Effective topological spaces II: A hierarchy

Annals of Pure and Applied Logic 29 (2):207-224 (1985)
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Abstract

This paper is an investigation of definability hierarchies on effective topological spaces. An open subset U of an effective space X is definable iff there is a parameter free definition φ of U so that the atomic predicate symbols of φ are recursively open relations on X . The complexity of a definable open set may be identified with the quantifier complexity of its definition. For example, a set U is an ∃∃∀∃-set if it has an ∃∃∀∃ parameter free definition using only recursively open predicate symbols. Since X is not equipped with a natural pairing apparatus such a U need not be an ∃∀∃-set. Let Σ denote the class of all ∃-sets, ∃∃-sets, ∃∃∃-sets etc. We show that an open set is in Σ iff it is equivalent modulo a nowhere dense set to a recursively enumerable open set . Thus Σ = e.r.e. Indeed we show the existence of a universal Σ -set as well as the existence of universal sets for higher levels of the definability hierarchy

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Citations of this work

Effective topological spaces I: A definability theory.Iraj Kalantari & Galen Weitkamp - 1985 - Annals of Pure and Applied Logic 29 (1):1-27.
Effective inseparability in a topological setting.Dieter Spreen - 1996 - Annals of Pure and Applied Logic 80 (3):257-275.

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References found in this work

Recursive predicates and quantifiers.S. C. Kleene - 1943 - Transactions of the American Mathematical Society 53:41-73.
Arithmetical Predicates and Function Quantifiers.S. C. Kleene - 1956 - Journal of Symbolic Logic 21 (4):409-410.
Hierarchies of Number-Theoretic Predicates.S. C. Kleene - 1956 - Journal of Symbolic Logic 21 (4):411-412.
Recursive constructions in topological spaces.Iraj Kalantari & Allen Retzlaff - 1979 - Journal of Symbolic Logic 44 (4):609-625.
Recursive Functionals and Quantifiers of Finite Types II.S. C. Kleene - 1971 - Journal of Symbolic Logic 36 (1):146-146.

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