Inner Derivation Set And Logic Metric

In the Point Set Topology, derived set is an important concept of topological spaces, its fundamental properties and relationships between it and other concepts are discussed in detail. In general, there are two ways to define derived set,one of which is defined by closure, through the deep research on basic concepts of Point and Set topology and relationships among them, In this paper, the definition of internal derived set is first introducedby using the thought and method of dual category, its fundamental properties and relationships between it and other concepts are studied.Derived set not only provides a way to determine topology, but it also provides a new way to do research on the properties of topological spaces;Based on the concept of internal derived set, the concept of correlative subset is proposed. As the application of internal derived set and correlative subset, several equivalent characterizations of disconnected space are given. Otherwise, through the deep research on the operations "(?),⊥" defined on FI algebras, the definition of DFI algebra is proposed, it is proved that DFI algebra is regular FI algebra, BL algebra, and their converse propositions aren't true through examples, and some conditions when a FI Algebracome to be a Boole Algebra are discussed, then the logical metric structure on Boole algebra is described; Then the logical metricsρL,ρ0,ρG,ρπon MV unit interval, R₀ unit interval, standard G algebra, standard product algebra are established respectively, and ([0,1],ρL), ([0,1],ρ0), ([0,1],ρG), ([0,1],ρπ) become logical metric spaces, in this paper, it is proved thatρL is the common metric, and the properties of the other three metrics are discussed, and at last the specific structures of them are discussed by using the knowledge of topology, and some good results are gotten.The main points of this paper is as follows:1.Preparatory Kownledge.In the first and second sections, some fundamental concepts and conclusions of Point Set Topology and Logical Algebra are mainly introduced ;In the third section, some basic properties of derived set and the way to determine topology are discribed.Otherwise, the generalized isolated point set is proposed based on isolated point set, which provides another way to determines topology.2.Internal derived set and its applications.Through the deep research on basic concepts of Point and Set topology and relationships among them, the concept of internal derived set is introduced in Point Set topology, by using the thought and method of dual category. Based on the concept of internal derived set, the concept of correlative subset is proposed, and its fundamental properties and relationships between it and other concepts are studied, and get some good results. At last, as the application of internal derived set and correlative subset, several equivalent characterizations of disconnected space are given.3.The logical metric structure on logical algebras.through the deep research on the operations " (?),⊥" defined on FI algebras, the definition of DFI algebra is proposed, its properties are discussed, and some equivalent forms of Boole algebra are given, then the logical metric structures on Boole algebra and [0, 1] are discussed.Then the logical metricsρL,ρ0,ρG,ρπon MV unit interval, R₀ unit interval, standard G algebra, standard productalgebra are established respectively, and ([0,1],ρL), ([0,1],ρ0), ([0,1],ρG), ([0,1],ρπ) become logical metric spaces, and the properties of the other three metrics are discussed, and at last the specific structures of them are discussed by using the knowledge of topology,and some good results are gotten.