About Non-regular N-Spaces And Characterization Of Metrizable Spaces In Terms Of G-function

The class of generalized metric spaces is generalization of the class of metric spaces. It is very helpful for characterizing the metrizability by studying generalized metric spaces. By weakening the conditions of metrizable theroms, some new types of spaces are obtained. These kinds of spaces include σ-spaces and N-spaces and Lasnev-spaces etc.. Compared with a base of a space, a network and a k-network possess more delicate and more variable structures. One aim of studying generalized spaces is to investigate the interior structures of these spaces. However, when we study general topological spaces, different axioms of separation are used. The weaker the axiom of separation is, the wider the application of the theorem is. In this thesis, We devote to the research of weakening axiom of separation on some special spaces.In this thesis, one of the main aims is to investigate the characterization of N-space in different conditions. By weakening the regular condition, N-space is redefined. We call it non-regular N-space. Firstly, we get the characterization of a non-regular N-space with a g-function. Secondly, a necessary and sufficient condition for a Frechet non-regular N-space is given. In the last part of the thesis, we obtained a new characterization of metric space in terms of g-function. The main results are illustrated as follows:Theorem I A Frechet space X is non-regular N-space iff it has a network F_n such that each F_n is a locally finite closed cover of X, and such thatfor each xGX and Fn with x G Fn G Tn for all n G N,{Fn\n G iV} is either a network at x or hep.Theorem 2 Let X be a metric space, then X has a g-function g satisfying: (ks) xn —> x and xn G g(n,yn) for all n G iV imply yn —> x; (h) {p(n, a:)|a: G X} is hep for each n.