Limit Theory And Weight Metric Space On Partial Order Set

Topology is a major area of mathematics concerned with the most basic prop-erties of space, such as connectedness, separation. More precisely, topology studiesproperties that are preserved under continuous deformations. There are many sub-fields in topology, such as Point-set topology(general topology), Algebraic topol-ogy, Diferential topology and Geometric topology. And together as a basis forall disciplines closely with many areas of mathematics. topology has graduallypenetrated into almost all modern branch of mathematics, and can be applied tophysics, chemistry, biology and economics, etc. In this article, we only concern thegeneral topology.Domain theory is another area, which studies special kinds of posets. Con-sequently, domain theory can be considered as a branch of order theory. Domaintheory has major applications in some area. Such as computer science, where it isconsider as the denotation semantics of functional programming languages. Theintuitive ideas of convergence in domain theory are closely relative to topology.In this thesis, we consider three topics of Domain theory, i.e. the topological ofconvergence classes, powerdomain, metric space.This thesis consists of the following three parts.The first part includes Chapter3and Chapter4. The topological of conver-gence classes are considered in this part. At first, we consider the convergence ofnets in quasi-continuous domain, i.e. the S-convergence and quasi-liminf conver-gence of a net. Using it, we describe the quasi-continuous domain. The followingresults are obtained:1. For a dcpo L, the S-convergence is topological if and only if L is quasi-continuous and the topology generating by it coincides with the Scott topology.2. Under certain assumption, the quasi-liminf convergence is topological if andonly if the dcpo is quasi-continuous and the topology generating by it coincideswith the Lawson topology.In what follows, the first part we generalize these conclusions to poset and givethe definition of liminf convergence in poset. We also present the necessary andsufcient condition for a poset to be continuous. Finally we give the definitionsof liminf of convergence of Z-continuous posets and discussed the relation betweenthe topology generating by it and the Z-Lawson topology.The second part includes Chapter5, we discussed possibility measures and some basic properties of possibility powerdomain, and we compare the possibilitymeasures combined with domain theory. In particular, we give a new integral ofcontinuous functions with possibility measure.The third part include Chapter6, combining with Domain theory, we gener-alizes the concept of metric spaces, and define weight metric space. The represen-tation theorem of weight metric spaces is given.