Insertion Property Of Continuous Map On Chains And Lattices

A map f : X - R from a topological space X into the real numbers space R is called lower (upper] semicontinuous if for every real number r R, the set {x X : f(x) > r}(the set {x 6 X : f(x) < r}) is open. Using the two concepts, many topological properties are characterized. Dieudonne -Hahn-Tong Theorem-Insertion Theorem is one of most well-known result. This theorem states that a space X is normal if and only if for every upper semicontinuous map g : X - R and every lower semicontinuous map h : X - R with f g, there exists a continuous map / : X - R such that g f h. Thus, the normality of space is conpletely characterized by lower semicontinuous function and upper semicontinous function. This theorem have been generalized in many ways. There are mostly two aspects in them, One is replacing the space R in insertion theorem by completely distributive lattices with some countability, i.e., Liu and Luo in [6] try to use lower(upper) semicontinuous map well defined to define the normality of fuzzy topological space and gain a series of good results[6]. However the other is abandoning the countability and strengthening the "order",i.e., replacing the space R in insertion theorem by complete chains( connected complete chains) and gain a series of good results also. Recently the possibility of the monotonization of insertion theorem has been investigated, which requires the inserted continuous funtion to increase if the two semicontinuous funtions increase. In fact, it turns out that the monotone versions also characterize significant topological properties. In this paper, we will discuss the changed form and monotonization of insertion theorem after it has been gemeralized on chains and lattices. The paper consists of four chapters. The results of study on the generalization of insertion theorem and its changed form on chains and lattices have been introduced in the first chapter. Furthermore, the work which will do has been listed. In the second chapter, the lattice-valued fuzzy sets have been introduced simply, furthermore, the vertion of (monotone) insrtion theorem on lattice in fuzzy topological space have been shown. Thestrong insertion theorem and the monotonization of insertion theorem after the monotonically strong zero-dimension has been defined have been discussed in the third chapter. In the forth chapter, we discussed the monotonization of insertion theorem on lattice after we have defined monotonical normality on fuzzy topology space.