Effective operators and continuity revisited

Abstract

In programming language semantics different kinds of semantical domains are used, among them Scott domains and metric spaces. D. Scott raised the problem of finding a suitable class of spaces which should include Scott domains and metric spaces such that effective mappings between these spaces are continuous. It is well known that between spaces like effectively given Scott domains or constructive metric spaces such operators are effectively continuous and vice versa. But, as an example of Friedberg shows, effective mappings from metric spaces into Scott domains are not continuous in general.

In a joint paper P. Young and the author presented a condition which under fairly general effectivity assumptions forces effective mappings between separable countable topological T _o-spaces to be effectively continuous. In this paper the condition is weakened. Moreover, a large class of separable countable T _o-spaces is given, and it is proved that a mapping between spaces of the class is effectively continuous, iff it is effective and satisfies the condition. A modification of Friedberg's example shows that the result is false without the extra condition.

Among others the class of spaces contains all recursively separable recursive metric spaces in which one can effectively pass from convergent normed recursive Cauchy sequences to their limits and all Scott domains that can be obtained via product and function space constructions from flat domains with at least three elements. The topology of the spaces in this class is effectively equivalent to the topology generated by those elements in the distributive lattice of all completely enumerable subsets of the space which possess a pseudocomplement and are regular with respect to this operation.