What you are missing is that it is using the topology of K as a W(k) module (not as a topological field), so basis is like $p^{-n}W(k)$ for $n\in\mathbb{Z}$. So the basis of W(R)[1/p] is rather consisted of things like $p^{-n}$-scaled $p^{N}W(R)+W(I)$. Topology of inductive limit coincides with tensor product because inductive limit and tensor product commute and the inductive limit mentioned above is just W(R) tensored with the inductive limit W(k)->W(k)->... which is precisely expressing the basis of neighborhoods of 0 of K we chose, $p^{-n}W(k)$.

What you are missing is that it is using the topology of K as a W(k) module (not as a topological field), so basis for K is like $p^{-n}W(k)$ for $n\in\mathbb{Z}$. So the basis of W(R)[1/p] is rather consisted of things like $p^{-n}$-scaled $p^{N}W(R)+W(I)$. Topology of inductive limit coincides with tensor product because inductive limit and tensor product commute and the inductive limit mentioned above is just W(R) tensored with the inductive limit W(k)->W(k)->... which is precisely expressing the basis of neighborhoods of 0 of K we chose, $p^{-n}W(k)$.

What you are missing is that it is using the topology of K as a free W(k) module (not as a topological field), so basis is like $p^{-n}W(k)$ for $n\in\mathbb{Z}$. So the basis of W(R)[1/p] is rather consisted of things like $p^{-n}$-scaled $p^{N}W(R)+W(I)$. Topology of inductive limit coincides with tensor product because inductive limit and tensor product commute and the inductive limit mentioned above is just W(R) tensored with the inductive limit W(k)->W(k)->... which is precisely expressing the basis of neighborhoods of 0 of K we chose, $p^{-n}W(k)$.

What you are missing is that it is using the topology of K as a W(k) module (not as a topological field), so basis is like $p^{-n}W(k)$ for $n\in\mathbb{Z}$. So the basis of W(R)[1/p] is rather consisted of things like $p^{-n}$-scaled $p^{N}W(R)+W(I)$. Topology of inductive limit coincides with tensor product because inductive limit and tensor product commute and the inductive limit mentioned above is just W(R) tensored with the inductive limit W(k)->W(k)->... which is precisely expressing the basis of neighborhoods of 0 of K we chose, $p^{-n}W(k)$.