Classes of Polish spaces under effective Borel isomorphism

We study the equivalence classes under Δ 1 1 \Delta ^1_1 isomorphism, otherwise effective Borel isomorphism, between complete separable metric spaces which admit a recursive presentation and we show the existence of strictly increasing and strictly decreasing sequences as well as of infinite antichains under the natural notion of Δ 1 1 \Delta ^1_1 -reduction, as opposed to the non-effective case, where only two such classes exist, the one of the Baire space and the one of the naturals. A key tool for our study is a mapping T ↦ N T T \mapsto \mathcal {N}^{T} from the space of all trees on the naturals to the class of Polish spaces, for which every recursively presented space is Δ 1 1 \Delta ^1_1 -isomorphic to some N T \mathcal {N}^{T} for a recursive T T , so that the preceding spaces are representatives for the classes of Δ 1 1 \Delta ^1_1 isomorphism. We isolate two large categories of spaces of the type N T \mathcal {N}^{T} , the Kleene spaces and the Spector-Gandy spaces and we study them extensively. Moreover we give results about hyperdegrees in the latter spaces and characterizations of the Baire space up to Δ 1 1 \Delta ^1_1 isomorphism.