G-Functions, Weak Base G-Functions And Applications Of Cardinal Functions

General topology is one of the fundamental branch of Mathematics with long history. Although its development was much later than some other old mathematical courses such as Analytics, Algebra, Euclidean Geometry and Number theory, after one hundred years' development, especially the rapid development from the 1950's to the 1970's, general topology is now becoming quite mature . In the research and development of general topology, the metrizable problem of a topological space is a main task. This is because that metric spaces have a lot of good properties, and they have many important applications in other field of Mathematics. The first metrization theorem was published in 1923 by Alexan-droff and Urysohn, and states that a T₀ topological space is metrizable if and only if it has a regular development. Two years later, in 1925, Urysohn proved his classical theorem that every regular space with a countable base is metrizable. After the publications of Urysohn's result the so-called "general metrization problem" was posed. This problem asks for a metrization theorem from which the Urysohn result follows as an easy and natural corollary. The general metrization problem remained unsolved until the early 1950's. at which time it was solved independently by three mathematicians, Nagata-Smirnov's characterization of metrizability in terms of aσ—locally finite base[l950],[1951]and Bing's characterization of metrizability in terms of aσ—discrete base[1951]. From then on. more and more topologists used all kinds of tools and methods to study metrization problem, such as G_δ-diagonal. point countable base(it is the generalization ofσ—locally finite base andσ—discrete base ).θ—base, g-functions and weak base g—functions defined in recent years. And they got so many metrizations results which enriched the study of metrization problem.At the same time, we should notice that only a very small part of topological spaces can be metrizable among the so many important topological spaces, therefore, it is of great importance to research the properties of generalized metric spaces which are tightly relative to metric spaces. This paper does not definite new generalized metric spaces because with the rapid development of general topology there are already so many generalized metric spaces. We want to use an existed tool in the research of generalized metric spaces—g—functions to study two very important topological spaces.In the first chapter, we use weak base g—functions to study metrization problem, and the main results include three parts . In section 1.3, we discuss the relationsheep between the two metrization theorems given by Z.M.Gao and L.X.Peng, and point out that they don't imply each other. Naturally, we propose the following question: Do there exist two theorems that generalize the metrization theorems given by Z.M.Gao and L.X.Peng respectively ? Then in section 1.4, through the study of the above question , we defined some new conditions about weak base g—functions which are weaker than some known conditions, then give an affirmative answer to the above question. At last, in section 1.5, we go on the study of metrization problem, and give a weaker condition of a metrization theorem given by A.M.Mohamad, then get a metrization theorem which generalizes A.M.Mohamad's metrization theorem.In Chapter two, we investigate the properties of the two important generalized metric spaces—σ—space and W N—space. First in Section 2.3, we use g—functions and CF-family to characterizeσ- space in the regular Frechet space, through the characterization we can see the difference betweenσ—space and Lasnev space. Then in section 1.4. we do some research on the expandability of W N—space, and get a condition when WN- space is expandable in the normal space. We have known that in q-space W N—space is equal to MCP space, so the above result is very important to resolve the question"Is MCP space expandable?" given by Chris Good in 2000.A mapping from a topological space to a cardinal set is called a cardinal function if for any topological space X, there is a cardinal f(X) such that if X is homeomorphisic to Y, then f{X) = f(Y). Cardinal functions extend some important topological properties to high cardinal condition. R. Hodel[1984] pointed out: cardinal function is one of the most efficient and important concepts in set topology.In Chapter three, we use cardinal functions to study the properties of L—fuzzy Order -preserving spaces.In a topological space, weight, character and density are very important cardinal functions, and they often reflect global and local properties of a topological space. So in this Chapter, we give the definitions ofω—weight,ω—character andω—density in L—fuzzy Order-preserving spaces and study their basic properties. Some theorems in this Chapter are generalizations of S.L.Chen's in 2004.