= Solution
Let $I(L/K)$ be the <inertia group> and let $\varphi$ denote <Frobenius> on the residue-field extension. The <Relative Weil group> is
$$
W(L/K)=\{\sigma\in\operatorname{Gal}(L/K):\bar\sigma=\varphi^n\text{ for some }n\in\mathbb Z\}.
$$
Its <Weil-group topology> makes $I(L/K)$ an open <profinite group> with its usual topology and gives $W(L/K)/I(L/K)$ the discrete topology. Thus every inertia coset is an open copy of $I(L/K)$.
Take $L=K^{\mathrm{nr}}$, the <maximal unramified extension> of $K$. Then
$$
\operatorname{Gal}(L/K)\cong\widehat{\mathbb Z},
\qquad
W(L/K)\cong\mathbb Z.
$$
The subgroup $\{0\}\subset\mathbb Z$ is open in the discrete Weil-group topology, but it is not open in the <profinite topology> inherited from $\widehat{\mathbb Z}$.
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