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Question. Can a countable group have an uncountable number of distinct Hausdorff group topologies?

By a group topology one understands a topology with respect to which the group operations are continuous. By distinct I mean non-isomorphic as topological groups.

Motivation. The examples I have in mind are the pro-$p$ topologies on (e.g.) free groups, which typically yield an infinite (but countable) number of distinct topologies. We can take the uncountably many subsets $\varpi$ of the primes and look at the pro-$\varpi$ topologies, but I'm not certain that they are all distinct.

Note that a countable set can be topologised in uncountably many ways.

(Interestingly, it seems that a kind of dual question has been asked before, which popped up in the list of similar questions.)

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Yes, and the example you have in mind works. In more detail, the topologies on $\mathbb{Z}$ given by its diagonal inclusions into $\prod_{p \in S} \mathbb{Z}_p$ for all sets $S$ of primes are all distinct, and there are uncountably many of them. You can recover the set $S$ from the topology as follows: look at all of the sequences that converge to $0$. Given such a sequence, look at the set $S'$ of primes $p$ for which the $p$-adic norm of the sequence converges to $0$. Then $S$ is the intersection of all such $S'$.

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  • $\begingroup$ Very slick. Thank you. $\endgroup$ – James Sep 3 '14 at 2:40

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