Let $T$ be a finite set of primes and let $K$ be the maximal extension of $\mathbf{Q}$ unramified outside $T$.
We have three Galois groups:
$G_{\mathbf{Q}} = \mathrm{Gal}(\overline{\mathbf{Q}}/\mathbf{Q})$
$G_{T} = \mathrm{Gal}(K/\mathbf{Q})$
and for any prime number $p$
$G_p = \mathrm{Gal}(\overline{\mathbf{Q}_p}/\mathbf{Q}_p)$
Are these compact topological groups?
Also, are there any canonical maps between these groups? I think $G_T$ maps to $G_p$ if $p$ is in $T$. Is that correct?