A Serre‐Swan Theorem for Bundles of Topological Modules

Let R be a unital topological ring whose set of invertible elements is open and inversion is continuous, and let X be a compact Hausdorff space admitting continuous R ‐valued partitions of unity. Considering bundles over X of fibre type a projective finitely generated R ‐module, we prove a Serre‐Swan type theorem: namely, the category of these bundles is equivalent to the category of projective finitely generated modules over the ring of continuous R ‐valued functions on X .