Computable Real-Valued Functions on Recursive Open and Closed Subsets of Euclidean Space

In this paper we study intrinsic notions of “computability” for open and closed subsets of Euclidean space. Here we combine together the two concepts, computability on abstract metric spaces and computability for continuous functions, and delineate the basic properties of computable open and closed sets. The paper concludes with a comprehensive examination of the Effective Riemann Mapping Theorem and related questions.