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Suppose that $G$ is a topologically finitely generated profinite group, and let $H$ be a subgroup of countably infinite index. Is $H$ necessarily topologically finitely generated with the subspace topology inherited from $G$?

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  • $\begingroup$ Note that it is no restriction to assume that $H$ is dense (because its closure is necessarily open). On the other hand when $H$ is dense but not assumed of countable index, there are counterexamples, e.g. $\bigoplus_n Alt_n\subset \prod_n Alt_n$. $\endgroup$
    – YCor
    Oct 10, 2015 at 21:01

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