Effective topological spaces II: A hierarchy

This paper (which is the continuation of the preceding paper 7]) is an investigation of definability hierarchies on effective topological spaces. An open subset $\text{U}$ of an effective space X is definable iff there is a parameter free definition φ of $\text{U}$ 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 $\text{U}$ 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 $\text{U}$ 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.