# Can a countable group have uncountably many distinct Hausdorff group topologies?

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.

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

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 $S'$.