More generally, let $(R,m)$ be a Noetherian local domain with fraction field $K$. The $m$-adic topology turns $R$ into a topological ring. When $R$ is a discrete valuation ring, this topology extends to a field topology on $K$, in such a way that $R$ is open in $K$. I wonder how much of this can be done for more general $R$. The first natural question is:

Is there a field topology on $K$ inducing the $m$-adic topology on $R$?

On the negative side, if $\dim(R)≥2$, there is not even a *ring* topology on $K$ such that $R$ is an open subring. Indeed, for any nonzero $f\in R$, multiplication by $f$ in $K$ must be a homeomorphism, so $fR$ would have to be open in $R$, which it is not unless $f\in R^\times$.

On the other hand, there is a ring topology on $K$ making $R$ closed: view $K$ as the union of all principal fractional ideals $f^{-1}R$ ($f\in R\smallsetminus\{0\}$), each endowed with its $m$-adic topology (as a free $R$-module). The transition maps are closed embeddings, and it is easy to see that the *colimit topology* on $K$ works. If we choose this topology on $K$, the question becomes:

Is this a field topology? In other words, is the inversion map on $K^\times$ continuous?

For instance, take $R=k[[x,y]]$ where $k$ is a field, and consider the sequence $n\mapsto \frac{x}{x+y^n}$. Does this sequence converge to $1$ for the above topology? Equivalently, does the sequence $n\mapsto \frac{y^n}{x+y^n}$ converge to $0$?