The Limit Operators And Fuzzy Co-topologies

Limit operator is introduced as the abstract of usual limit, it is the extend of usual limit. We can get the co-topology of limit operator in the set X through the closure operator of a set in the power set of the set X by the limit operator of the set X. If the result affected by the limit operator in any sequences. of a set X is unique,the co-topology is a usually co-topology; else it is not.The followings are the construction and the main contents of this paper:In Chapter 1, we introduce the two spaces:φ~*-space and L~*-space, closure operator and other operators (interior, boundary, derive operator). Firstly, we study some properties of closure operator, give the relation between usual co-topology and theφ~*-space (L~*-space). Secondly, we study some properties of the other operators and the relations of each of them with closure operator, and give the conclusion which is they are equivalent.Thirdly we study the ordered relation of the set composed by limit operators in X, and naturally we define the partially ordered set and the operations in it. Finally we study the relations of all kinds of operators, and give implications among them. And we also prove implications if they exist, give counterexamples or explains if they do not exist.In Chapter 2, we study closed subspace and product space inφ~*- space. Firstly, we define restriction of limit operator, get closed subspace and give some properties in it; Finally we study some properties of product space, get closure, interior and boundary of it, give the relation among co-topologies of two spaces and their product space.In Chapter 3, we study the continuous mapping, closed mapping, open mapping. Firstly we define continuous mapping in X, give the equivalent conditions, gluing theorem, homeomorphic mapping. Secondly, we define closed mapping in X, give the equivalent conditions, discuss relations with continuous mapping. Finally, we study open mapping, give the equivalent conditions, discuss relations with continuous mapping, closed mapping.In Chapter 4, we study the connected property and separated property. Firstly, we define connection of X in space, prove some properties of it.Secondly we study connected branches. Finally, we define separation, give the conclusions about Frechet co-topology.In Chapter 5, we study limit operators in fuzzy set. Firstly, we define L~*-space, a closure operator and a Frechet L-co-topology can be induced by a limit operator in -spaces. Furthermore, the concept of is defined on and the relations between Frechet L-co-topologies and sequential L-co-topologies are discussed. An equivalent condition for a sequential L-co-topology to be a Frechet L-co-topology is obtained. Secondly, we study some properties of the other operators and the relations of each of them with closure operator, and give the conclusion which is they are equivalent. Thirdly, we study the ordered relation of the set composed by limit operators in X, and naturally we define the partially ordered set and the operations in it. Finally we study the relations of all kinds of operators, and give implications among them. And we also prove implications if they exist, give counterexamples or explains if they do not exist.