Effective topological spaces. II: A hierarchy
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.
scientific article; zbMATH DE number 3943799
scientific article