Given a topological ring $R$, under what conditions and in what way, can one induce a topology on the $R$-points of a scheme $X$?  For example, if $X$ is $P^n$ or $A^n$, one has natural topology on the $R$-points.

If $G$ is a group scheme/A  and $R$ is $A$-algebra (still a topological ring), will the induced topology on $G\left(R\right)$ (as above) automatically make $G$ into a topological group.

For number theorists, if $G$ is an algebraic group/Q, we can consider the adelic points $G\left(A_{K}\right)$ for any number field $K$.  Is the induced topology on $G\left(A_{K}\right)$ that of a restricted direct product?  Under what conditions will $G\left(A_{K}\right)$ be locally compact or satisfy other nice properties?