The Research Of Fuzzifying Semi-open Topological Theory

In this paper, we establish the theory of fuzzifying semi-topology using the idea of Lukasiewicz's logics. First, we introduce the concepts of fuzzifying semi-topology and fuzzifying semi-topological space. As a tool of handling local problems in topological theory, we establish semi-neighborhood system which harmonizes with fuzzifying semi-topology and make it possible to handle the local problems. As a necessary part of fuzzifying semi-topology theory, we also define the concepts of fuzzifying semi-topological operator including fuzzifying semi-interior and fuzzifying semi-closure and then describe them essentially. Then we discuss the relationship between fuzzifying semi-topological operators and fuzzifying topological operators. As a bridge relating various fuzzifying topological objects and fuzzifying semi-topological objects, we introduce the concept of fuzzifying semi-continuous mapping and give its equivalent descriptions. We also define the concept of fuzzifying semi-topological category . Our theory remains most of the important properties of general topology when the lattice value is 0 or 1. Besides, we illustrate an example to point out the disadvantage of foreign scholar's same theory and the difference between us.This paper is composed of six sections:To begin with the paper, the introduction section introduces relating background knowledge. It also shows the developement of the theory of fuzzifying semi-topology and point out the problem to solve.The first section introduces several other definition of fuzzifying semi-topology and the symbols used in this paper.The second section introduces the most important and basic concept of this paper-fuzzifying semi-topology and discusses its properties. It also illustrate an example to show the difference mentioned above.The third section establishes semi-neighborhood system and discusses its properties.The fourth section defines the concepts of fuzzifying semi-topological operator including fuzzifying semi-interior and fuzzifying semi-closure and describes them essentially.The fifth section introduces the concepts of fuzzifying semi-continuous mapping and gives its equivalent descriptions. Then we define the concept of fuzzifying semi-topological categories.The sixth section proposes several unsloved problems.