Let $K$ be a non archimedean local field and let $K^{\mathrm{unr}}$ be its maximal unramified extension in a fixed separable closure. The Weil group $W_K$ is then defined to be the subgroup of elements of the absolute Galois group $G_K$ that restrict to integer powers of the Frobenius element in $\mathrm{Gal}(K^{\mathrm{unr}}/K)$. In other words, $W_K$ is the inverse image of $\mathbb{Z}\subseteq \widehat{\mathbb{Z}}$ under the surjection $G_K\to \mathrm{Gal}(K^{\mathrm{unr}}/K)\cong\widehat{\mathbb{Z}}$. My question is that what exactly is the topology on $W_K$? I learn that $W_K$ is endowed with a topology so that the exact sequence of (abstract) groups $$1\to I_K\to W_K\to \mathbb{Z}\to 1$$ becomes an exact sequence of topological groups and that the inclusion map $W_K\to G_K$ is continuous. Since $\mathbb{Z}$ is discrete, it is necessary that the inertia group $I_K$ with its usual profinite topology is open in $W_K$. So, is the desire topology on $W_K$ the coarsest one that contains $I_K$ and the subspace topology inherited from $G_K$? I know this is a basic thing but I am really confused, since many sources (for example the Wiki page or the Galois representations note by R. Taylor) just simply say that $W_K$ is equipped with a topology in which $I_K$ is an open subgroup.
Could you please explain this to me? Thank you.