Study On Some Properties Of LF-closure Spaces

In 1968,the concept of fuzzy topological space was introduced firstly by C.L.Chang, and some concepts in general topology were extended. Moreover, related properties were investigated. In 1983 and 1985,crisp-closure space and F-closure space were separately presented by American mathematician,Mashhour and Ghanim. Afterwards, a new space called LF-closure space for completely distributive lattic and fuzzy lattic was introduced in paper [5] and [6].Closure space is an important concept considering the range of topology,lattic theories and pseudo-matrix theories. It contains many basic problems on topology. Meanwhile,LF-closure space is not only the extention of the L-fuzzy topological space and the power topological molecule lattic,but closure space .Therefore, the study to the problems of LF-closure space is especially important. At present,we have already achieved a series of good results.In a whole,there are not many papers related the topological properties in LF-closure space presently. Connectedness property in LF-closure space was respectively investigated by different angles. But there are not any papers studied the connectedness property in LF-closure space in layer angle,the related work is significant. Compactness is the important content in topology. And We can extend the definition of strong F compact property of LF-closure space and study their equivalent conditions and the basic properties,the same work are also in paper [16],[17],[19] and [22],the investigation on strong F compactness,equivalent conditions and basic properties, and so on are necessary parts for riching LF-closure space. Similarly, lindel(o|¨)f property is also an essential topology property. Of course, new lindel(o|¨)f property is defined in different angel,these are the essential part of riching the theory. But,above work is imperfect. Firstly,we extend the strong F compact property of L-fuzzy topology space to LF-closure space,and discuss its equivalent conditions and the basic property. Finally, we introduce a new weak-lindel(o|¨)f property on the base of some papers, discuss its basic properties,prove it is a weak-topology invariant property,give the relation of them, and so on.The basic contents of this paper is following:Chapter one. Preparation: The main signals,concepts and important conclusions needed in this paper are given;Chapter two.Layer connectedness property of LF-closure space:In the first part,the concept of layer connectedness is defined and the equivalent conditions are discussed. In the second part,some of its basic properties are given.Then,it is proved weakly-invariant under continuous L-valued Zadeh function and multiplication.Chapter three. Strong F compact property of LF-closure space:The strong F compact property is extended.In addition,the equivalent conditions and the basic properties of it in LF-closure space are discussed.Then,it is proved that the main results of strong F compact property in LF-closure space such as weakly-invariant under continuous L-valued Zadeh function and limited multiplication.Chapter four. Weak-lindel(o|¨)f property of LF-closure space:The concept of the lindel(o|¨)f property and weak-lindel(o|¨)f property are defined in LF-closure space. Some basic properties, the equivalent conditions and the relation between the weak lindel(o|¨)f property and other lindel(o|¨)f property are studied.