Completely unaware of the Bohr topology, I recently [asked][1] whether or not there was a Hausdorff group topology on the integers $\mathbb{Z}$ which made the group fail to be first countable. For me, this topological group is a bit extreme since there are no non-trivial convergent sequences. I'm very interested to know if there is a sequential example.

If $\mathbb{Z}$ is given a Hausdorff group topology which makes it a sequential space, must it be first countable?

  [1]: http://mathoverflow.net/questions/114816/hausdorff-group-topologies-on-finitely-generated-groups