A constructive and functorial embedding of locally compact metric spaces into locales

The paper establishes, within constructive mathematics, a full and faithful functor M from the category of locally compact complete metric spaces and continuous functions into the category of formal topologies (or equivalently locales). The functor preserves finite products, and moreover satisfies f ≤ g if, and only if, M ( f ) ≤ M (g) for continuous f ,g : X → R. This makes it possible to transfer results between Bishop’s constructive theory of metric spaces and constructive locale theory.