A remark on topologies for rational point sets

Let k be a ring, X be a k-scheme and R be a k-algebra endowed with an arbitrary topology. In this text, we introduce the fine topology on X(R), which is based on Grothendieck's definition of a topology for affine k-schemes. We prove that the fine topology is functorial in both X and R and that it coincides with Grothendieck's topology for affine k-schemes, with the strong topology for k-varieties over topological fields k and with the adelic topology for k-varieties over a global field k. In some concluding remarks, we explain how properties of the topology of R are reflected in geometric properties of the fine topology, and discuss a possible application to higher local fields.