Effective topological spaces II: A hierarchy

Abstract

This paper (which is the continuation of the preceding paper [7]) 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 (such sets are said to be essentially recursively enumerable or e.r.e.). 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.