In number theory, an adele is a kind of product of elements of the completion at each prime. For function fields, we take (a kind of) product of the completion at each point, and at non-singular points, this completion is a ring of formal power series. The adeles are intuitively defined so that we have an injective homomorphism from the ring of functions on a given open set (Zariski open for a number field) to the adeles for that open set.

My question is, what if we replace formal power series by *convergent* Laurent series (i.e. in the complex topology)? This is what I might call a holomorphic adele. There is a natural injective map from the ring of meromorphic functions on an open set to the ring of adeles on that set. We can also put a topology on these "adeles" similar to how it's normally done. Is this construction ever consisdered? Might it be useful? Is there a connection between the adelic topology and the complex topology?