Are there any interesting cases (interesting here is interpreted rather loosely here) where you can show $X$ has property $P$ whenever all $X(K)$ have property $P$ where $K$ runs through all fields?
To clarify, let me state different versions of the questions.
Assume $X$ is defined over a base field $k$ and $K$ runs through all field extensions of $k$.
Version 1 Consider the sets $X(K)$ as having just a topology and assume $X$ is reduced.
I'm concerned with topological properties: being irreducible, being connected, counting irreducible or connected components, being an open subscheme of another scheme, being non empty,...
Version 1' If $X$ is not necessarily reduced what properties are insensitive to being reduced or not?
Version 2 Consider $X(K)$ as a topological space with the additional data of a sheaf of rings where for every $U(K) \subset X(K)$ we consider the ring generated by set maps $r_K: U(K) \to K$ for $r \in \mathcal{O}_X(U)$ where $r_K$ is defined by setting $r_K(\phi) = \phi^\#(r)$ for every $\phi \in U(K)$.
I'm more interested in Version 1 but perhaps more can be said about Version 2.
Motivation: Sometimes the field valued points of a scheme are much easier to understand than general $R$ points. For example a split reductive group $G$ you have the Bruhat decomposition of $G(K)$ which tell you the double cosets for a Borel $B \subset G$ are indexed by permutation matrices. But in general the cosets in $G(R)$ for a $K$-alebra $R$ are more complicated.
