Point Countable Cover Their Mapping Of The Nature Of Research

General topology has gone through over one hundred years'development since it was initiated to become a independent science branch by Poincare from the end of the 19th century to now. Although its independence and development were late more relative to some other antique mathematical course such as Analytics, Algebra, Euclidean Geometry and Number theory, through over one hundred years, especially the vivid development from the 1940s to the 1970s , general topology are getting increasingly mature and perfect. To now, the theories, fruits and methods of topology have already applied or seeped into almost every important field of mathematics even into physics, chemistry,bio-logy and engineering. In the research and development of general topology the metriz-able problem of the topological spaces was a central task interminally,this is because that metric spaces have a lot of good proverties,and they have important application in the field of math. However, there are only extremely tiny part can be metriz-able in many important topological spaces, therefore, it will have important significance to research generalized metric spaces tightly relative to metric spaces. In the research of the theory of generalized metric spaces, the method of mapping and covering is a very important tool, especially the point-countable covers play a more important role. For example, the concept of paracompactness was introduced by way of using a class of special point-countable cover by the famous French mathematician Dieudonne in 1944. Fairly important covers have point-countable bases, weakly bases, k-networks, sequence-neighbourhood networks and so on. Fairly important mappings have quotient s-mappings, closed s-mappings,open mappings, open-and-closed mappings, com-pact-and-open mappings, perfect mappings, countable bi-quotient mappings, compact mappings. Therefore, it will have important meaning to research the relations among some covers, the relations among some mappings as well as the proverty of the covers under mappings.The paper don't attempt to definite new generalized metric space classes and new covers and mappings. This is because in the development of revent several decades in topology,the space classes were definited by all sorts of formal generalizations have reached a flooded extent, continual introduction of new spaces and over tiny division have made topology develop to an empty theorical margin. The paper tyies to use topological spaces and theories than have existed to discuss some important problems in topology and set theory,and resolved a open problem that was proposed in document[l], this is also the most wonderful part of the paper. At the same time,it is relatively colorfully that this paper makes use of some proverties of topological spaces to discuss the problems about countable ordinal exponentiation arithmetic. Besides, this paper gave out plentiful examples of topological spaces , thereby made its theories be not empty and nothing. Notice that all these topological spaces in the discussion of the paper satisfy the axiom of T2 separation.The chapter 2 of thid paper mainly include two-aspect contents:semi-stratifiable spaces and point-countable covers. In 1970, Creede introduced an important class of generalized metric space - semi-stratifiable space. In this chapter, we gave out an equivalent description of semi-stratifiable spaces:Topological space (X,T) is semi-stratifiable if and only if exists a function g : N x X -> T satisfying the two conditions as follows:(2)if x e A', {xn} C X satisfying n e N, x g(n,xn),then xn -> x. As g-functions have important effect in the stratifiable spaces and the semi-stratifiable spaces, we proved the iquivalence of two conditions about g-functions:Let g be a g-function of space(X,T),then the two conditions as follows are equiva-IVlent:(l)if y e X \ H € 7~,then it exists an m e N satisfying y 0 g(m, H) - \J g(m, x).(2)For a point x of X and sequence {xn} C X,if x 6 g(n, a;n),then xn -> x.The main results about point-countable cove...