Some Topological Properties On Generalized Paratopological Rough Groups

Inspired by the generalization of topological groups to generalized topological groups and generalized paratopological groups,we extend topological rough groups to generalized paratopological rough groups in the first part(Chapter 2).In this part,we study some topological properties of generalized paratopological rough groups,which consists of three sections.In Section 1,we firstly discuss how to derive the generalized neighborhood base at arbitrary point from the generalized neighborhood base at the rough identity element in generalized paratopological rough groups(Theorem 2.1.4),further discuss how to derive the generalized neighborhood of arbitrary subset from the generalized neighborhood at the rough identity element in generalized paratopological rough groups(Theorem 2.1.6).Secondly,we discuss the generalized paratopologized problem of rough groups(Theorem 2.1.8).In Section 2,we mainly study rough subgroups of generalized paratopological rough groups.We construct an example to illustrate there exists a generalized open rough subgroup in a generalized paratopological rough group which is not generalized closed(Example 2.2.5).And we propose when a generalized open rough subgroup of a generalized paratopological rough group is generalized closed(Question 2.2.6),and partially answer this question.We prove that the generalized open rough subgroups of order 3 in generalized paratopological groups are generalized closed(Theorem 2.2.12).In addition,we study the closure operation of subsets of generalized(paratopological)topological rough groups.In Section 3,we study generalized separation properties of generalized topological rough groups,such as generalized T0,generalized T1 and generalized T2.In particular,we prove that a generalized topological rough group is generalized T1 if its upper approximation is generalized T0(Theorem 2.3.5);a generalized topological rough group is generalized T2 if its upper approximation is generalized T1(Corollary 2.3.13).A generalized topological isomorphism theorem for generalized paratopological groups has been obtained in the literature.In the second part(Chapter 3),we continue to study the generalized topological isomorphism of generalized paratopological groups on this foundation,establishing a new generalized topological isomorphism theorem.Namely,let G and G be generalized paratopological groups,f:G→ G be a surjective generalized continuous and generalized open homomorphism,and let H be a normal subgroup of G.Set H=f-1(H)and N=Kerf.Then the generalized paratopological groups G/H,G/H and(G/N)/(H/N)are mutually generalized topologically isomorphic(Theorem 3.4).