De Groot Duality for Represented Spaces.

We explore de Groot duality in the setting of represented spaces. The de Groot dual of a space is the space of closures of its singletons, with the representation inherited from the hyperspace of closed subsets. This yields an elegant duality, in particular between Hausdorff spaces and compact T 1 -spaces. As an application of the concept, we study the point degree spectrum of the dual of Baire space, and show that it is, in a formal sense, far from being countably-based.