Domain Theory Research On T₀ Space

Domain theory plays a fundamental role in the denotational semantics of pro-gramming languages.Since Domain theory has been established by Scott in the early 1970’s,having achieved fruitful achievements.It is characterized by the close con-nection and interaction between orders and topologies,which makes it the common study field of both computer experts and mathematicians,and having wide applica-tions.When Scott topology was raised,the study of Domain theory has been closely associated with T0 spaces.Especially since 2013,many scholars have studied Domain theory from T0 spaces.At present,domain theory on T0 spaces has abundant research results.At the same time,it needs to be further study in many ways.This thesis is to further investigate the theory of domain on T0 spaces.The structure of this thesis is organized as follows:Chapter One Preliminaries.This chapter reviews some preliminaries about several commonly used classes of lattice theory,topology,domain theory as well as categorical theory.Chapter Two SI-continuous spaces.Firstly,some equivalent characterizations of the primes and the co-primes of the SI-topology of a T0 space are given.Secondly,the relations between the category of SI-continuous spaces and the category of domains are studied.It is proved that the category of SI-continuous spaces and the category of domains is isomorphism-dense.Finally,an adjunction between the category of domains and the category of directed SI-continuous spaces is constructed.Chapter Three Quasicontinuous spaces.Firstly,the concept of quasicontinuous spaces is introduced,and some equivalent characterizations of irreducible complete spaces to be quasicontinuous spaces are given by using Rudin’s lemma in topological spaces.Secondly,the relations between quasicontinuous spaces and SI-continuous spaces are discussed.Finally,an adjunction between the category of quasicontinuous spaces and the category of quasicontinuous domains is constructed.Chapter Four K-bounded sober spaces.Firstly,some properties of k-bounded sober spaces are introduced.It is proved that the SI-topology space of SI-continuous space is k-bounded sober.Secondly,a counterexample is given to present that the same method of the standard sobrification of a T0 space is not true for k-bounded sobrification.Finally,the definition of k-bounded sauber spaces is given by using cut-operations and prove that the category of k-bounded sober spaces is a reflective subcategory of the category of k-bounded sauber spaces.Chapter Five Irreducibly(order)convergence in T0 spaces.Firstly,the definition of irreducible(order)convergence is introduced,and then the definition of irreducible(order)topology is obtained naturally.Some characterizations of irreducible(order)open sets are given.Secondly,the definition of irreducible(order)continuous spaces is give,and some properties of irreducible(order)continuous spaces are obtained.It is proved that every SI-continuous space is an irreducible continuous space.Finally,it is proved that a T0 space X is an irreducibly continuous space if and only if the irreducible(order)convergence is topological.Chapter Six I2-convergence in T0 spaces.Firstly,the definition of I2-convergence in T0 spaces is given by using cut-operations,and then the concept of I2-topology is also given.The relations between I2-topology and irreducible topology on X are discussed.Secondly,the concept of I2-continuous spaces is introduced.It is proved that I2-convergence to be topological in I2-continuous spaces.Finally,a class special space,called IDC-spaces,is introduced and some examples of IDC-spaces are given.It is presented that a sufficient and necessary condition for I2-convergence to be topological in IDC-spaces.