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Sep 2, 2019 at 18:21 comment added KConrad This example as written needs $p > 2$. If $p = 2$ then $U_n$ is all $p$-adic integers with absolute value $1/p^n$, so $U = \mathbf Z_2 - \{0\}$ and $\overline{U} = \mathbf Z_2$ is open in $\mathbf Z_2$. For the case $p = 2$ you could use $U_n = 2^n + 2^{n+2}\mathbf Z_2$ instead.
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Sep 2, 2019 at 15:00 history answered Heiko CC BY-SA 4.0