Notice: the following statements about the product topologies are all Cartesian product topology, we are in the category of topology not the category of schemes.
In this page of sober space, it said any product of sober spaces is sober. What does it mean by "any"? Any index or any finite products?
Fact 1. Could anyone give a proof about this fact that the product $X\times Y$ of any two sober spaces $X,Y$ is sober?
I am considering the product topology of two spectral spaces. Let $X=\operatorname{Spec} A$, $Y=\operatorname{Spec} B$ where $A,B$ commutative rings. Then I wonder to know
Does there exist a canonical choice of a ring $C$ such that $\operatorname{Spec} C$ is cannonically isomorphic to the Cartesian topology of $X\times Y$?
The topologies of $X,Y$ are quasicompact, and they have bases consisting of quasicompact opens and the intersection of any two quasicompact open is quasicompact open. Those above properties are preserved in $X\times Y$ (if I am right). So by Fact 1, the product topology of $X\times Y$ is a spectral space, thus it can be realized as a spectrum of a commutative ring (see the same wiki article).
Moreover, if we are considering the product topology $X\times_Z Y$ (the induced topology of the product topology $X\times Y$), where $X,Y,Z$ are affine schemes the maps $X\to Z$, $Y\to Z$ are induced by the ring maps, what will happen in this case, is $X\times_Z Y$ a spectral space, etc?
Thanks!