A Discussion Of The Function Space On Quasicontinuous Domain And The KF-continuous Spaces

With the rapid development of theoretical computer science and improvement of basic disciplines of mathematics,a new research field has emerged.Domain theory,an important field generated by the cross combination of theoretical computer science and some basic disciplines of mathematics,has developed vigorously and effectively.At present,Domain theory is one of the most active periods.Since the 1970s,a lot of results have been obtained in Domain theory which can be combined with many other branches of mathematics,such as category theory,logic,lattice theory,topology,algebra.But there are still a number of interesting problems about the function spaces.For example,the continuity,lawson compactness and topology properties of function space form three of the most important and heavily studied problems.Quasicontinuous domain is not only an important domain structure,but also a reasonable generalization of continuous domain.Its main idea is to extend the approximation relation between points to the sets.It has the characteristics similar to the continuous domain,and the results have generalized the important conclusions of continuous domain to quasicontinuous domain and obtained a series of new results.In particular,it is proved that the function space constructed by the quasicontinuous domains is not quasicontinuous domain.So naturally,the quasicontinuity of function space is worth studying.The main work of the initial part of this thesis is to solve the above mentioned problems.We obtain a series of results on the structure and topology aspect of the function space as follows.In Chapter 3,we mainly prove that:(?)Let P be a bounded complete quasicontinuous domain(quasicontinuous lattice)and X a SI-continuous space.Then the function space[X?P]is a quasicontinuous domain(quasicontinuous lattice)with respect to the pointwise order.(?)Let P be a bounded complete quasicontinuous domain(quasicontinuous lattice)and X a SI-continuous space.Then the Scott topology and Isbell topology agree on the function space[X?P].i.e.Is[X?P]=?[X?P].It is well known that every bounded complete dcpo or complete lattice has the property M*.In Chapter 4,we generalize the range part to dcpo with the property M*and come to some results:(?)let P be a quasicontinuous domain with property M*and X a bounded complete domain.Then the function space[X?P]is a quasicontinuous domain with respect to the pointwise order.And a counterexample is given to illustrate that the property M*is a necessary property.(?)Let P be a quasialgebraic domain with property M*and X a bounded complete algebraic domain.Then the function space[X?P]is a quasialgebraic domain with respect to the pointwise order.By the above conclusions,we have known that the algebraicity is necessary in the domain.Furthermore,we also give a counterexample to illustrate the influence of the algebraicity in the domain of the quasialgebraicity of the function space.(?)Let P be a quasicontinuous domain with property M*and X a bounded complete domain.Then the Scott topology and Isbell topology agree on the function space[X ? P].i.e.Is[X?P]=?[X ?P].Directed sets and irreducible sets are two important concepts in Domain theory.As we all know,every directed set is irreducible.A non-empty set is irreducible with respect to the Alexandroff topology if and only if it is a directed set.Continuous domains and continuous posets are defined by directed sets.By replacing the directed set,the irreducible-derived topology(SI-topology)is defined by means of the irreducible sets.SI-topology is a generalization of the notion of Scott topology in T0-space.Base on the SI-topology,Zhao propose and study SI-continuous spaces.An important conclusion that a C-space is an SI-continuous space if and only if it is a continuous poset under the specialization order is proposed.In classical Domain theory,the sober spaces,well-filtered spaces and d-spaces form three of the most important classes of topological spaces.Every sober space is a well-filtered spaces and every well-filtered space is a d-spaces.Sober spaces(resp.d-spaces)is characterized by the irreducible sets(resp.directed set).But well-filtered spaces has been less studied and characterized.In recent years,in order to study and characterize well-filtered spaces,some scholars obtain a characterization for the well-filtered spaces using KF-set,which is between directed sets and irreducible sets.In Chapter 5 of the second part of this paper,we propose and investigate KFcontinuous spaces and KF-quasicontinuous spaces by KF-set.Some topological properties are studied in KF-continuous spaces and KF-quasicontinuous spaces.What is more,we conclude that a topological space is a KF-meet continuous and quasicontinuous space if and only if it is KF-continuous.The convergence theory is one of the most important theories in Domain theory.In Chapter 6,we study the convergence of net based on KF-topology in the T0-space,and give the necessary and sufficient conditions for the KF-convergence to be topological.For some T0-space X with condition m*,we prove that the KF-convergence is a topological convergence for the KF-topology if and only if X is a K-continuous space.