Studies On Some Properties Of Generalized Paratopological Groups

Generalized topological spaces are an important concept in topology.For nearly 20 years,many topologists have studied their some properties and obtained some results.This paper consists of two parts to study some properties of generalized topological spaces and generalized paratopological groups.In the first part,we construct several examples of mixed generalized closed sets in the generalized topological spaces to point out the gaps in literature.In this part,we have the following results.Result 1(Example 2.2)Let ?1,?2 be two generalized topologies on X.The union of two(?1,?2)-generalized closed sets is not necessarily a(?1,?2)-generalized closed set.The intersection of two(?1,?2)-generalized open sets is not necessarily a(?1,?2)-generalized open set.Result 2(Example 2.5)Let ?1,?2 be two generalized topologies on X,which a subset A of X is a(?1,?2)-generalized closed set and A is ?2-closed can not deduce that c?2(A)\A is ?1-closed.In the second part,we introduce the notion of generalized paratopological groups to generalize generalized topological groups.After that,we discuss the generalized continuity of mappings,generalized neighborhood bases,closure operation,generalized separation axioms,generalized compactness,subgroups,quotient groups,generalized product properties of generalized paratopological groups and so on.At the same time,we raise some questions.In this part,we have the following results.Result 3(Corollary 3.1.6)Every generalized paratopological group is a generalized homogeneous space.Result 4(Theorem 3.2.4)Let X be a generalized paratopological group,u be a family of subsets of X and x be a point of X.Then the following conditions are mutually equivalent:(1)u is a generalized neighborhood base of the neutral element e.(2){xU:U?u is a generalized neighborhood base of x.(3){Ux:U?u is a generalized neighborhood base of x.Result 5(Theorem 3.2.8)Let(X,·)be a group,u be a family of subsets of X and the neutral element e??u.If the following conditions hold:(1)For every U?u,there exists V1,V2 ?u such that V1V2(?)U.(2)For every U?u and g ? X,there exists V ?u such that gVg-1(?)U.Then there exists a unique generalized paratopological group(X,-,?)such that u is a generalized neighborhood base of the neutral element e of(X,?).Result 6(Theorem 3.3.13)Let X be a generalized paratopological group,g and h be two points of X.If K is a generalized compact subset of X,then gKh is a generalized compact subset of X.Result 7(Theorem 3.4.2)Let(X,·,?)be a generalized paratopological group.Then(1)Every ?-open subgroup of X is ?-closed in X.(2)Every subgroup which contains some generalized neighborhood of the neutral element e is ?-open in X.Result 8(Theorem 3.4.7)Let X be a generalized paratopological group and H be a subgroup of X.If IntXH?O,then H is generalized open in X.Result 9(Corollary 3.4.8)Let X be a generalized paratopological group and U be a generalized open neighborhood of the neutral element e.Then the subgroup Un=1?(U?U-1)n generated by U is a generalized open subgroup of X.Result 10(Theorem 3.5.5)Let X be a generalized paratopological group and H be a normal subgroup of X.Then X/H is generalized T1 if and only if H is generalized closed.Result 11(Theorem 3.5.8)Let X,Y be generalized paratopological groups,let f be a generalized continuous and generalized open homomorphism from X onto Y.Then Y and X/Kerf are generalized topologically isomorphic.Result 12(Theorem 3.6.2)A generalized closed onto homomorphism between generalized paratopological groups is generalized open.Result 13(Theorem 3.6.6)Let X,Y be generalized paratopological groups and let f:X?Y be a homomorphism.If f is generalized open at some point of X,then f is generalized open.Result 14(Theorem 3.7.3)Let {(X?,·,??):?? A} be a family of generalized paratopological groups and the subset Y?=U?? of X? be closed with respect to the binary operation,that is,every x,y ? Y? implies x-y ?Y? for each ??A.Then X=???AX? is a generalized paratopological group.Especially,the product of a family of strong generalized paratopological groups is also a generalized paratopological group.Result 15(Corollary 3.7.4)For every ??A,let X? be a strong generalized paratopological group and H? be a normal subgroup of X?,then ???A(X?/H?)and???AX?/???AH? are generalized topologically isomorphic.