Classes of Polish Spaces under Effective Borel Isomorphism.
- 1st ed.
- 1 online resource (102 pages)
- Memoirs of the American Mathematical Society Series ; v.240 .
- Memoirs of the American Mathematical Society Series .
Cover -- Title page -- Preface -- Chapter 1. Introduction -- 1.1. A few words about the setting -- 1.2. The basic notions -- 1.3. Theorems in the general context -- 1.4. The Cantor-Bendixson decomposition -- Chapter 2. The spaces \spat -- 2.1. Definition and properties -- 2.2. Elementary facts about the classes of \del isomorphism -- Chapter 3. Kleene spaces -- 3.1. Definition and basic properties -- 3.2. Expanding the toolbox -- 3.3. Chains and antichains in Kleene spaces under \dleq -- 3.4. Analogy with recursive pseudo-well-orderings -- 3.5. Incomparable hyperdegrees in Kleene spaces -- Chapter 4. Characterizations of \ca up to \del isomorphism -- 4.1. Copies of the complete binary tree -- 4.2. A characterization in terms of the perfect kernel -- 4.3. A measure-theoretic characterization -- 4.4. The tree of attempted embeddings -- Chapter 5. Spector-Gandy spaces -- 5.1. The Spector-Gandy Theorem -- 5.2. Application to the spaces \spat -- 5.3. Chains and antichains in Spector-Gandy spaces under \dleq -- Chapter 6. Questions and related results -- 6.1. Parametrization -- 6.2. Kleene spaces -- 6.3. Incomparable and minimal hyperdegrees -- 6.4. Connections with lattice theory -- 6.5. Attempted Embeddings and incomparability -- 6.6. Spector-Gandy spaces -- 6.7. Connections with \pii equivalence relations -- Bibliography -- Index -- Back Cover.
The author studies the equivalence classes under \Delta^1_1 isomorphism, otherwise effective Borel isomorphism, between complete separable metric spaces which admit a recursive presentation and he shows the existence of strictly increasing and strictly decreasing sequences as well as of infinite antichains under the natural notion of \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.