Somebody asked a question earlier regarding this proof but I'm confused about a different part. I understand everything but the line "As the zero set of finitely many polynomials, $R$ is a closed subset of $S \times S$".
Can somebody help explain to me what this statement means? What does it mean to be the zero set of finitely many polynomials? What polynomials is $R$ the zero set of? Do the polynomials mentioned come from the determinants of the minors? Thank you for the help.