1. What are the computable functions on topological algebras?

2. What methods exist to axiomatically specify functions on topological algebras?

3. Can all computable functions be specified?

Such a theory seems to be in its infancy: there are many approaches to computability theory on general and specific spaces, and few approaches to specification theory. In some earlier papers, we have studied the questions 1 and 2 with the needs of data type theory in mind, and built a bridge between computations and specifications to try to answer 3. In this paper, we extend and combine several of our results, to prove new theorems that

(i) show the equivalence of some six deterministic or nondeterministic models of computation on various metric algebras and, in particular, on spaces R of real numbers;

(ii) provide finite

(iii) show the existence of finite universal algebraic specifications of computably approximable finite dimensional deterministic dynamical systems.

A technical issue is the localisation of uniform continuity using exhaustions of open sets. We use specifications composed of conditional equations, inequalities and, for convenience, new exhaustion primitives, that define functions uniquely up to isomorphism.