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A topological group is a group $G$ together with a topology on the elements of $G$ such that the group operation and group inverse function are both continuous (with respect to the topology).

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Are convolution algebras ever "topologically noetherian"?

The completed group ring of a compact $p$-adic Lie group with coefficients in a field of characteristic $p$ is topologically Noetherian. In fact, it is even abstractly Noetherian. This is a theorem of …
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