Effectivity on Subsets and Continuous Functions in Computable T0-spaces

We investigate aspects of effectivity and computability on open, closed and quasi-compact sets as well as partial continuous functions in computable T0spaces. A computable T0-space is a second countable T0-spaces with a notation of a base whose domain is recursive and computable intersection on the base. As we do not suppose our space to be a Hausdorff space, we don’t talk about compact subsets. A set is called quasi-compact, if every open cover has a finite subcover. We use the framework of the representation approach, TTE, where continuity and computability on finite and infinite sequences of symbols are defined canonically and transferred to abstract sets by means of notations and representations. Computable Analysis connects Computability/Computational Complexity with Analysis/Numerical Computation by combining concepts of approximation and of computation. During the last 70 years various mutually non-equivalent models of real number computation have been proposed (Chap. 9 in [8]). Among these models the representation approach (Type-2 Theory of Effectivity, TTE) proposed by Grzegorczyk and Lacombe [5, 6] seems to be particularly realistic, flexible and expressive. So far the study of computability on sets of points, sets (open, closed, compact) and continuous functions has developed mainly bottomup, i.e., from the real numbers to Euclidean space and metric spaces [11, 2, 9, 8, 12, 1, 13]. But often generalizations to more general spaces are needed (locally compact Hausdorff spaces [3], non-metrizable spaces [10], second countable T0spaces [7, 4]). This work is a generalization of the concepts introduced in [8] for the Euclidean case. Whenever reasonable, we transfer a representation to computable T0-spaces and discuss its properties and their relations to each other. We use the concept of multi-valued partial functions. For a partial multifunction f : ⊆ A ⇒ B, f(a) is interpreted as the set of all results which are “acceptable” on input a ∈ A. Any concrete computation will produce on input a ∈ dom(f) some element b ∈ f(a), but usually there is no method to select a specific one. For a representation δ : ⊆ Σ → M , if δ(p) = x then the point x ∈ M can be identified by the “name” p ∈ Σ. We will have applications where a sequence p ∈ Σ contains information about a point x which is sufficient for some computation, although p does not identify x. We arrive at the concept of multi-representation δ : ⊆Σ ⇒ M . A multi-representation can be considered as a naming system for the points of a setM where each name can encode many