Separation In Computable Topology Space

Computable analysis is a branch of computability theory studying real numbers and real functions which can be computed by digital computers. Computable analysis is based on the one hand on analysis and numerical analysis and on the other hand on computability theory and computational complexity theory. We explore the foundation of computable topology in the framework of Type-2theory of effectivity, TTE, where continuity and computability on finite and infinite sequences of symbols are defined canonically and transferred to abstract sets by means of notation and representations.For computable topology spaces from To to T4, we introduce a number of com-putable versions of the topological separation axioms, and systematically investigate the implication relationship between axioms.Chapter1makes a sketch of Type-2theory of effectivity. Chapter2presents some basic concepts and some lemmas which are necessary for latter chapters. Chapter3explores the foundation of computable topology. Chapter4investigates the implication relationship between computable separation axioms.