The Nature And Application Of Open Remote Domain In Topological System

Steven Vickers has proposed the topological system by synthesizing the methods of topology and the theory of logics,which is a new topological research object.The research results of this theory for one thing enrich the content of topology,for another provide mathematical foundation for computer science.Professor Wang Guojun proposed the concept of open remote neighborhoods in topological molecular lattices by using closed elements.In point set topological space,for a point,its open remote neighborhood is a subset of closed sets without that point.Based on this idea and method,in this paper,we propose the concept of open remote neighborhoods in topological system by using open element in frame and study its properties.At the same time,the concept of open remote neighborhood system have been proposed through open remote neighborhoods in topological system,and its structure and properties have been studied.Secondly,this paper presents the equivalent characterization of continuous mapping and separatability based on the open remote system.Finally,the structure and some basic properties of meet and join of open remote neighborhood of points are discussed.It is proved that the set of all meet or join of open remote neighborhood systems with the defined order is a complete lattice.On the basis of above consideration,the main contents of this paper are as follows:Chapter 1:Preliminaries.In this chapter,we recapitulate the basic concepts and relevant conclusions of poset,lattice,topological space and topological system which will be used throughout this thesis.Chapter 2:Open remote neighborhood,open remote neighborhood system and its basic properties in topological systems.In this chapter,firstly,we propose the concept of open remote neighborhood,and the concept of open remote neighborhood system is given from open remote neighborhood;At second,the structure and basic properties of open remote neighborhood system are also studied;At last,the method of determining topological system is given by open remote neighborhood systems.Chapter 3:Equivalent characterization of continuous mapping and separation.This chapter gives the equivalent description of continuous mapping between topo-logical systems based on the open remote neighborhood system.And the equivalent description of To topological systems and T1 topological systems is given based on the open remote neighborhood system.Chapter 4:The structure and properties of meet and join of open remote neighborhood systems in topological systems.In this chapter,the structure and some basic properties of meet and join of open remote neighborhood of point are discussed.It is proved that the set of all join J1（D）or meet 32（D）of open remote neighborhood systems with the defined order is a complete lattice.And the concrete forms of meet and join in J1（D）and J₂（D）have been given.