The Research Of Lowen Functor And Alexandrov Topology

The content of this paper is divided into two parts.In part one,as the future research of classical Lowen functors,a pair of functors between the category of classical topological spaces and that of L-topological spaces are introduced,namely,stratifbrm Lowen functors.They have good properties, and form an adjunction between these two categories above.There is a natural relation between classical Lowen functors and stratiform Lowen functors introduced in this paper,and some of their applications are also provided.In part two,it is presented,by means of Fuzzifying Alexandrov topology for Fuzzy preordered sets and specialization order for Fuzzifying topological spaces,a systematic investigation of the interrelationship between Fuzzy preordered sets and Fuzzifying topological spaces.In details,there can be a Fuzzifying topology for a given Fuzzy preorder,namely,Fuzzifying Alexandrov topology.On the other hand,a powerful method generating Fuzzy preorder for an arbitary Fuzzy mapping is provided,by which it is proved that each Fuzzifying topology is able to generate a Fuzzy preorder.At last,a pair of adjoint functors between the category of Fuzzy preordered sets and that of Fuzzifying topological spaces are established categorically.This paper is composed of five chapters:To begin with this paper,the introduction chapter introduces relating background knowledge.It also shows the development of classical Lowen functors and the close relaton between preorder and topology and point out the problems to solve. The preliminary chapter introduces the common symbols used in this paper as well as some basic concepts.In 2.1,the main conception of this paper-stratiform Lowen functors is introduced, with discussions of their properties and close relation to the classical Lowen functors.In 2.2,some applications of stratiform Lowen functors are provided,especially for their property of stratifrom structure to replace the classical Lowen functors in some situations.In 3.1,it is offered a systematic way to induce a Fuzzifying topology by a Fuzzy preorder with a description of category.In 3.2,it is provided a powerful method generating Fuzzy preorder as the way of which to be induced by a Fuzzifying topology,and its description categorically.The last chapter is the conclusion of this paper for the importance of our research.