Given a topological ring R$R$, under what conditions and in what way, can one induce a topology on the R$R$-points of a scheme X$X$? For example, if X$X$ is P^n$P^n$ or A^n$A^n$, one has natural topology on the R$R$-points.
If G$G$ is a group scheme/A and R$R$ is A$A$-algebra (still a topological ring), will the induced topology on G(R)$G\left(R\right)$ (as above) automatically make G$G$ into a topological group.
For number theorists, if G$G$ is an algebraic group/Q, we can consider the adelic points G(A_K) for$G\left(A_{K}\right)$ for any number field K$K$. Is the induced topology on G(A_K)$G\left(A_{K}\right)$ that of a restricted direct product? Under what conditions will G(A_K)$G\left(A_{K}\right)$ be locally compact or satisfy other nice properties?
