HE Chun-li,LIN Xue,CAI Zhang-yong,YE Pei-qi

(1.School of Mathematics and Statistics,Nanning Normal University,Nanning 530100,China; 2.Shangrao No.2 Middle School,Shangrao 334000,China)

Abstract: In this paper,we continue to study the generalized topological isomorphism of generalized paratopological groups and obtain a new generalized topological isomorphism theorem.

Key words: generalized paratopological group;generalized open homomorphism;generalized topological isomorphism

1 Introduction

In 2020,Cai Zhang-yong and Ye Pei-qi in [1] introduced generalized paratopological groups and quotient groups of generalized paratopological groups.In 2021,Cai Zhang-yong and Ye Pei-qi[2]continued to study some properties of generalized paratopological groups,proposed the definition of generalized topological isomorphism of generalized paratopological groups and obtained a generalized topological isomorphism theorem of generalized paratopological groups.On this basis,we continue to study the generalized topological isomorphism of generalized paratopological groups and establish a new generalized topological isomorphism theorem.

2 Generalized topological isomorphism theorem of generalized paratopological groups

Definition2.1[3]LetXbe a nonempty set.If a subfamilyτof the power set P(X) ofXsatisfies the following two conditions:

(1) Ø∈τ;

(2) for everyi∈I,ifGi∈τthen ∪i∈IGi∈τ;

thenτis called a generalized topology onXand (X,τ) is a generalized topological space,where every element ofτis called a generalized open set,the complement of each generalized open set is called a generalized closed set.The family composed of all generalized open subsets ofXis denoted asτ(X).

Definition2.2[3]LetX，Ybe generalized topological spaces andf:X→Ybe a mapping.Then

(1)fis called generalized continuous,iff-1(V)∈τ(X) for everyV∈τ(Y);

(2)fis called a generalized open mapping,if the image of every generalized open subset ofXis a generalized open subset ofY;

(3)fis called a generalized homeomorphism,iffis bijective andf,f-1are generalized continuous.

Definition2.3[1]Let (X,·) be a group and (X,τ) be a generalized topological space.If the mapping

op2:X×X→Xdefined byop2(x,y)=xy,∀x,y∈X,

is generalized continuous,then (X,·,τ) is called a generalized paratopological group.

Definition2.4[2]Let (X,·,τ) and (Y,∘,π) be generalized paratopological groups.A mappingf:X→Yis called a generalized topological isomorphism between (X,·,τ) and (Y,∘,π),iffis both a generalized homeomorphism between the generalized topological spaces (X,τ) and (Y,π) and an isomorphism between the groups (X,·) and (Y,∘).

LetGbe a group,Hbe a normal subgroup ofG(denoted byH◁G) and the natural mapping

φH:G→G/Hbe defined byφH(g)=gH,∀g∈G.

Then

(1)φHis an epimorphism;

(2)φH-1(φH(g))=gH,∀g∈G.

We recall the following result(see [4]).

LetX,Ybe groups,f:X→Ybe an epimorphism and the mappings，

φKerf:X→X/Kerfbe defined byφKerf(x)=xKerf,∀x∈X.

Definition2.5[1]Let (X,·,τ) be a generalized paratopological group andH◁X.We define the mappingη:X→X/Hbyη(x)=xH,∀x∈X.

We get the quotient space ofXwith respect toη,denoted by (X/H,τ(X/H)), whereτ(X/H) is the quotient topology ofX/Hwith respect toη,that is,

τ(X/H)={B⊆X/H:η-1(B)∈τ(X)}.

Lemma2.6[2]LetX,Ybe generalized paratopological groups andfbe a generalized continuous and generalized open homomorphism fromXontoY.ThenYandX/Kerfare generalized topologically isomorphic.

Now we prove the main theorem in this paper.

We note first that: it follows fromN◁H◁GthatH/N◁G/N.

Corollary2.8 LetHandNbe normal subgroups of a generalized paratopological groupG,andN⊆H.Then the generalized paratopological groupsG/Hand (G/N)/(H/N) are generalized topologically isomorphic.

ProofNote that it follows fromN◁H◁GthatH/N◁G/N;the mappingφN:G→G/Nis a surjective generalized continuous and generalized open homomorphism.