= Weil-group topology
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The topology of the absolute <Weil group> $W_K$ of a non-Archimedean <local field> makes the <inertia group> open with its profinite topology, and makes the quotient $W_K/I_K\cong\mathbb Z$ discrete. In particular inertia cosets are open. This topology is finer than the subspace topology inherited from the <absolute Galois group>: the induced topology on the integer quotient would otherwise be its nondiscrete profinite topology.
For a finite <Galois extension> $L/K$, the <Relative Weil group> instead has the open subgroup $L^\times$ with its usual topology and finite quotient $\operatorname{Gal}(L/K)$. Keeping the absolute and relative constructions separate prevents a finite relative Galois quotient from being confused with the infinite residue-action quotient.<Arithmetic Duality Theorems by James S. Milne (2006)>[ChapterI, §8, printed p.102 and Example8.1(a)]{ref}
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