COMPUTABILITY OF POLISH SPACES UP TO HOMEOMORPHISM
Matthew Harrison‐Trainor, Alexander Melnikov, Keng Meng Ng · Journal of Symbolic Logic · 2020
Abstract We study computable Polish spaces and Polish groups up to homeomorphism. We prove a natural effective analogy of Stone duality, and we also develop an effective definability technique which works up to homeomorphism. As an application, we show that there is a $\Delta ^0_2$ Polish space not homeomorphic to a computable one. We apply our techniques to build, for any computable ordinal $\alpha $ , an effectively closed set not homeomorphic to any $0^{(\alpha )}$ -computable Polish space; this answers a question of Nies. We also prove analogous results for compact Polish groups and locally path-connected spaces.