8. Some Extensions

Abstract

In this book, we study computability concepts which are induced by notations and representations. Although every representation δ :⊆ Σ^ω → M of a set induces a computability concept, only very few of them are useful. As an important class we have studied representations constructed from computable topological spaces (Sect. 3.2). Remember that every T_o-space with countable base can be extended to a computable topological space by an injective notation of a subbase. Many important To-spaces with countable base can be generated from separable metric spaces (Definition 2.2.1). Therefore, it is useful to study computability on metric spaces separately. The reader who is not familiar with the mathematical concepts used in this section is referred to any standard textbook on real analysis, for example, Rudin [Rud64].