Galois group of a global field

Let k be a global field and s a separable closure of k.

For each place v of k, let k_v be a completion k_v of k at v, and s_v
a separable closure of k_v containing s.

Let G (resp. G_v) be the Galois group of s/k (resp. of s_v/k_v), and
H_v the image of the restriction morphism from G_v to G.

Is G topologically generated by the H_v?
.