Let $K$ be a local field complete with discrete valuation $v$ with finite residue field $k$. I am just wondering how does one know that the the inertia subgroup $I_v$ is a closed subgroup of $Gal(\overline{K}/K)$?
And if it talks about finite quotient $I_v/J$ may I assume automatically that $J$ is a closed subgroup of $I_v$?
I am asking this question because it came up in the book I am reading and the book doesn't talk about this stuff... Thank you very much.