Let $R$ be a topological ring such that the topology is not discrete. Then the ring of Laurent power series $R((t))$ can be defined in the following way:
$$R((t))=\varinjlim_{r}t^r R[[t]]=\varinjlim_{r}\varprojlim_{m}\frac{t^rR[[t]]}{t^{r+m}R[[t]]}\,.$$
Since $ \frac{t^rR[[t]]}{t^{r+m}R[[t]]}\cong \oplus^m R$, we can start with the product topology on the right of the above equation and through an ind-pro process we can construct a topology on $R((t))$. It was proven that with such a topology $R((t))$ is not a topological ring anymore since the multiplication is not continuous as function in two variables.
In this post, it is clear that the subspaces $t^m R[[t]]$ are not open in $R((t))$. Are they closed? Moreover, what is a local basis for $R((t))$? Is there a way to express it explicitly?
Assuming the topology on $R$ is $T_0$ (which to me is part of the definition of "topological ring"), then $t^mR[[t]]$ is closed in $R((t))$. Indeed, to check that it's closed, you just have to check that its intersection with $t^rR[[t]]$ is closed for all $r\in\mathbb{Z}$. The space $t^rR[[t]]$ just has the product topology when you identify it with $R^\mathbb{N}$ via its coefficients, and $t^mR[[t]]$ consists of those elements which have vanishing coefficients corresponding to powers of $t$ less than $m$. This is just the subset of $R^{\mathbb{N}}$ on which certain coordinates are $0$, which is closed in the product topology as long as $\{0\}$ is closed in $R$.
I would not expect there to be a nice local basis for $R((t))$; direct limits of non-open inclusions of spaces usually don't have nice local bases. For instance, if $R$ is $T_0$ and not discrete, the space $R((t))$ is not sequential and in particular cannot have a countable local basis. Here's a sketch of the proof: there must be a net converging to $0$ which is eventually not contained in any single $t^rR[[t]]$, since $R$ is not discrete so every neighborhood of $0$ must intersect $R\setminus t^rR[[t]]$ for all $r$. But there is no sequence converging to $0$ which is eventually not contained in any $t^rR[[t]]$, since any set which contains only finitely many points in each $t^rR[[t]]$ is closed.