Proposition (I.6.7,HAG): Every nonsingular quasi-projective curve $Y$ is isomorphic to an abstract nonsingular curve.
The first paragraph of the proof establishes a bijective map $\phi: Y \rightarrow U$, where $U$ is a subset of the discrete valuation rings of $K/k$, where $k$ is the algebraically closed base field and $K$ is the field of rational functions of $Y$. The bijection $\phi$ takes a point $P \in Y$ to the DVR $\mathcal{O}_{P,Y}$.
The second paragraph of the proof provides an argument that actually $U$ is an open set in the topology of DVRs of $K/k$ defined at page 42. However, is that not obvious, since $\phi$ is a bijection and $Y$ is infinite (by ex. 4.8)?
$|U| = \infty$ does not necessarily imply that $U$ is open - by definition a subset $\emptyset \ne S \subseteq C_K$ is open iff $|C_K \setminus S| < \infty$ (consider an infinite subset of $C_K$ whose complement is also infinite).