On WM Spaces And Weak-stratifiable Spaces

In this paper, we study the theory of g-functions which is one of the most important parts of the theory of generalized metric spaces. We discuss g-function characterizations of some generalized metric spaces and some related problems. The paper contains three parts.Chapter1is devoted to the introduction of the formation and development of the theory of g-functions and the research background of this paper.In chapter2, we present characterizations of wM spaces in terms of g-functions. The main result is as follows:Theorem1For a spaces X, the following are equivalent:(1) Xis a wM space;(2) There is a g-function g for X such that if〈yn〉has a cluster point and g (n, xn)∩g(n, yn)≠Φ for all n∈N,then 〈xn〉 has a cluster point;(3) There is a g-function g for X such that if yn∈g (n, p) and g (n, xn)∩g(n, yn)≠Φ for all n∈N, then 〈xn〉 has a cluster point;(4) There is a g-function g for X such that if g(n,p)∩g(n,yn)≠Φ and g(n,xn)∩g (n,yn)≠Φ for all n∈N,then(xn) has a cluster point;(5) There is a g-function g for X such that if g(n,p)∩g(n,yn)≠Φ and xn∈g(n,yn) for all n∈N,then (xn) has a cluster point;(6) There is a g-function g for X such that if p∈g(n,zn),g(n,zn)∩g(n,yn)≠Φ and xn∈g(n,yn) for all n∈N,then (xn) has a cluster point.Chapter3is devoted to the study of weak weak stratifiable spaces and w-MCP spaces which were introduced in [23] by Peng. We show that a space with a countable closed k-network need not be a weak strateftable space which partially answers a question posed by Lin in [26]. Moreover, we present a sufficient condition for a topological space to be a weak stratifiable space.The main results are as follows:Example2There exists a non-weak-stratifiable space which has a countable close k-nerwork.Proposition3A space X that has a σ-cushioned pair sk-network is a weak stratifiable space.