Applications of effective methods in computable mathematics

Abstract

This thesis is divided into 5 chapters. The first chapter gives a brief introduction to computability theory and type-2 theory of effectivity that will used in the subsequent chapters. In chapter 2, we define some new computable separations axioms and investigate some of its properties with respect to those computable axioms introduced in the literature. Chapter 3 introduces soft set theory and some new soft separation axioms are defined and some of their properties are investigated. In chapter 4, we introduce a new notion of computable soft topological spaces and effectivize the soft separation axioms introduced in chapter 3 and investigate their properties. In chapter 5, A new notion of computable categoricity for effective topological spaces is introduced where the notion of "homeomorphism" is used instead of that of "isomorphism" that used in computable structure theory. In chapter 6, the notion of degree of categoricity for strong reducibilities is introduced and studied as an analogue to the degree of categoricity in computable structure theory.