How to prove the spectrality of valuation spectrum of a commutative ring?

Question

Please help me to understand the proof of spectrality of valuation spectrum of a commutative ring. I'm trying to understand the proposition 4.7 in page 25 here. Sorry I spent a lot of time on this proof but I really can't understand.

1. How do these topologies are defined over $SPV(A)$ and why the clopen subsets of the topology induced by $P(A\times A)$ are of the form $SPV(A)(f/s)$?

2. How the topology induced by $P(A\times A)$ is defined? (What are the open sets here?)

3. Why $SPV(A)$ is $T_0$? (How is the open sun set containing one of $f$ or $s$ but not the other one defined?)