Finitely generated $k-$algebras of regular functions on an algebraic variety

Question

I am reading Q. Liu's "Algebraic geometry and arithmetic curves". In the proof of Lemma 4.3. at page 61 (the closed points of an open subset $U$ of an algebraic $k-$variety $X$ are closed in $X$), he says that $k(x)$ is a finite field extension of $k$, and he seems to use the fact that $k(x)$ is the quotient by a maximal ideal of a finitely generated $k-$algebra. If we write $k(x)=\mathcal{O}_{X,x} / \mathfrak{\tilde{m}}_x$, this does not seem to work, because at page 56 in Remark 3.48. he says that Spec $\mathcal{O}_{X,x}$ is in general not an algebraic variety, in particular not an affine variety, in particular $\mathcal{O}_{X,x}$ is in general not a finitely generated $k-$algebra. But we could also obtain $k(x)$ as $\mathcal{O}_{X}(U) / \mathfrak{m}_x$, since $U$ can be assumed to be an open affine subscheme of $X$. Still, I think it cannot be assumed to be an affine subvariety, so I cannot see why $\mathcal{O}_{X}(U)$ should be finitely generated as a $k-$algebra.

Answer 1

I am answering my own question with the solution given by danneks in the comments, so that I can mark it as closed. The point is that the localization of a finitely generated $k-$algebra at an element is again finitely generated. Hence the principal opens subsets, that form a basis of any affine variety, are again affine varieties. So pick an affine variety $V$ that contains $x$, consider the open subset $x \in U \cap V \subseteq V$ and choose a principal open subset $x \in D(f) \subseteq U \cap V \subseteq V$. Since $D(f) \subseteq U$, $x$ is closed in $D(f)$, so we can argue as we wanted.