Computability on subsets of Euclidean space I: closed and compact subsets

Abstract

In this paper we introduce and compare computability concepts on the set of closed subsets of Euclidean space. We use the language and framework of Type 2 Theory of Effectivity (TTE) which supplies a concise language for distinguishing a variety of effectivity properties and which admits highly effective versions of classical theorems. In particular, Type 2 Theory of Effectivity allows to separate topological from computational aspects of effectivity. We consider three different computability concepts on the set of closed subsets, each of which is characterized by several representations which are proved to be equivalent. The three induced types of computable closed sets have already been considered by many authors, however, under different and partly inconsistent names. Our characterizations show that they can be regarded as straightforward generalizations of the r.e., co-r.e., and recursive subsets of natural numbers. Therefore, we suggest to call them the recursively enumerable, the co-recursively enumerable, and the recursive closed subsets of Euclidean space. Open subsets obtain the dual names. We extend the investigation by introducing several natural representations of the compact subsets of Euclidean space and proving equivalences. The paper extends and generalizes earlier definitions, adds new ones and compares them in a single framework. The resultant canonical computability concepts induce computability of objects as well as computability of operators on the space of closed and compact subsets.