Another Topology on the Prime Spectrum of a Ring Let's fix a commutative ring $R$. We'll write $X = \operatorname{Spec}R$ for the set of prime ideals of $R$. Finally, let's write $\beta X$ for the Stone-Čech compactification of $X$.
We can define a function $\lim  : \beta X \to X$ using the following recipe:
$$ f \in \lim \mu \iff \mu(\{p \in \operatorname{Spec} R : f \in p \}) = 1$$
Indeed, $\lim \mu$ is prime by Łoś's theorem, or an easy direct verification.
I believe this function actually endows $X$ the structure of a $\beta$-algebra.  In other words, this endows $X$ with a compact Hausdorff topology. In particular, it must be distinct from the Zariski topology.
My questions: does this topology have a name? Is there somewhere I could read about this?
 A: This is the constructible topology, the topology generated by both the Zariski open sets and the basic Zariski closed sets (i.e., the closed sets defined by principal ideals).  Another way to say it is that it is the topology generated by declaring the distinguished Zariski open subsets $D(f)$ to be clopen, rather than just open.  Or, it is what you get by considering $\operatorname{Spec} R$ as a subset of $\mathcal{P}(X)\cong \{0,1\}^X$ and giving it the product topology, where you put the discrete topology on $\{0,1\}$.  (This description is immediate from your description of what limits look like, since you are just taking pointwise limits in $\{0,1\}^X$.)
It can also be described purely topologically from the Zariski topology: it is the topology generated the sets that are compact and open in the Zariski topology, together with their complements.  This topology is useful in the abstract theory of spectral spaces (spaces that are homeomorphic to the spectrum of a ring with the Zariski topology); see https://stacks.math.columbia.edu/tag/08YF for instance.  It is also of interest because the constructible sets (in the sense of algebraic geometry) are exactly the clopen subsets in the constructible topology (so the constructible topology makes $\operatorname{Spec} R$ the Stone space of the Boolean algebra of constructible sets).
