A characterization of effective topological spaces

Starting with D. Scott's work on the mathematical foundations of programming language semantics, interest in topology has grown up in theoretical computer science, under the slogan "open sets are semidecidable properties". But whereas on Scott domains all such properties are also open, this is no longer true in general. In this paper we present a characterization of effectively given topological spaces that says which semidecidable sets are open. We consider countable topological To-spaces that satisfy certain additional topological and computational requirements which can be verified for a general class of Scott domains and metric spaces, and we show that the given topology is the recursively finest topology generated by semidecidable sets which is compatible with it. From this general result we derive the above mentioned theorem about the correspondence of the semidecidable properties with the Scott open sets. This theorem, in its turn, is a generalization of the Rice/Shapiro theorem on index sets of classes of recursively enumerable sets. Moreover, characterizations of the canonical topology of a recursively separable recursive metric space are derived. It is shown that it is the recursively finest effective T3-topology that can be generated by semidecidable sets the topological complement of which is also semidecidable.