In Miranda's Algebraic Curves and Riemann Surfaces the definition of a general Riemann surface in projective space is as follows:
Let $X$ be a Riemann surface, which is a subset of a projective space $\mathbb P^n$. We say that $X$ is holomorphically embedded in $\mathbb P^n$ if for every point $p$ on $X$ there is a homogeneous coordinate $z_j$ such that:
a. $z_j\neq 0$ at $p$;
b. for every $k$, the ratio $z_k/z_j$ is a holomorphic function on $X$ near $p$; and
c. there is a homogeneous coordinate $z_i$ such that the ratio $z_i/z_j$ is a local coordinate on $X$ near $p$.
A Riemann surace which is holomorphically embedded in projective space is called a smooth projective curve.
In Bézout's theorem (IV.2.13) it is used that $X$ is compact. Now I'm guessing that a smooth projective curve is always compact. Since $\mathbb P^n$ is compact, it suffices to show that $X$ is closed. I wouldn't know how to show this. Does anyone know? Or should it have been added to the definition?
Miranda simply forgot to assume compactness (this is not the only place in the book where this assumption is forgotten). His definition does not imply compactness. The simplest example is to take $n=1$ and $X=P^1\setminus \{p\}$.