| This thesis mainly includes three parts. Part I includes Chapter 2 and Chapter 3. In this part, the degrees of the Urysohn, completely Hausdorff, completely regular, regularity, and normality axioms are introduced in (L, M)-fuzzy topological spaces. The relations among them are discussed.Part II is Chapter 4. In this part, the notions of the degrees of countable compactness and the Lindelof property in (L, M)-fuzzy topological spaces are introduced, the relations among them and the degree of compactness are studied.Part III is Chapter 5. In this part, the degrees to which a mapping is continuous, open or closed are introduced in (L, M)-fuzzy topological spaces by using implication operation, some characterizations of them are presented. Also their relations with the degrees of compactness, connectedness, separation axioms in (L, M)-fuzzy topological spaces are discussed.In the following, let me explain explicitly what I have done.Chapter 1 is the foundations for the whole paper. In this chapter, a general survey of the topic as well as some notions and results will be used in this paper are given.In Chapter 2, the Urysohn, completely Hausdorff and completely regular axioms in L-topological spaces are generalized to (L, M)-fuzzy topological spaces. Firstly, the degrees of the Urysohn, completely Hausdorff and completely regular axioms are introduced in (L, M)-fuzzy topological spaces by using (L, M)-fuzzy interior operators and (L, M)-fuzzy closure operators. Each (L, M)-fuzzy topological space can be endowed with the Urysohn, complete-ly Hausdorff and completely regular axioms to some degrees. Secondly, some properties and characterizations of them are investigated. At last, the relations among the Urysohn, com-pletely Hausdorff, completely regular, T31/2 and T2 axioms in (L, M)-fuzzy topological space are discussed.Chapter 3 is regularity and normality axioms. Firstly, the degrees of regularity and normality axioms are introduced in (L, M)-fuzzy topological spaces. Each (L, M)-fuzzy topo- logical space can be endowed with regularity and normality to some degrees. Secondly, some characterizations of them are studied by using (L, M)-fuzzy interior operators and (L, M)-fuzzy closure operators on X. Some properties of them are investigated. Thirdly, the relations among the degrees of T1, T2,T3,T4, completely regular and regularity axioms in (L, M)-fuzzy topological spaces are investigated. Finally, it is shown that the degrees of T1, T2, T3 and T4 separation axioms are equal in (L, M)-fuzzy metric spaces.Chapter 4 is the Lindelof property and countable compactness. We introduce the notions of the degrees of countable compactness and the Lindelof property in (L, M)-fuzzy topological spaces. Some properties and characterizations of them are studied. Moreover, the relations among them and the degrees of compactness are given.Chapter 5 is continuous mapping. Firstly, the degrees to which a mapping is contin-uous, open or closed are introduced in (L, M)-fuzzy topological spaces by using implication operation. Some properties and relations among them are studied. Secondly, some character-izations of them are presented by using (L, M)-fuzzy quasi-coincident neighborhood system, (L, M)-fuzzy neighborhood system, (L, M)-fuzzy interior operator and (L, M)-fuzzy closure operator. Moreover, the characterization of the degrees to which a mapping is continuous is presented by using (L,M)-fuzzy convergence structure. At last, their relations with the degrees of compactness, connectedness, T1, T2, T3, T4, the Urysohn, completely Hausdorff, completely regular, regularity and normality axioms in (L, M)-fuzzy topological spaces are discussed.In Chapter 5, conclusion remarks and expectation are made. |
