I am reading lecture 3 of Conrad notes (link : https://math.stanford.edu/~conrad/Perfseminar/ ), in which he proves space of valuations is a spectral space. Last theorem of lecture 3.
We have a map $j: \text{Spv}(A) \rightarrow \text{PowerSet}({A \times A}) $ defined by $ j(v)= \{ (a_1 , a_2) \in (A \times A) \; | \; v(a_1) \leq v(a_2) \}$. Here, $\text{Spv}(A)$ is set of all equivalence classes of valuations on $A$. I want to prove image of this map $Img(j)$ is closed subset. In step (4) of the proof, he says
The axioms listed in Step 2 are closed conditions on $(A \times A)$ (check!)
But I am not sure how to prove that. I know open subsets of $\text{PowerSet}({A \times A})=\{0,1\}^{A \times A} $ are of the form $U_{S_1,S_2}=\{S \subseteq A\times A \; | \; S \subseteq S_1 \; \text{and} \; S \cap S_2=\phi \}$ for some finite subsets $S_1, S_2 \subseteq (A\times A))$. But I don't know how to use this. Can you please help with this?
