On the Lattices of Effectively Open Sets

The lattice E of c.e. sets is a popular object of study in Computability Theory [So87]. A principal fact about this lattice is the undecidability of its first-order theory Th(E). Moreover, Th(E) is known [HN98] to be m-equivalent to the first-order arithmetic Th(N) where N := (ω; +,×). The lattice E may be considered as the lattice of effectively open subsets of the discrete topological space ω. The lattices of effectively open sets Σ1(X) of various “effective” topological spaces X are of interest in Computable Analysis [We00] and Effective Descriptive Set Theory, hence it is natural to ask which results about E hold true for the lattices of effectively open sets of “natural” effective topological spaces. Here we attempt to make apparently first steps in this direction.