Gaetan Chenevier wrote: > Is there an example of such a group? > Yes, take the subfield of C generated by all the square roots of integers, call it K.
OK, the final sentence of my posting on this was incorrect. Let me try to understand the claim. If V is the direct sum of countably many copies of the cyclic group of order 2, we can consider V as a vector space over the field with two elements. Its dual vector space V* is the cartesian product of countably many cyclic groups of order two. The double dual V** will be bigger than V, which is the Pontryagin dual of V*. That's because V* is uncountable, and, not very constructively, one can choose a basis and assign values of a linear functional on it. Therefore there are many subgroups of index 2 in V* that aren't closed in the product topology.
In the original context V* occurs as a Galois group Gal(K/Q), as one checks by taking the linearly disjoint subfields made by adjoining sqr(-1) and the sqr(p) for p prime. This Galois group is a quotient of the Galois group of the algebraic numbers over Q.