I would like to know if there's any structure theorem classifying all compact abelian groups, any reference is appreciated.
I do know there is a structure theorem for compact abelian Lie groups, and for locally compact abelian groups, however the first one is too specific and the second one is too general.
I know there are pro-finite groups, and Lie groups. Recently i discovered there are also Pro-Lie groups (which are not necessarily compact). Is every compact abelian group a direct product of these groups? If so, what can we say about compact Pro-Lie groups?