# closure of units of number fields in the finite idele topology

Let $K$ be a number area. Represent by $\mathcal O _K^\times$ its rings of devices and also by $\mathcal O _{K,+} ^\times$ its ring of entirely favorable devices. More allow us represent by $\mathbb A _{K,f}^\times$ the ideles of $K$ along with its common idele geography and also by $\widehat{\mathcal O}_K^\times$ its typical portable part of indispensable, limited, ideles.

What can be claimed concerning the closures $\overline {\mathcal O _K ^\times}$ and also $\overline{\mathcal O _{K,+}^\times}$ absorbed $\widehat{\mathcal O}_K^\times$?

Is it feasible to define the closures clearly?

For me one of the most intriguing instance are actual square areas for the start. If I bear in mind appropriately for fictional square areas the devices need to be distinct.

Thanks significantly beforehand for your aid!

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