I was reading this question [The connected component of the idele class group](https://mathoverflow.net/questions/350676/the-kernel-of-the-global-class-field-theory-homomorphism) but I am very confused about the structure of the *solenoids* $(\widehat{\mathbb{Z}}\times\mathbb{R})/\mathbb{Z}$, (where $\mathbb{Z}$ acts diagonally), more specifically why it is connected. More generally, if we replace $\widehat{\mathbb{Z}}$ by $\mathbb{Z}\_p$, is the resulting quotient group $(\mathbb{Z}\_p\times \mathbb{R})/\mathbb{Z}$ still connected?


Hoping if someone can give me any hints or answers!!



Is $(\mathbb{Z}_p\times \mathbb{R})/\mathbb{Z}$ connected?
No originality here, but I would tell the story as follows. Consider the subgroup $(\mathbb{Z}\times\mathbb{R})/\mathbb{Z}$ of $(\mathbb{Z}\_p\times\mathbb{R})/\mathbb{Z}$. It is dense, because $\mathbb{Z}$ is dense in $\mathbb{Z}\_p$. It is also connected, because it is isomorphic to $\mathbb{R}$ (with a coarser topology than the standard one). Therefore, in the group $(\mathbb{Z}\_p\times\mathbb{R})/\mathbb{Z}$, the connected component of the identity is dense, whence it equals $(\mathbb{Z}\_p\times\mathbb{R})/\mathbb{Z}$.


The same proof works for $(\widehat{\mathbb{Z}}\times\mathbb{R})/\mathbb{Z}$.


