:

Connected component of the Idele group

Let $K$ ba a number field with $r_1$ real embeddings and $r_2$ pairs of complex embeddings. Let $I_K$ be the group of ideles of $K$ and let $H$ be the connected component of identity. How to show that $H$ is contained in all the open subgroups of $I_K$ and further $H=(\Bbb{C}^{\ast})^{r_2} \times (\Bbb{R}_+^{\ast})^{r_1}$. The problem is here
It will be helpful if anyone gives me reference for this fact or the way of proving the result.

A:

Every non archimedean topology is totally disconnected.
Hence every multiplicative group of non archimedean local field and its unit group is totally disconnected. 
since product of totally disconnected space is also totally disconnected.,
By the definition of idele, connected component of idele group is product of connected component of archimedean local field. Now claim follows directly.

