Effective topological spaces. II: A hierarchy
type: Item
instance of: scholarly article
title: Effective topological spaces. II: A hierarchy
zbMATH Open document ID: 0588.03028
DOI: 10.1016/0168-0072(84)90025-3
author: Iraj Kalantari
author: Galen Weitkamp
published in: Annals of Pure and Applied Logic
publication date: 1985-01-01T00:00:00Z
review text: The work (which is the continuation of Part I [ibid. 29, 1-27 (1985; Zbl 0569.03018)]) is a study of definability hierarchies on effective topological spaces. Formulas concerning effective topological spaces are investigated with respect to the quantifier complexity of their prenex form. For example, a set U is an \(\exists \exists \forall \exists\)-set if it has an \(\exists \exists \forall \exists\) definition using only recursively open predicate symbols. If \(\Sigma\) is the class of all \(\exists\)-sets, \(\exists \exists\)-sets, \(\exists \exists \exists\)-sets etc., it is shown that \(\Sigma =e.r.e\). where e.r.e. is the class of essentially recursively enumerable sets, i.e. of those sets which are equivalent modulo a nowhere dense set to a recursively enumerable open set. The existence of a universal \(\Sigma\)-set is proven as well as the existence of universal sets for higher levels of the definability hierarchy.
Mathematics Subject Classification ID: 03D45
zbMATH DE Number: 3943799
zbMATH Keywords: definability hierarchies
zbMATH Keywords: quantifier complexity
zbMATH Keywords: essentially recursively enumerable sets
zbMATH Keywords: universal sets
reviewed by: Marius Zimand
MaRDI profile type: Publication
full work available at URL: 0168-0072(84)90025-3
OpenAlex ID: W2065683845
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description: scientific article; zbMATH DE number 3943799