Let $X$ be a smooth (integral) projective curve over a field $k$ and let $K$ be the function field of $X$.
I'd like to prove that there is a bijective correspondence between the closed points of $X$ and the set of places (i.e. equivalent absolute values on $K$) which are trivial on $k$.
Clearly for every closed point $p$ I have the local ring $\mathcal O_{X,p}$ which is a DVR, therefore I get a desired valuation on $K$, hence a metric.
But what about the inverse of this map? From an absolute value of $K$ I want to get a closed point of $X$ inverting the above construction.
Valuations are the same as places, which are just maps from a valuation ring into its quotient field. The center of a place is the prime ideal of all functions on which the place is zero. In the case of a curve this will be a maximal ideal, and if the field is algebraically closed you get a point of the curve. This theory is exposited in Zariski and Samuel: Commutative Algebra Vol.II. In general many places will correspond to a given point, but if the point is simple there will be only one place for that point.