Some Problems Study In Lω-Spaces

In paper[1],the concept of L—order-preserving operator space(This article is abbreviation called the Lω—space)is introduced by means of an L—order-preserving operatorω.It has general sense which can induce different spaces,such as L—topological spaces,L—closure spaces,when given properly different L—order-preserving operatorsωand it has many good properties.Domestic scholars such as Shui-Li Chen,Zhao-xia Huang,Hong-Xia Han and so on have done a lot of work.In this paper the quasiω—Lindel(?)f property is introduced and stutied firstly in Lω—spaces.Secondly theω—semi—set is defined and theω—semi—continuous certain properties are studied.Finally theωS—separation property and theωS—connected property are defined in the Lω—space,some properties are discussed systematicaly,the content of this artical are mainly including as follows:1.The concept of quasiω—Lindel(?)f properties are introduced in Lω—spaces,the basic properties of quasiω—Lindel(?)f are discussed,and several equivalent conditions of quasiω—Lindel(?)f properties are given.2.In this section,the definition ofω—semi—set,ω—semi—continuous,ω—strong—semi—continuous are introduced,and the relation ofω—continuous andω—semi—continuous are discussed.3.The concept ofωS—separations properties are given,includingωS_-1,ωS₀,ωS₁,ωS₂ separation properties,and the relation of them are discussed.4.The concept ofω—semi—connected property are given.the basic topological properties ofω—semi—connected are discussed.Some equivalent conditions are given and the Fan Ji Theorem ofω-semi-connected properties are given.