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. (shrink)

The axiom of symmetry (A ℵ 0 ) asserts that for every function F: ω 2 → ω 2 there is a pair of reals x and y in ω 2 so that y is not in the countable set $\{(F(x))_n:n coded by F(x) and x is not in the set coded by F(y). A(Γ) denotes axiom A ℵ 0 with the restriction that graph(F) belongs to the pointclass Γ. In § 2 we prove A(Σ 1 1 ). In § (...) 3 we show A(Π 1 1 ), A(Σ 1 2 ) and $^\omega 2 \nsubseteq L$ are equivalent. In § 4 several effective versions of A(REC) are examined. (shrink)