Rice-Shapiro theorem in Computable Topology

We provide requirements on effectively enumerable T0–spaces which guarantee that the Rice-Shapiro theorem holds for the computable elements of these spaces. We show that the relaxation of these requirements leads to the classes of effectively enumerable T0– spaces where the the Rice-Shapiro theorem does not hold. We propose two constructions that generate effectively enumerable T0–spaces with particular properties from wn-families and computable trees without computable infinite paths. Using them we propose examples that give a flavor of this class.