On CSS Spaces And K-CSS Spaces And The Union Of Some Generalized Metric Spaces

It is well known that the Metrization Theorem is one of the most important theorem in General Topology. We can reduce some conditions of the Metrization Theorem to get some generalized metric spaces. For example, we can get M₁-spaces, M₂-spaces and M₃-spaces by weakening the conditions of Nagata-Smirnov-Bing's Metrization Theorem. A regular, T₁-space X is called a M₃-space, if X has aσ-cushioned-pair-base. In 1966, Borges used stratifiable operator and g function to describe M₃-space.Stratifiable operator and g function urged scholars to use other methods to study generalized metric spaces. Scholars of General Topology have introduced some different spaces such as semi-stratifiable spaces, k-semistratifiable spaces. We will discuss CSS spaces which are similar to semi-stratifiable spaces. At the same time, these spaces have enriched the content of the theory of generalized metric spaces.In 1973, H.W.Martin introduced the class of CSS. CSS spaces are similar to semi-stratifiable spaces. X is called a CSS space, if its compact sets are uniformly G_δ-sets. In the second part of this paper, we mainly prove that if space X has a quasi-G_δ(2) diagonal, then X is a CSS space. If X is the union of countable family of closed CSS subspaces, then X is a CSS space. We also show that the countable product of CSS spaces is a CSS space.In the third part of this paper, we mainly show that if X is the union of countable family of closedβ-subspaces (semi-stratifiable spaces), then X is aβ-space (semi-stratifiable space).In the fourth part of the paper, we generalize the conception of CSS space and get another class of spaces, we denote it by k-CSS space. In this part, we discuss the basic property of k-CSS spaces. k-CSS spaces are hereditary and can be preserved by a perfect mapping. A union of two closed k-CSS spaces is a k-CSS space. In addition, we mainly give a characterization of k-CSS spaces in terms of certain g function.T₂ space X is a k-CSS space if and only if X has a g function satisfying:(b) If a sequence {x_n} of X converges to x∈X and y_n∈g(n,x_x) for each n∈N, if {y_n} has a convergent subsequence {y_nk},then {y_nk} converges to x.We prove that the countable product of k-CSS spaces is a k-CSS space by g function. If X is submesocompact locally and k-CSS space, then X is k-CSS. And we discuss the relationship between k-CSS spaces and CS spaces. We have a conclusion: if X is first countable and k-CSS, then it is a CS space.At last, we get a conclusion on regular G_δ-diagonal. We have that if X isωθandβ-space and has a regular G_δ-diagonal, then X is developable and has a strong development.