Function Characterizations Of Semi-Stratifiable Spaces And Generalized Continuity

In the past several years, the problem of the insertion of a continuous function between pairs of semi-continuous functions in a monotone manner was investigated extensively. The resolution of the problem presents some characterizations of certain spaces, such as eountably paracompact spaces, stratifiable spaces, etc. Enlightened by these theorems, we present some characterizations of semi-stratifiable spaces in this paper. Besides, generalized continuity is another problem which has been paid closed attention to. In this paper, we introduce another several forms of generalized continuity, and investigate the interrelations between them and some defined before.The paper can be divided into four parts.The first chapter, introduction, gives the origin of the problem we discuss.Chapter 2 is devoted to the characterizations of seini-stratifiable spaces. In this chapter, we give some properties of semi-continuous functions and present, some characterizations theorems of semi-stratifiable spaces with them. The main results is: A space X is semi-stratifiable if and only if there is an order-preserving map φ: LSC(X) → USC(X) such that for any h∈LSC(X). 0 ≤ φ(h) ≤ h. and 0 < φ(h)(x) < h(x) whenever h(x) > 0.In chapter 3. we introduce weakly semi-continuous mappings and weakly pre-continnous mappings to generalize weak α-continuity. Some of their properties are investigated, and we make some comparisons between certain weak forms of continuity. The main results is: A mapping f : (X.T) → (Y.U) is w.s.c.(resp. w.p.c.) if and only if f is weakly quasi-continuous (resp. almost weakly continuous).Chapter 4 deals with the decomposition of continuity. In this chapter, we present some decomposition theorems of A-continuity and continuity. The main results is: For a mapping f : (X,T) → (Y,U), the following are equivalent.(a) f is continuous.(b) f is both α-continuous and A-continuous.(c) f is both α-continuous and LC-contiuuous.(d) f is both α-continuous and C-continuous.