The Notion Of Covering Rough Membership Function And Stratified L-convergence Spaces

In this paper, we introduce the notion of covering based rough membership function,and then establish a connection between rough sets and fuzzy sets in a natural way; wealso discuss some categorical properties and some topological properties of stratifiedL-convergence spaces. The contents are arranged as follows:Firstly, the original rough membership function is defined by using equivalence classes,and then is extended to topological spaces. Note that both equivalence classes andtopological spaces form some coverings on a universe of discourse. Based on theseobservations, the notion of rough membership function is generalized to arbitrarycoverings. The properties of this conception are studied, and an axiomatic study is alsopresented.Secondly, for levelwise stratified L-convergence spaces, a strong left-continuouscondition is introduced. Then it is proved the category of strong left-continuous levelwisestratified L-convergence spaces can reflectively embed into the category of levelwisestratified L-convergence spaces; and can describe the category of strong stratifiedL-convergence spaces rightly.Thirdly, for stratified L-convergence spaces, the notions of bounded (compacted) setand locally bounded (compacted) space are introduced. These notions can be regarded asthe natural extensions of J ger’s corresponding notions. It is proved that the general Lowenfunctor (by which the category of convergence spaces embeds coreflectively into thecategory of L-convergence spaces) both preserves and reflects boundedness (compact)and locally boundedness (compact);the category of locally bounded(compacted) stratifiedL-convergence spaces is a coreflective subcategory of the category of stratifiedL-convergence spaces.