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(R) (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(A_K) for any number field K. Is the induced topology on G(A_K) that of a restricted direct product? Under what conditions will G(A_K) be locally compact or satisfy other nice properties?