Highlights of the Rice-Shapiro Theorem in Computable Topology

Computable topological spaces naturally arise in computer science for continuous data type representations that have tools for effective reasoning about quite complex objects such as real numbers and functions, solutions of differential equations, functionals and operators. Algebraic and continuous domains, computable metric spaces, computable Polish spaces have been successfully used in the theoretical foundation of computer science. In this paper we consider generalisations of the famous Rice-Shapiro theorem in the framework of effectively enumerable topological spaces that contain the weakly-effective \(\omega \)–continuous domains and computable metric spaces as proper subclasses. We start with the classical case when the spaces admit principal computable numberings of computable elements and one can investigate arithmetical complexity of index sets. We provide requirements on effectively enumerable topological spaces which guarantee that the Rice-Shapiro theorem holds for the computable elements of these spaces. It turns out that if we relax these requirements then the Rice-Shapiro theorem does not hold. Then we discuss the perspective of extensions of the Rice-Shapiro theorem to spaces that do not have computable numberings of computable elements, in particular to computable Polish spaces.