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For a prime number $p$, the $p$-adic topology on the set $\omega$ of non-negative integers is generated by the base consisting of the arithmetic progressions $x+p^n\omega:=\{x+p^ny:y\in\omega\}$ where $x,n\in\omega$.

Question. Let $p,q$ be two distinct prime numbers. Is the set $\{q^n:n\in\omega\}$ closed in the $p$-adic topology on $\omega$?

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    $\begingroup$ No. It's indeed dense in an open subgroup of $\mathbf{Z}_p^\times$, the group of invertible elements in the $p$-adics. Hence its closure in $\omega$ contains a whole arithmetic progression. Indeed, the $\mathbf{Z}_p^\times$ is known to be isomorphic to $\mathbf{Z}_p\times F(p)$ for some finite group $F(p)$, with $F(p)\simeq\mathbf{Z}/(p-1)\mathbf{Z}$ for odd $p$. $\endgroup$
    – YCor
    Commented Nov 27, 2019 at 8:58
  • $\begingroup$ @YCor Thank you for the comment. Could you write an arithmetic progression in the closure of the set $\{7^n:n\in\omega\}$ in the 3-adic topology on $\omega$? $\endgroup$
    – Taras Banakh
    Commented Nov 27, 2019 at 9:16
  • $\begingroup$ $7$ equals $1$ mod $3$, so the closure is exactly $3\mathbf{Z}_3+1$ in this case. (So the closure in $\omega$ is $\omega\cap (3\mathbf{Z}+1)$.) $\endgroup$
    – YCor
    Commented Nov 27, 2019 at 9:19
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    $\begingroup$ (I should have said that $7$ is not equal to $1$ mod $9$ in my previous comment, to deduce the conclusion.) Yes it's true. I don't know a reference of this and never saw it. What's standard is the description of $\mathbf{Q}_p^\times$ or $\mathbf{Z}_p^\times$ as topological group. It's very standard and hence probably in several basic books, but I don't have a particular reference in mind. $\endgroup$
    – YCor
    Commented Nov 27, 2019 at 9:34
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    $\begingroup$ BTW all this makes uses the fact that the closure of every submonoid of a compact group is a subgroup. (Used here in an easy case: the submonoid is generated by a single element and the ambient compact group is $\mathbf{Z}_p\times$ (finite). $\endgroup$
    – YCor
    Commented Nov 27, 2019 at 10:14

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