Effective Representations

Abstract

It cannot be expected that arbitrary representations have interesting constructive or computational properties. In this chapter we introduce some types of “effective” representations. For any topological space with T_o-topology and countable base a distinguished equivalence class of effective representations (the admissible representations) can be introduced by natural topological requirements. These requirements correspond to the axiomatic characterization of the numbering φ of the partial recursive functions by the smn-theorem and the utm-theorem (see Chapter 2.1). Final topologies which play an important role in this theory are studied in advance. Admissible representations have several natural properties; especially for admissible representations of T_o-spaces topological continuity coincides with continuity w.r.t. the representations. For any given numbering of a base of a T_o-space, we define the computationally admissible representations. For separable metric spaces the Cauchy-representations are introduced which turn out to be admissible. The concept of admissible representations is sufficiently powerful in order to define constructivity and computability in a natural way in functional analysis and measure theory.