The Study On Normal-Closed,Compactness,Separation In Topology

The thesis consists of three relatively independent papers:(1)《The equivalence of the first countable normal-closed spaces》in classical topology;(2)《Almost ultra-fuzzy compactness in L-Fuzzy topological spaces》;(3)《Strong semi-separation axiom in fuzzifying topological spaces》. The primary studies in three papers are the following:(1)Let P be some topological property in classical topological spaces, A P-space X is called P-closed , if X is a closed subspace of Y , when X inserts every P-space Y. Several topologists discussed problem of P-closed equivalence and P-closed extension ,when P is the first countable T2,the first countable zero dimension,the first countable complete regularity,the first countable Urysohn,the first countable weekly regularity in the paper [3-7] ; The paper [8] studied the equivalence of the first countable semi-regular-closed spaces .On this foundation , we will discuss the same topic for the first countable normal-closed spaces,and investigate some nice conclusions.(2)Ultra-fuzzy compactness[9,10 ]is a compactness in L-Fuzzy topological spaces , the paper [11] introduced Ultra-fuzzy compactness again by the concepts of relative-R-neighborhood family . The paper [12] introduced the characterizations of almost nice compactness ,and studied topological properties . In this paper , we will introduce a almost Ultra-fuzzy compa-ctness by the concepts of relative-R-neighborhood family and almost relative-R-neighborhood family , The relationships between this compactness and Ultra-fuzzy compactness,nearly Ultra-fuzzy compactness,almost nice compactness are discussed.It indicates that almostUltra-fuzzy compacttness and other various kinds of fuzzy compactness make up more completely compactness theory systems in L-Fuzzy topological spaces. The characterizations of it are obtained , and some properties of it are discussed.(3)In classical topology,Levine introduced the concept of semi-open sets at first , and studied fundamentally theory of semi-open sets .Then Guojun Wang,Zhongqiang Yang discussed the theory of semi-open sets in the further way . Azard and Jinming Fang extended the theory of semi-open sets in I-topology and topological molecule lattices. MingshengYing introduced fuzzifying topology ,then F.H.Khedr defined degree of semi-open set A in fuzzifying topological spaces as follows: S₀(A)=∨A^°( x), A^°(x) = N_x( A) This definition contradicts with classical conclusion(such as ,the intersecttion of semi-open set and open set is semi-open set). Yue Zuo introduced a new semi-open sets by Lukasiewicz logical,defined as follows: It satisfies the classical conclusion .Guoliang Zhao introduced strong semi-open sets,strong semi-neighborhood,strong semi-closure and strong semi-interior in fuzzifying topological spaces by the new semi-open sets. In this paper,we will introduce strong semi-separation axiom by strong semi-open sets,strong semi-neighborhood and so on,and equivalences of them are given,the relations of each other are discussed.