Rough-Induced Topologies And Their Properties Based On Binary Relations

Rough sets depend on binary relations to closely adhere to topologies.The binary relations are related to reflexive,symmetric,transitive relations,and these three basic features construct a systematic structure with a three-layer network,which contains seven type-s of binary relations(i.e.,the reflexive,symmetric,transitive,reflexive and symmetric,reflexive and transitive,transitive and symmetric,equivalent relations);thus,the relevant combination s-tudy of rough sets and topologies has great significance.In fact,rough approximation operators can induce topologies,and corresponding rough-induced topologies already have studies on the reflexive and symmetric,reflexive and transitive,equivalent relations,while the rough-induced topology becomes trivial for the symmetric relation.Thus,this thesis mainly researches the rough-induced topologies and their properties based on the surplus three types of binary rela-tions(i.e.,the reflexive,transitive,transitive and symmetric relations),to complete the system-atic construction of the rough-induced topologies based on the three-layer network of binary relations.Concretely,this thesis includes three parts as follows.1.Based on the reflexive relation,approximations are defined,and the rough-induced topology is constructed by the lower approximation;the inclusion relationships of the upper and lower approximations with the interior and closure are obtained.The rough-induced topology based on the reflexive relation satisfies conditions of(ref)and(clop).The approximations of the interior/closure and the interior/closure of the approximations are established,and the inclusion order relationships among the interior,closure,approximation operators are revealed.By constructing a reflexive relation,the interior and closure of a topology that satisfies condition(sepa)respectively become the lower and upper approximations based on the reflexive relation.The topology that satisfies condition(sepa)is the rough-induced topology based on a reflexive relation,and for a single point set over the universe,its closure is exactly itself.2.Based on the transitive relation,approximations are defined,and the rough-induced topology is constructed by the lower approximation;the lower approximation becomes the in-terior,and the closure is included in the upper approximation.According to the rough-induced topology,the minimum base is established.The rough-induced topology based on the transitive relation satisfies the conditions of(ref)and(clop).The approximations of the interior/closure and the interior/closure of the approximations are established,and the inclusion order relation-ships among the interior,closure,approximation operators are discussed.By constructing a transitive relation,the lower/upper approximation based on the transitive relation and the inte-rior/closure of a topology that satisfies condition(COMP)exhibit the inclusion relationships.Moreover,a topology that satisfies condition(COMP)and the rough-induced topology based on a transitive relation also offer the inclusion relationships.3.Based on the symmetric and transitive relation,the approximations of rough sets are defined,and the rough-induced topology is constructed by that the single point set includes its upper approximation;the interior and closure of the rough-induced topology are constructed.According to the rough-induced topology,the base and neighborhood base are established to gain the relevant countability properties,including the second and first coutability,separability,and Lindelof feature.Finally,an example is used to effectively illustrate the rough-induced topology and its countability.In this study,the three-layer and seven-type network based on three basic binary relations is fully utilized,and thus the systematic construction of rough-induced topologies is completed.Corresponding properties and results deeply reveal the close connection between rough sets and topologies,mainly from the binary relations viewpoint.