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Let $G$ be a totally disconnected locally compact group, $U$ a closed and open subset of $G$. Is there a neighbourhood $V$ of the identity such that for all $g \in U$ we have $gV \subset U$?

Equivalently (by Van Danzig's theorem), is every closed and open subset of $G$ a union of cosets of some compact open subgroup of $G$?

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  • $\begingroup$ If $V$ exists, then since $e \in V$, you get $g \in gV \subset U$ for all $g \in G$. $\endgroup$
    – Paul Frost
    Commented Dec 3, 2019 at 13:57
  • $\begingroup$ @PaulFrost sorry I meant for all $g \in U$. It should be correct now $\endgroup$
    – frafour
    Commented Dec 7, 2019 at 15:35

2 Answers 2

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I think the answer is no.

For example, let $G$ be your favorite non-discrete tdlc group such that there exists $U_0\supseteq U_1\supseteq \dots$ a decreasing sequence of clopen neighborhoods of the identity, with $\bigcap_{n=0}^\infty U_n = \{e\}$, but $U_n\neq \{e\}$ for all $n$. For example, you could take $G$ to be the automorphism group of the complete binary tree and $U_n$ to be the set of all automorphisms which fix the ball of radius $n$ around the root.

Now consider $G\times \mathbb{Z}$, which is also tdlc. Let $U = \{(x,n)\mid n\geq 0, x\in U_n\}$. This set is clopen in $G\times \mathbb{Z}$. But for any neighborhood $V$ of the identity $(e,0)$, there is some $n$ such that $V\not\subseteq U_n$. So letting $g = (e,n)$, we have $gV\not\subseteq U$.

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  • $\begingroup$ Note that it's not really important for such a decreasing sequence to exist--if it doesn't, you can just take a product with a larger discrete group than $\mathbb{Z}$ (as long as $G$ is not discrete so $\{e\}$ itself is not clopen). $\endgroup$ Commented Dec 7, 2019 at 16:09
  • $\begingroup$ @EricWofsey Yes, good point. $\endgroup$ Commented Dec 7, 2019 at 16:10
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No. For instance, let $G=\mathbb{Z}_p\times\mathbb{Z}$ and consider the set $U=\bigcup_{n\in\mathbb{N}} p^n\mathbb{Z}_p\times \{n\}$. More generally, given any non-discrete totally disconnected locally compact group $H$, you can take a product of $H$ with a discrete group and then take a union of shrinking clopen subsets of different cosets of $H$ in the product.

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  • $\begingroup$ You beat me by 14 seconds! $\endgroup$ Commented Dec 7, 2019 at 16:06

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