Approximate L-open Sets In L-pretopological Spaces And Related Connectedness And Categorical Properties

L- pretopological space is a generalization of L- topological space. In this paper, some akin forms of L-open sets in L -pretopological spaces and related connectedness and categorical properties are studied deeply for L a C-lattice with an order-reversing involution. The arrangement of this paper is as followsChapter one, preliminary knowledge, in which we give the basic concepts and results of the theory of L- pretopological space and that of category which are used in the whole paper.Chapter two, approximate L- open sets in L - pretopological spaces. Firstly, the definitions ofθ- L-open set.δ- L-open set.α- L-open set and L-open set in an L-pretopological space are introduced, and properties of these notions and relationships between them are discussed. Secondly, the notions of strong 9-continous mapping. 9-continous mapping, strong 6-continous mapping.δ-continous mapping, strong a-continous mapping and n- continous mapping be-tween two L - pretopologieal spaces are defined. Then relationships between them are discussed.Chapter three, connectedness related to approximate L-open sets in L-pre topological spaces. First.θ-connectedness.δ-connectedness and n-connect-ednes of an L - pretopological space are introduced. Second. Based on these results, it is s hown that connectedness.θ-connectedness.δ-connectedness and n-connectedness of an L-pretopological space is the same if 1 is a join irreducible elementChapter four, categorical properties related to approximate L - open sets in L - pretopological spaces. First of all. the notions ofθ- L - pretopological space.δ- L-pretopological space,α- L,- pretopological space are defined. And then It is proved thatθLTop (the category of allθ- L- topological spaces and con tinuous mappings) is a reflective subeategory of LTopsθ(the category of all L-topological spaces and strongθ-continuous mappings). LTop (the category oi all L-topological spaces and continuous mappings) and LTopθ(the category of all L-topological spaces andθ-continuous mappings).δLTop (the category of allδ- L topological spaces and continuous mappings) is a reflective subcategory of LTopsδ(the category of all L-topological spaces and strongδ-continous mappings) and LTopsδ(the category of all L -topological spaces andδ-continuous mappings). αLTop (the category of allα-L-topological spaces and continuous mappings) is a coreflective subcategory of LTopsα(the category of all L-topological spaces and strongα-continous mappings) and LTopα(the category of all L-topological spaces andα-continuous mappings) for L a powerset lattice:(?)LPTop (the cat-egory of allθ-L-pretopological spaces and continuous mappings) is a reflective subcategory of LPTopsθ(the category of all L-pretopological spaces and strongθ-continuous mappings), LPTop (the category of all L- pretopological spaces and continuous mappings) and LPTopθ(the category of all L-pretopological spaces andθ-continuous mappings), andδLPTop (the category of allδ-L-pretopological spaces and continuous mappings) is a reflective subeategory of LPTops (the cate-gory of all L-pretopological spaces and strongδ-continous mappings) and LPTop"δ(the category of all L-pretopological spaces andδ-continuous mappings).