Computability of Polish spaces up to homeomorphism

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 Polish space not homeomorphic to a computable one. We apply our techniques to build, for any computable ordinal, an effectively closed set not homeomorphic to any -computable Polish space; this answers a question of Nies. We also prove analogous results for compact Polish groups and locally path-connected spaces