Is $\operatorname{Spec}R_S$ homeomorphic to $ \left\{ \mathfrak p\in \operatorname{Spec}R:\mathfrak p\cap S=\emptyset \right\}$? The correspondence theorem for localizations gives a bijection between $\operatorname{Spec}R_S$ and $ \left\{ \mathfrak p\in \operatorname{Spec}R:\mathfrak p\cap S=\emptyset \right\}$. According to the answers to this question, the latter set is open iff the saturation of $S$ is finitely generated. Are these sets always homeomorphic in this case?
For one direction I think it's enough to show $\uparrow a= \left\{\mathfrak p\in \operatorname{Spec}R_S:\mathfrak p\ni a \right\}$ pulls back to a closed set but I'm having trouble with this. I am not sure about the other direction and can't think of a counterexample either.

It definitely is. This is EGA I, Corollaire (1.2.6), with $S := A \backslash\mathfrak{p}$ which is multiplicative as $\mathfrak{p}$ is prime. It is everywhere on the internet and in almost every algebraic geometry book. (For instance, it is not in Hartshorne's classic "Algebraic Geometry".)
