I'm working with the proposition II.6.7 of Hartshorne's book. I have a question in the detail of his proof, and so far I haven't found any clues of this online.
For the notion, a curve over an algebraically closed field $k$ is an integral separated scheme of finite type over $k$, of dimension $1$. Moreover, a curve is complete if it's proper over $k$.
The proposition is said as follows:
Let $X$ be a nonsingular curve over $k$ with function field $K$. Then the following conditions are equivalent:
(i) $X$ is projective; (ii) $X$ is complete; (iii) $X\cong t(C_K)$, where $C_K$ is the abstract nonsingular curve of (I, §6), and $t$ is the functor from varieties to schemes of (2.6).
In the proof of (ii)$\Rightarrow$(iii), Hartshorne demonstrates a fact that the closed points of $X$ are in 1-1 correspondence with the points of $C_K$, and thus it is clear that $X\cong t(C_K)$. How can he conclude this? Maybe it's really trivial. But I still don't know how to show this.
I even have no idea how to construct a continuous map between these two schemes. Does anyone help me? Thanks in advance!
As a topological space, $t(C_K)$ is all the points of $C_K$ plus an additional point we call the generic point, as $C_K$ is irreducible. We can define a bijection between $t(C_K)$ and $X$ by sending the generic point to the generic point and using the 1-1 correspondence between the closed points of $X$ and the points of $C_K$, and if we can show that the bijection and it's inverse are continuous, we will have shown that these two spaces are homeomorphic.
The topology on $t(C_K)$ is the following: a set is closed iff it's a (possibly empty) finite collection of points from $C_K$ or the whole space. This is the same topology on $X$: a set is closed iff it's a (possibly empty) finite collection of closed points or the whole space. Since both directions of our bijection sends these sets to each other, we see that the bijection is actually a homeomorphism.