# Onα-Layer Ideal Topologies

Authors: Xiu-Yun Wu; Li-Li Xie
Journal: Advances in Fuzzy Systems
 (2012)

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## Abstract

The purpose of this paper is to study theory of two different kinds ofα-layer order-preserving operator space, namely, ωα-opos and ωα*(ℑ)-opos. The former kind of space is formed by α-layer function in L-fuzzy order-preserving operator space. The later kind of space is derived by local α-remote neighborhood function, which is related with ωα-opos and α-ideal. We study characteristic properties of the two kinds of spaces, respectively, and give some applications to show the intimate relations under two different ωα*(ℑ)-oposs.


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## Body

## 1. Introduction

In general topology, Vaidyanathaswamy firstly defined concepts of local function and its derived ideal topology from initial topology and ideal [1]. Some interesting extensive works were done by Janković and Hamlett [2]. After M. E. Abd El-Monsef introduced the concept of I-open set, many researchers were devoted to research on local semitopology. There are many local semiopen sets, such as β-I-open set, strong β-I-open set, and δ-I-open set [3–5]. All of these local semiopen sets are given by comparing interior and closure operators in the initial topology and its ideal derived topology.By utilizingq-neighborhood which is mentioned in [6], Sarkar generalized the concepts of local function and derived topology into fuzzy topology in 1997 [7].As there is a layer structure inL-fuzzy topology, fuzzy local functions and their derived fuzzy ideal topology must be more complex. Hence, in this paper, we will analyze the ideal topological properties in terms of layer structure of L-fuzzy topology and reveal the inner relations between α-layer ideal space and fuzzy ideal topological space.In the first part of the paper, we establish the theory ofωα-opos. We introduce the concept of α-layer function in L-fuzzy order-preserving operator space. Then, based on its basic properties, we form ωα-opos. We also prove it preserves many good properties. In the second part of the paper, we establish the theory of ωα*(ℑ)-opos. It is a α-layer space with local topological properties. We introduce the concept of local α-remote neighborhood function via an ωα-opos and an α-ideal. On the basis of its basic properties, we form ωα*(ℑ)-opos. It is finer than the old one. We observe the structures of local α-remote neighborhood functions under different α-ideals and ωα-opos as well as the relations of the correspondent ωα*-opos. We also obtain some equivalent conditions of ωα*-opos under compatibility of α-ideal and ωα-opos. Finally, as an application, we define four kinds of connectivity and reveal their inner relations.


## 2. Preliminaries

In a general topological space(X,τ), ℐ⊂X is an ideal, U⊂X. The concept of local function of U with respect to τ and ℐ is given by U*={x∈X:U∩V∉ℐ,V∈𝒩(x)}, in which 𝒩(x) is the neighborhood system of x [1].In this paper, an latticeL is called a completely distributive lattice with an order reserving involution′.X,Y will always denote nonempty crisp sets, A mapping A:X→L is called an L-fuzzy set.LX is the set of all L-fuzzy sets on X. An element e∈L is called an irreducible element in L, if p∨q=e implies p=e or q=e, where p,q∈L. The set of all nonzero irreducible elements in L will be denoted by M(L) (see [1]). If x∈X, α∈M(L), then xα is called a molecule in LX. The set of all molecules in LX is denoted by M*(LX). If A∈LX, α∈M(L), take A[α]={x∈X:A(x)≥α}. If E⊂X, the complement of E, denoted by E', and E'=X-E={y∈X:y∉E} [8].AnL-fuzzy order-preserving operator space and some related conceptions are given in the following.LetX be an nonempty set. An operator ω:LX→LX is called a L-fuzzy order preserving operator in LX, if it satisfies (1) ω(1X)=1X, (2) for all A,B∈LX and A≤B implies ω(A)≤ω(B). A set A∈LX is called an ω-set, if ω(A)=A. The set of all ω-sets in LX is denoted by Ω. And (LX,Ω) is called an order-preserving operator space (briefly, L-fopos). A molecule xα∈M*(LX),P∈Ω, P is called an ω-remote neighborhood of xα, if xα∉P. The set of all ω-remote neighborhood of xα is denoted by ωη(xα). Let xα∈M*(LX),A∈LX, xα is called an ω-adherent point of A, if for all P∈ωη(xα), A∉P. The union of all ω-adherent points of A is called the ω-closure of A, denoted by Aω-. A set A∈LX is called ω-closed, if Aω-=A. The set of all ω-closed sets in LX is denoted by Ωω-. Ωω- is finite union and infinite intersection preserving [9].Similar concepts in a general topology are defined as follows.LetX be a nonempty set, and let 𝒫(X) be the family of all subsets of X. An operator σ:X→X is called an order-preserving operator in X, if it satisfies (1) σ(X)=X, (2) for all A⊂B⊂X implies σ(A)⊂σ(B). A set A⊂X is called a σ-set, if σ(A)=A. The set of all σ-sets in X is denoted by Δ. (X,Δ) is called an order-preserving operator space on X (briefly, opos). Let x∈X,P∈Δ, P is called a σ-remote neighborhood of x, if there is Q⊂X, such that x∉Q,P⊂Q. The set of all σ-remote neighborhood of x is denoted by ση(x). Let x∈X,A⊂X, x is called a σ-adherent point of A, if  for all P∈ση(x), A⊄P. The union of all σ-adherent points of A is called the σ-closure of A, denoted by Aσ-. A set A⊂X is called σ-closed, if Aσ-=A. The set of all σ-closed sets in X is denoted by Δσ-. Δσ- is finite union and infinite intersection preserving.Let(X,Δ) be an opos, and let L be an fuzzy lattice, for all α∈L. A fuzzy set A:L→X is called a lower continuous function, if {x∈X:A(x)≤α}∈Δσ-. Then, the set of all the lower continuous functions, denoted by ωL(Δ), consists an L-fuzzy cotopology in LX. The space (LX,ωL(Δ)) is called the induced L-fopos by (X,Δ).A nonempty subfamilyℑ of LX is called an α-ideal, if ℑ satisfies the following conditions:(1)
A∈ℑ, and B[α]⊂A[α] implies B∈ℑ,(2)
A,B∈ℑ implies A∨B∈ℑ.It is easy to check ifB∈LX, A∈ℑ, such that B[α]=A[α], then B∈ℑ. Moreover, ℑ of LX is an α-idea if and only if  ℑ[α]={A[α]:A∈ℑ} is an ideal on X. If B∈LX, A∈ℑ, such that B[α]=A[α], then B∈ℑ. If ℑ,𝔍 are two α-ideals, then ℑ∨𝔍={I∨J:I∈ℑ,J∈𝔍} and ℑ∩𝔍 are α-ideals too.In this paper, if𝔉⊂LX, we denote 𝔉[α]={A:A∈𝔉} and  𝔍′={A′:A∈𝔉}.


## 3.ωα-Closed Set and ωα-opos

In this section, we list out the main results in [10], which we will use in the following sections. Proofs of the theorems in the following can be found in [10] as well.Definition 1.
Let(LX,Ω) be an L-fopos, α∈M(L). An operator ωα:LX→LX is defined by, for all A∈LX,(1)ωα(A)=⋀{G∈Ωω-:G[α]⊃A[α]}.Theorem 2.
Let(LX,Ω) be an L-fopos, α∈M(L). A,B∈LX, then the following statements hold:(1)
A[α]⊂B[α] implies ωα(A)≤ωα(B);(2)
A[α]⊂ωα(A)[α];(3)
ωα(A∨B)=ωα(A)∨ωα(B);(4)
ωα(ωα(A))=ωα(A).Definition 3.
Let(LX,Ω) be an L-fopos, α∈M(L). A set A∈LX is called ωα-closed, if ωα(A)[α]=A[α]. The set of all ωα-closed sets in LX is denoted by ωα(Ω). (LX,ωα(Ω)) is called an ωα-order-preserving operator space (briefly, ωα-opos).Theorem 4.
Let(LX,Ω) be an L-fopos, α∈M(L). Then,(1)
1X∈ωα(Ω);(2)
if{Ai:i=1,2,…,n}⊂ωα(Ω), then ⋁i=1nAi∈ωα(Ω);(3)
if{Ai:i∈I}⊂ωα(Ω), then ∧i∈IAi∈ωα(Ω).Remark 5.
The theorem shows thatωα is finite union and infinite intersection preserving.Corollary 6.
Let(LX,Ω) be an L-fopos, α∈M(L). Then,  ωα(Ω) consists an L-fuzzy cotopology on LX.Lemma 7.
Let(X,Δ) be an opos, (LX,ωL(Δ)) be an L-fopos  induced by (X,Δ) and let E⊂X. Then, E∈Δσ-  if and only if   χE∈(ωL(Δ))ω-.Theorem 8.
Let(X,Δ) be an opos, (LX,ωL(Δ)) an L-fopos  induced by (X,Δ). A∈LX, α∈M(L). Then, A∈ωα(ωL(Δ)) if and only if  A[α]∈Δσ-.Theorem 9.
Let(X,Δ) be an opos, (LX,ωL(Δ)) is an L-fopos  induced by (X,Δ). Then, A∈ωL(Δ)  if and only if  for all α∈M(L), A∈ωα(ωL(Δ)).Definition 10.
Let(LX,ωα1(Ω1)), (LY,ωα2(Ω2)) be ωα1-opos, ωα2-opos, respectively. An L-fuzzy homomorphism f→:LX→LY is called (ωα1(Ω1),ωα2(Ω2))-continuous, if for all B∈ωα2(Ω2), then f←(B)∈ωα1(Ω1).Theorem 11.
Let(LX,ωα1(Ω1)), (LY,ωα2(Ω2)) be ωα1-opos, ωα2-opos, respectively. f→:LX→LY is an L-fuzzy mapping. Then, the following statements are equivalent:(1)
f→ is (ωα1(Ω1),ωα2(Ω2))-continuous;(2)
for allB∈LY, ωα1(f←(B))[α]⊂f←(ωα2(B))[α];(3)
for allA∈LX, f→(ωα1(A))[α]⊂ωα2(f→(A))[α].Definition 12.
Let(LX,ωα(Ω)) be an ωα-opos, xα∈M*(LX),A∈ωα(Ω). A is called an ωα-closed remote neighborhood of a, if x∉A[α]. The set of all ωα-closed remote neighborhood of xα will be denoted by ηωα(xα). An fuzzy point xα is called ωα-adherent point of B∈LX, if for every P∈ηωα(xα), B([α])⊄P([α]).Remark 13.
P
∈
η
ω
α
(
x
α
)
⇔
(
P
[
α
]
)
'
∈
𝒩
(
x
), where 𝒩(x) is the neighborhood system of x in (X,ωα(Ω)[α]). Hence, (ηωα(xα)[α])'=𝒩(x).Theorem 14.
Let(LX,ωα(Ω)) be an ωα-opos, xα∈M*(LX),A∈LX, then(1)
xα is an ωα-adherent point of A∈LX if and only if  x∈ωα(A)[α],(2)
ωα(A) is the union of all ωα-adherent points of A.


## 4.ωα-ℑ-Closed Sets

Let(LX,ωα(Ω)) be an ωα-opos. ℑ⊂LX be an α-ideal. xα∈M*(LX), ηωα(xα) be the ωα-remote family of xα. For any A∈LX, take
(2)A*(ℑ,ωα)=⋁{xα:A[α]⊄P[α]∪I[α],∀P∈ηωα(xα),I∈ℑ}.
Then, A*(ℑ,ωα) is called the local ωα-remote function of A with respect to ℑ and ωα(Ω), simply denoted by A*(ℑ) or A*.Remark 15.
IfL=[0,1], ω is the closure operator, then (A[α]*)' is the local function of A[α] in (X,ωα(Ω)[α]), and 𝒜*=⋁α∈L(A[α]*)' is the fuzzy local function of A in [7]. Furthermore, we have (A[α]*)'=(A[α])'*.Theorem 16.
Let(LX,ωα(Ω)) be an ωα-opos. ℑ,𝔍⊂LX be two α-ideals. A,B∈LX. Then, the following statements hold.(1)
ℑ[α]={∅}⇒A*(ℑ)[α]=ωα(A)[α], ℑ[α]=𝒫(X)⇒A*(ℑ)[α]=∅.(2)
A[α]⊂B[α]⇒A*(ℑ)[α]⊂B*(ℑ)[α].(3)
ℑ[α]⊂𝔍[α]⇒A*(𝔍)[α]⊂A*(ℑ)[α].(4)
A*(ℑ)[α]=ωα(A*(ℑ))[α]⊂ωα(A)[α].(5)
(A*(ℑ))*(ℑ)[α]⊂A*(ℑ)[α].(6)
A∈ℑ⇒A*(ℑ)[α]=∅.(7)
(A∨B)*(ℑ)[α]=A*(ℑ)[α]∪B*(ℑ)[α].(8)
(A*-B*)[α]=((A-B)*-B*)[α]⊂(A-B)[α]*.(9)
B∈ℑ⇒(A∨B)[α]*=A[α]*=(A-B)[α]*.(10)
P∈ωα(Ω)⇒P[α]*⊂P[α].Proof.
(1) Supposeℑ[α]={∅}, by Theorem 14, we have
(3)xα≤A*(ℑ,ωα)⟺xα≤⋁{xα:A[α]⊄P[α],∀P∈ηωα(xα)}⟺xα≤ωα(A).
IfX∈ℑ[α], then there must be B∈ℑ, such that A[α]=B[α]={∅}. Therefore, A*(ℑ)[α]=∅.
(2), (3), and (6) Easy.
(4)x∉ωα(A)[α] implies ωα(A)∈ηα(xα). By A[α]⊂ωα(A)[α], we have x∉A[α]*. So A[α]*⊂ωα(A)[α]. Thus, ωα(A)[α]*⊂ωα(ωα(A))[α]=ωα(A)[α]. Besides, x∈ωα(A*)[α] implies for all  P∈ηωα(xα), A[α]*⊄P[α]. Take y∈A[α]*-P[α], so P∈ηωα(yα) and for all  Q∈ηωα(yα),I∈ℑ, A[α]⊄Q[α]∪I[α]. Particularly, let P=Q. So A[α]⊄P[α]∪I[α]. This means x∈A[α]*. Hence, ωα(A*)[α]⊂A[α]*. Therefore, ωα(A*)[α]=A[α]*.
(5) By (4),(A*(ℑ))*(ℑ)[α]=(ωα(A*(ℑ)))*(ℑ)[α]=ωα(ωα(A*(ℑ)))[α]=(ωα(A*(ℑ)))[α]=A*(ℑ)[α].
(7) By (2),(A∨B)*(ℑ)[α]⊃A*(ℑ)[α]∪B*(ℑ)[α]. Conversely, if and only if  x∈A*(ℑ)[α]∪B*(ℑ)[α], then for all P∈ηωα(xα),I∈ℑ, A[α]⊄P[α]∪I[α], and B[α]⊄P[α]∪I[α]. Hence, (A∨B)[α]⊄P[α]∪I[α]. This implies x∈(A∨B)*(ℑ)[α]. So (A∨B)*(ℑ)[α]⊂A*(ℑ)[α]∪B*(ℑ)[α].
(8) For eachx∈(A*-B*)[α], we may remark x∈(A-B)[α]*. If not, there are P∈ηωα(xα),I∈ℑ, such that (A-B)[α]⊂P[α]∪I[α]. Thus, A[α]⊂P[α]∪I[α]∪B[α]. Since x∉B[α]*, there are Q∈ηωα(xα),J∈ℑ, B[α]⊂Q[α]∪J[α]. This shows
(4)A[α]⊂P[α]∪I[α]∪B[α]⊂P[α]∪I[α]∪Q[α]∪J[α]=(P∨Q)[α]∪(I∨J)[α].
Consequently, x∉A[α]*. But this contradicts with x∈(A*-B*)[α]. Therefore, (A*-B*)[α]⊂((A-B)*-B*)[α]. The reverse inclusion is obvious. So (A*-B*)[α]=((A-B)*-B*)[α].
(9) Obviously,(A∨B)[α]*=A[α]*∪B[α]*=A[α]*. And (A-B)[α]*⊂A[α]*. Conversely, x∉(A-B)[α]*, there are P∈ηωα(xα),I∈ℑ, (A-B)[α]⊂P[α]∪I[α]. So A[α]=(A-B)[α]∪B[α]⊂P[α]∪I[α]∪B[α]=P[α]∪(I∨B)[α]. Thus, x∉A[α]*. Then, (9) holds.
(10) Suppose there isx∈P[α]*-P[α]. So P∈ηωα(xα). By P[α]⊂(P∨I)[α] for each I∈ℑ, we have x∉P[α], a contradiction.Theorem 17.
Let(LX,ωα(Ω)) be an ωα-opos. Let ℑ⊂LX be an ideal. A∈LX. Take ωα*(A)=A∨A*, then(1)
ωα*(A)[α]=∅, if and only if  A[α]=∅,(2)
ωα*(A∨B)[α]=ωα*(A)[α]∪ωα*(B)[α],(3)
ωα*(ωα*(A))[α]=ωα*(A)[α],(4)
(ωα*(A*))[α]*=ωα*(A*)[α]=(ωα*(A))[α]*=(ωα(A*))[α]*=A[α]*.Proof.
According to (5), (7) in Theorem16, the proof is trivial.
By Theorem17, we know if and only if  we take ωα*(ℑ,Ω)={A∈LX:ωα*(A)[α]=A[α]}, then ωα*(ℑ,Ω) (simply denoted by ωα*(ℑ)) consists an L-fuzzy cotopology on LX. So it is called ωα-cotopology formed by the ideal ℑ and ωα(Ω). The pair (LX,ωα*(ℑ,Ω)), simply denoted by (LX,ωα*(ℑ)), is called ωα*(ℑ)-opos. By Theorem 16 (1), we have ℑ[α]={∅}⇒ωα*(A)[α]=ωα(A)[α], thus ωα*(ℑ,Ω)=ωα(Ω), and ℑ[α]=𝒫(X)⇒ωα*(A)[α]=A[α], thus ωα*(ℑ,Ω)=LX. But in general, {∅}⊂ℑ⊂𝒫(X), A[α]⊂ωα*(A)[α]⊂ωα(A)[α], consequently, ωα(Ω)⊂ωα*(ℑ,Ω)⊂LX. Moreover, if and only if   ωα(Ω)[α]={∅,X}, then ωα*(Ω)={A:A[α]∈ℑ[α]}.Example 18.
LetX={x,y},L={0,1/2,1}. An L-fuzzy set G∈LX satisfying G(x)=a,G(y)=b will be denoted by (a,b). Let Ω={(0,0),(0,1/2),(1,1/2),(1,1)} be a cotopology on LX. α=1/2. Put ωα(A)=⋀{E∈Ω:E[α]⊃A[α]}. Then, ωα(Ω)={A:ωα(A)[α]=A[α]} consists an ωα-cotopology, and ηωα(xα)={(0,0),(0,1/2),(0,1)},ηωα(yα)={(0,0)}. Totally, there are four kinds of α-ideals in LX, namely, 𝔏={(0,0)}, 𝔐=LX, ℑ={(0,0),(0,1/2),(0,1)} and 𝔍={(0,0),(1/2,0),(1,0)}. 𝔏 and 𝔐=LX are the trivial α-ideals. Let us study ℑ and 𝔍.
DenoteB=(0,1/2),D=(1/2,0),E=(1/2,1/2). Then, A*(ℑ)=B*(ℑ)=(0,0),D*(ℑ)=E*(ℑ)=(1/2,1/2). So A,B,E∈ωα*(ℑ), But D∉ωα*(ℑ). In addition, ωα*(ℑ,Ω)=ωα(Ω). However, D*(𝔍)=(0,0), which means D∈ωα*(𝔍). It is easy to check ωα*(𝔍,Ω)=LX.
In this example, we see(ωα(B))*(ℑ)[α]=B*(ℑ)[α]=∅. However, ωα*(B)[α]=(B∨B*(ℑ))[α]={y}. Thus, (ωα(B))*(ℑ)[α]≠ωα*(B)(ℑ)[α]. Furthermore, (ωα(Ω))*(ℑ)={A*(ℑ):A∈ωα(Ω)}={(0,0),(1/2,1/2)} is not an ωα-cotopology.Theorem 19.
Let(LX,ωα(Ωi))(i=1,2) be an ωα-opos. ωα(Ω1)⊂ωα(Ω2). ℑ is an α-ideal. A∈LX, then(1)
A*(ℑ,ωα(Ω2))[α]⊂A*(ℑ,ωα(Ω1))[α],(2)
A*(ℑ,ωα*(ℑ,Ω1))[α]⊂A*(ℑ,ωα(Ω1))[α],(3)
ωα*(ℑ,Ω1)⊂ωα*(ℑ,Ω2).Theorem 20.
Let(LX,ωα(Ω)) be an ωα-opos, an α-ideal ℑ⊂ωα(Ω). Then, ωα*(ℑ)=ωα(Ω).Proof.
Obviously,ωα*(ℑ)⊃ωα(Ω). If A∉ωα(Ω), that is, A[α]≠ωα(A)[α]. Take y∈ωα(A)[α]-A[α]. Let us prove y∈A*(ℑ)[α]. In fact, if y∉A*(ℑ), there are Q∈ηωα(yα),I∈ℑ, such that A[α]⊂(Q∨I)[α]. So
(5)y∈ωα(A)[α]⊂ωα(Q∨I)[α]=(ωα(Q)∨ωα(I))[α]=(Q∨I)[α].
It meansy∈I[α]. On the other hand, as A[α]⊂(Q∨I)[α], we have (A∧I)[α]⊂Q[α]. Then,
(6)ωα(A∧I)[α]=(ωα(A)∧ωα(I))[α]=ωα(A)[α]∩I[α]⊂ωα(Q)[α]=Q[α].
Notice that y∈ωα(A)[α]∩I[α], we get y∈Q[α], a contradiction. Hence, y∈A*(ℑ)[α]. So A*(ℑ)[α]⊄A[α], which implies A∉ωα*(ℑ). Therefore, ωα*(ℑ)⊂ωα(Ω). The proof is completed.Theorem 21.
Let(LX,ωα(Ω)) be an ωα-opos, and let  ℑ be an α-ideal. Take ωα(Ω)∨ℑ as the supremum ωα- cotopology generated by ωα(Ω)∪ℑ. Then, ωα*(ℑ)=ωα(Ω)∨ℑ.Proof.
Put𝔅={P∨I:P∈ωα(Ω),I∈ℑ}. It is easy to prove 𝔅⊂ωα*(ℑ) is a base of ωα*(ℑ).Lemma 22.
Let(LX,ωα(Ω)) be an ωα-opos. Let ℑ,𝔍⊂LX be two ideals. I∈ℑ, A∈LX. Then, (A-I)*(𝔍,ωα(Ω))=A*(𝔍,ωα*(ℑ)).Proof.
By Theorem21,
(7)x∉(A-I)*(𝔍,ωα(Ω))⟺∃P∈ηωα(xα),J∈𝔍,(A-I)[α]⊂(P∨J)[α]⟺∃P∈ηωα(xα),J∈𝔍,A[α]⊂(P∨J∨I)[α]⟺∃Q∈ηωα*(ℑ)(xα),J∈𝔍,A[α]⊂(Q∨J)[α]⟺x∉A*(𝔍,ωα*(ℑ)).Theorem 23.
Let(LX,ωα(Ω)) be an ωα-opos. Let ℑ,𝔍⊂LX be two α-ideals. A∈LX, then(1)
A*(ℑ∩𝔍)[α]=A*(ℑ)[α]∪A*(𝔍)[α],(2)
A*(ℑ∨𝔍,ωα(Ω))[α]=A*(ℑ,ωα*(𝔍))[α]∩A*(𝔍,ωα*(ℑ))[α].Proof.
(1) Supposex∉A*(ℑ)[α]∪A*(𝔍)[α], there are P,Q∈ηωα(xα), and I∈ℑ,J∈𝔍, such that A[α]⊂(P∨I)[α], and A[α]⊂(Q∨J)[α]. Thus, A[α]⊂(P∨I)∧(Q∨J)[α]. As (P∧Q)∨(P∧J)∨(I∧Q)∈ηωα(xα), and I∧J∈ℑ∩𝔍, it is clear that x∉A*(ℑ∩𝔍)[α]. So A*(ℑ∩𝔍)[α]⊂A*(ℑ)[α]∪A*(𝔍)[α]. The reverse inclusion is obvious according to (3) in Theorem 16. Therefore, (1) holds.
(2) Supposex∉A*(ℑ∨𝔍,ωα(Ω))[α]. There are P∈ηωα(xα), and I∈ℑ,J∈𝔍,K=I∨J, such that A[α]⊂(P∨K)[α]. So (A-J)[α]⊂(P∨I)[α], (A-I)[α]⊂(P∨J)[α]. Hence, by Lemma 22, we have x∉(A-J)*(ℑ,ωα(Ω))[α]=A*(ℑ,ωα*(𝔍))[α], and x∉(A-I)*(𝔍,ωα(Ω))[α]=A*(𝔍,ωα*(ℑ))[α]. This shows that x∉A*(ℑ,ωα*(𝔍))[α]∩A*(𝔍,ωα*(ℑ))[α]. Therefore, A*(ℑ∨𝔍,ωα(Ω))[α]⊃A*(ℑ,ωα*(𝔍))[α]∩A*(𝔍,ωα*(ℑ))[α].
Conversely. supposex∉A*(ℑ,ωα*(𝔍))[α]. Then, there exists  P∈ηωα*(𝔍)(xα), and I∈ℑ, such that A[α]⊂(P∨I)[α]. Because of P∈ωα*(𝔍,Ω)=ωα(Ω)∨𝔍, there is J∈𝔍, such that ωα(P-J)∈ηωα(xα). Hence, A[α]⊂(P∨I)[α]=(P-J)[α]∪(I∨J)[α]. This shows x∉A*(ℑ∨𝔍,ωα(Ω))[α]. Therefore, A*(ℑ∨𝔍,ωα(Ω))[α]⊂A*(ℑ,ωα*(𝔍))[α]. Similarly, we can prove A*(ℑ∨𝔍,ωα(Ω))[α]⊂A*(𝔍,ωα*(ℑ))[α]. So A*(ℑ∨𝔍,ωα(Ω))[α]⊂A*(ℑ,ωα*(𝔍))[α]∩A*(𝔍,ωα*(ℑ))[α]. The proof is completed.By Theorem20, we get two important results.Corollary 24.
Let(LX,ωα(Ω)) be an ωα-opos. Let  ℑ be an α-ideal, then A*(ℑ,ωα(Ω))[α]=A*(ℑ,ωα*(ℑ))[α], and as a result ωα*(ℑ)=ωα*(ωα*(ℑ))(ℑ).Proof.
By (2) in Theorem23, take ℑ=𝔍, clearly.Corollary 25.
Let(LX,ωα(Ω)) be an ωα-opos. Let ℑ,𝔍⊂LX be two α-ideals. Then,(1)
ωα*(ℑ∨𝔍)=ωα*(ωα*(𝔍))(ℑ)=ωα*(ωα*(ℑ))(𝔍),(2)
ωα*(ℑ∨𝔍)=ωα*(𝔍)∨ωα*(ℑ),(3)
ωα*(ℑ∩𝔍)=ωα*(𝔍)∩ωα*(ℑ).Proof.
(1) By (2) in Theorem23, easy.
(2) Sinceωα(Ω)⊂ωα*(𝔍) for every α-ideal 𝔍, by (1) and Theorem 20, we have ωα*(ℑ∨𝔍)=ωα*(ωα*(𝔍))(ℑ)=ωα*(𝔍)∨ℑ=ωα(Ω)∨ωα*(𝔍)∨ℑ=ωα*(𝔍)∨ωα*(ℑ).
(3) Clearly,ωα*(ℑ∩𝔍)⊂ωα*(𝔍)∩ωα*(ℑ). Conversely, P∈ωα*(𝔍)∩ωα*(ℑ). Then, P*(ℑ)[α]⊂P[α], and P*(𝔍)[α]⊂P[α]. So by (1) in Theorem 23, P*(ℑ∩𝔍)[α]=P*(ℑ)[α]∪P*(𝔍)[α]⊂P[α]. Therefore, P∈ωα*(ℑ∩𝔍).Theorem 26.
Let(LX,ωα(Ω)) be an ωα-opos, and let  ℑ be an α-ideal. A∈LX. Then, ωα*(A)[α]=ωα(A)[α] if and only if  ℑ⊂ωα(Ω).Proof.
Sufficiency. according to (4) in Theorem 16, ωα*(A)[α]⊂ωα(A)[α]. On the other hand, if x∉ωα*(A)[α], then y∉A[α]* and y∉A[α]. So there are P∈ηωα(yα), I∈ℑ, such that A[α]⊂(P∨I)[α]. Here, we may assume y∉I[α], since ℑ is an α-ideal. Therefore, ωα(A)[α]⊂ωα(P∨I)[α]=(P∨I)[α]. This implies y∉ωα(A)[α]. ωα*(A)[α]⊃ωα(A)[α].
Necessary. if A∈ℑ, then A[α]*=∅. Since A[α]=(A∨A*)[α]=ωα*(A)[α]=ωα(A)[α], A∈ωα(Ω).
Since(X,ωα(Ω)[α]) is an opos on X. ℑ[α] is an ideal on X. We have the following results.Theorem 27.
Let(LX,ωα(Ω)) be an ωα-opos, and let  ℑ be an α-ideal. Then, for every A∈LX, (A[α])*=A[α]*, and therefore ωα*(A)[α]=cl*(A[α]).Proof.
Since(8)x∉A[α]*⟺∃P∈ηωα(xα),I∈ℑ,A[α]⊂(P∨I)[α]⟺∃P[α]∈ωα(Ω)[α],x∈(P[α])′,I∈ℑ,A[α]∩(P[α])′=I[α]⟺x∉(A[α])*,
then (A[α])*=A[α]*, ωα*(A)[α]=(A∨A*)[α]=A[α]∪(A[α])*=cl*(A[α]).Corollary 28.
Let(LX,ωα(Ω)) be an ωα-opos, and let  ℑ be an α-ideal. Then, for every A∈LX, 𝒳A[α]*=𝒳A[α]*=𝒳(A[α])*.Proof.
x
∉
(
𝒳
A
[
α
]
*
)
[
α
]
⇔
there
exists
P
∈
η
ω
α
(
x
α
)
,
I
∈
ℑ
,
A
[
α
]
⊂
(
P
∨
I
)
[
α
]
⇔
x
∉
A
[
α
]
*
⇔
x
∉
(
𝒳
A
[
α
]
*
)
[
α
]. So 𝒳A[α]*=𝒳A[α]*. Besides, by Theorem 27, the later equation is easy.Corollary 29.
Let(X,Δ) be an opos  on X, ℐ⊂X an ideal. The induced ωα-opos  and α-ideal are denoted, respectively, by (LX,ωα(ωL(Δ))) and ℑ={𝒳I:I∈ℐ}. Then, ωα*(ℑ,ωα(ωL(Δ)))[α]=Δ*.Proof.
By Theorem27 and Corollary 28, ωα*(ℑ,ωα(ωL(Δ)))[α]⊂Δ*. On other hand, again by Corollary 28, for every U⊂X, ωα*(𝒳U)[α]=(𝒳U∨𝒳U*)[α]=U∪(𝒳U*)[α]=U∪U*=cl*(U).


## 5. Compatibility ofωα(Ω) with ℑ

Definition 30.
Let(LX,ωα(Ω)) be an ωα-opos. ℜ⊂ωα(Ω) is called an α-ωα-remote neighborhood family of A (briefly, α-ωα-RF of A, if for all x∈A[α], there is P∈ℜ, such that P∈ηωα(xα)).Definition 31.
Let(LX,ωα(Ω)) be an ωα-opos. Let ℑ be an α-ideal. ωα(Ω) is said to be compatible with ℑ, denoted by ωα(Ω)~αℑ, if for any A∈LX and x∈A[α], there are P∈ηωα(xα) and I∈ℑ, such that A[α]⊂(P∨I)[α], then A∈ℑ.Example 32.
In Example18, we know ωα={(0,0),(0,1/2),(0,1),(1/2,1/2),(1/2,1),(1,1/2),(1,1)}, ηωα(xα)={(0,0),(0,1/2),(0,1)},ηωα(yα)={(0,0)}. Take ℑ1=LX, ℑ2={(0,0)}, ℑ3={(0,0),(1/2,0),(1,0)},  ℑ4={(0,0),(0,1/2),(0,1)}.
It is easy to checkℑi,(i≤4) are all α-ideals, ωα(Ω)~αℑ1 and ωα(Ω)~αℑ2. Take G=(1/2,1/2), then x∈G[α] and R=(0,1/2)∈ηωα(xα),J=(1/2,0)∈ℑ3, such that G[α]⊂(R∨J)[α]. But G∉ℑ3, therefore ωα(Ω)≁αℑ3. Furthermore, for all A∈LX, if A[α]=∅ or A[α]={y}, then A∈ℑ4. If A[α]={x} or A[α]={x,y}, there does not exist E∈ηωα(xα),I∈ℑ4, such that A[α]⊂(R∨J)[α]. Therefore, ωα(Ω)~αℑ4.Theorem 33.
Let(LX,ωα(Ω)) be an ωα-opos, ℑ an ideal, ωα(Ω)~αℑ. Then, 𝔅=ωα*(ℑ).Proof.
For everyB∈ωα*(ℑ), B[α]*⊂B[α]. Put A=B-B*(ℑ). If x∈A[α], then x∉B[α]*. So there are P∈ηωα(xα) and I∈ℑ, such that B[α]⊂(P∨I)[α]. Thus, A[α]⊂B[α]⊂(P∨I)[α]. Since ωα(Ω)~αℑ, we get A∈ℑ. Besides, by (4) in Theorem 16,  B*∈ωα(Ω). Notice that B[α]=(A∨B*)[α]. Thus, B∈𝔅, and ωα*(ℑ)⊂𝔅. The reverse inclusion is obvious. Therefore, 𝔅=ωα*(ℑ).Lemma 34.
Let(LX,ωα(Ω)) be an ωα-opos, ℑ  an α-ideal. Then, for every A∈LX, (A-A*(ℑ))[α]∩(A-A*(ℑ))*(ℑ)[α]=∅.Proof.
Ifx∈(A-A*(ℑ))[α], so x∉A*(ℑ)[α]. That is, there are P∈ηωα(xα),I∈ℑ, such that A[α]⊂(P∨I)[α]. As (A-A*(ℑ))[α]⊂A[α], we have x∉(A-A*(ℑ))*(ℑ)[α]. Therefore, the conclusion holds.Theorem 35.
Let(LX,ωα(Ω)) be an ωα-opos, ℑ an ideal. Then, the following statements are equivalent.(1)
ωα(Ω)~αℑ.(2)
IfA∈LX has an α-ωα-RF ℜ, satisfying for all P∈ℜ, there is I∈ℑ, such that A[α]⊂(P∨I)[α], then A∈ℑ.(3)
For allA∈LX, (A∧A*(ℑ))[α]=∅⇒A∈ℑ.(4)
For allA∈LX, A-A*(ℑ)∈ℑ.(5)
For allA∈ωα*(ℑ), A-A*(ℑ)∈ℑ.(6)
For allA∈LX, if ∄∅≠B[α]⊂B*(ℑ)[α], B[α]⊂A[α], then A∈ℑ.Proof.
(1)⇒ (2) If A∈LX has a α-ωα-RF ℜ, satisfying the condition in (2), then for every x∈A[α], there are P∈ℜ and I∈ℑ such that P∈ηωα(xα) and A[α]⊂(P∨I)[α]. As ωα(Ω)~αℑ, we have A∈ℑ.
(2)⇒ (1) By Definitions 30 and 31, clearly.
(1)⇒ (3) For all A∈LX, (A∧A*(ℑ))[α]=∅. If A[α]=∅, then A∈ℑ. If x∈A[α], then x∉A*(ℑ)[α], there are P∈ηωα(xα) and I∈ℑ, such that A[α]⊂(P∨I)[α]. Because ωα(Ω)~αℑ, we have A∈ℑ.
(3)⇒ (1) if A∈LX satisfying, for every x∈A[α], there are P∈ηωα(xα) and I∈ℑ, such that A[α]⊂(P∨I)[α]. Then, x∉A*(ℑ)[α]. By (3), we have A∈ℑ. Hence, ωα(Ω)~αℑ.
(3)⇒ (4) By Lemma 34, obviously.
(4)⇒ (5) Directly.
(5)⇒ (1) If for every A∈LX and x∈A[α], there are P∈ηωα(xα) and I∈ℑ, such that A[α]⊂(P∨I)[α], this means (A∧A*(ℑ))[α]=∅. Put B=A∨A*, then B[α]*=(A∨A*)[α]*=A[α]*∪(A*)[α]*=A[α]*. So ωα*(B)[α]=(B∨B*(ℑ))[α]=B[α]. This means B∈ωα*(ℑ). By (5), we have B-B*∈ℑ. Since
(9)(B-B*)[α]=((A∨A*)-(A∨A*)*)[α]=(A∨A*)[α]-A[α]*=A[α],
we draw a conclusion: for every x∈A[α], x∉B[α]*=A[α]*. Therefore, A∈ℑ.
(4)⇒ (6) For any A∈LX, A=(A-A*)∨(A∧A*). So A[α]*=((A-A*)[α]*∪(A∧A*))[α]*. By (4), A-A*∈ℑ, so (A-A*)[α]*=∅. Thus A[α]*=(A∧A*)[α]*. Since (A∧A*)[α]⊂A[α]*, we have (A∧A*)[α]⊂(A∧A*)[α]*. Here by the assumption in (6), (A∧A*)[α]=∅. This implies A[α]=(A-A*)[α]*. Again by (4), A∈ℑ.
(6)⇒ (4) By Lemma 34, for any A∈LX, (A-A*)[α]∩(A-A*)[α]*=∅. Let's prove A-A* satisfies the condition in (6), In fact, if not, there is ∅≠B[α]⊂B[α]*, B[α]⊂(A-A*)[α], then B[α]⊂B[α]*⊂(A-A*)[α]*. We have ∅≠B[α]⊂(A-A*)[α]∩(A-A*)[α]*. A contradiction. Therefore, A-A*∈ℑ.Theorem 36.
Let(LX,ωα(Ω)) be an ωα-opos, ℑ be an α-ideal. Then the following statements are equivalent, and implied by ωα(Ω)~αℑ.(1)
For everyA∈LX, (A∧A*)[α]=∅⇒A[α]*=∅.(2)
For everyA∈LX, (A-A*)[α]*=∅.(3)
For everyA∈LX, (A∧A*)[α]*=A[α]*.Proof.
ω
α
(
Ω
)
~
α
ℑ
⇒
(
1
). By (3) in Theorem 35, If (A∧A*)[α]=∅⇒A∈ℑ. Thus A*(ℑ)[α]=∅.
(
1
)
⇒
(
2
) By Lemma 34, (A-A*(ℑ))[α]∩(A-A*(ℑ))*(ℑ)[α]=∅. Then by (1), we have (A-A*)[α]*=∅.
(
2
)
⇒
(
3
)
(
A
-
A
*
)
[
α
]
=
(
A
-
(
A
∧
A
*
)
)
[
α
]. By (8) in Theorem 16, (A-A*)[α]*=(A-(A∧A*))[α]*⊃A[α]*-(A∧A*)[α]*. According to (2), (A-A*)[α]*=∅. Thus, A[α]*=(A∧A*)[α]*.
(
3
)
⇒
(
1
) Straightforward.Theorem 37.
Let(LX,ωα(Ω)) be an ωα-opos, ℑ be an α-ideal. ωα(Ω)~αℑ. Then for every A∈ωα*(ℑ)  if and only if  there are B∈ωα(Ω) and I∈ℑ, such that A[α]=(B∨I)[α].Proof.
Necessity. suppose A∈ωα*(ℑ). So A[α]*⊂A[α] and A[α]⊂((A-A*)∨A*)[α]. By (4) in Theorem 2, and (5) in Theorem 17, we have A*∈ωα(Ω) and A-A*∈ℑ.
Sufficiency. suppose A[α]=(B∨I)[α], ωα(B)[α]=B[α] and I∈ℑ. Then A[α]*=B[α]*∪I[α]*=B[α]*⊂ωα(B)[α]=B[α]⊂A[α]. Therefore, A∈ωα*(ℑ).


## 6.ωα-Connectivity and ωα*-Connectivity

In this section, we shall introduce four kinds of connectedness and discuss their relations.Concepts ofωα*(ℑ)-separated sets and ωα*-connected are defined as:Definition 38.
Let(LX,ωα) and (LX,ωα*(ℑ)) be ωα-opos and ωα*(ℑ)-opos, respectively.(1)
A,B∈LX are called ωα-separated sets (resp. ωα*(ℑ)-separated set), if ωα(A)[α]∩B[α]=ωα(B)[α]∩A[α]=∅, (resp. ωα*(A)[α]∩B[α]=ωα*(B)[α]∩A[α]=∅).(2)
G∈LX is called an ωα-connected set (resp. ωα*-connected set), if there not exist ωα-separated sets B,C∈LX, such that, B[α]≠∅,C[α]≠∅, and G[α]=B[α]∪C[α], (resp. there not exist ωα*(ℑ)-separated sets B,C∈LX, such that, B[α]≠∅,C[α]≠∅, and G[α]=B[α]∪C[α]).Theorem 39.
Let(LX,ωα*(Ω)) be an ωα*-opos. A is ωα*-connected and A[α]*⊂B[α]*⊂ωα*(A)[α]. Then, B is ωα-connected.Proof.
LetB[α]=E[α]∪F[α], and E,F are ωα*-separated. Take
(10)E[α]Δ=A[α]∩E[α],F[α]Δ=A[α]∩F[α].
Then, A[α]=E[α]Δ∪F[α]Δ, and
(11)ωα(EΔ)[α]∩F[α]Δ=((EΔ)*∨EΔ)[α]∩F[α]Δ⊂ωα*(E)[α]∩F[α]=∅.
Similarly, we have ωα(FΔ)[α]∩E[α]Δ=∅. Thus, EΔ,FΔ are ωα*-separated.
AsA is ω[α]*-connected, so E[α]Δ=∅, or F[α]Δ=∅. Take E[α]Δ=∅, for example. Then, A[α]=F[α]Δ. This means A[α]⊂F[α]. As ωα*A[α]⊂ωα*F[α], and
(12)E[α]⊂E[α]∩ωα*(A)[α]⊂E[α]∩ωα*(F)[α]=∅.
Then, B is ωα*-connected.Theorem 40.
Let(X,Δ) be an opos, ℐ⊂X an ideal. Take ℑ={𝒳I:I∈ℐ}. The induced ωα-opos  and ωα*(ℑ)-opos  are denoted, respectively, by (LX,ωα(ωL(Δ))) and (LX,ωα*(ℑ,ωα(ωL(Δ)))). Then, for every A∈LX,(1)
A is ωα-connected in (LX,ωα(ωL(Δ))) if and only if    A[α]⊂X is connected in (X,Δ);(2)
A is ωα*-connected in (LX,ωα*(ℑ,ωα(ωL(Δ)))) if and only if  A[α]⊂X is connected in (X,Δ*);(3)
A[α]⊂X is connected in (X,Δ)⇒A[α]⊂X is connected in (X,Δ*).Proof.
(1)Necessity. suppose A[α] is not connected, then there are nonempty U,V⊂X, such that A[α]=U∪V, and Uσ-∩V=Vσ-∩U=∅. By Theorem 8, ωα(1U)[α]=Uσ-, ωα(1V)[α]=Vσ-. So
(13)ωα(1U)[α]∩(1V)[α]=ωα(1V)[α]∩(1U)[α]=∅.
This means 1U,1V are ωα-separated sets, and A[α]=(1U)[α]∪(1V)[α]. This is a contradiction with A is ωα-connected.
Sufficiency. suppose A is not ωα-connected. Then, there are ωα-separated sets B,C∈LX, such that B[α],C[α]≠∅, and A[α]=B[α]∪B[α]. Since ωα(B)∈ωα(ωL(Δ)), by Theorem 8, we have ωα(B)[α]∈Δσ-. Thus, (B[α])σ-⊂(ωα(B)[α])σ-=ωα(B)[α]. Therefore,
(14)(B[α])σ-∩C[α]⊂ωα(B)[α]∩C[α]=∅.
Similarly, we get (C[α])σ-∩B[α]=∅. This means A[α]⊂X is not connected. It is another contradiction.
(2) By Theorem27, similarly.
(3) Notice that(A[α])σ-⊃cl*(A[α]); easily.Theorem 41.
Let(LX,ωα*(ℑ)) be an ωα*(ℑ)-opos. A,B∈LX. Then,(1)
ωα-separated ⇒ωα*-separated,(2)
A and B are ωα*-separated ⇒A* and B* are ωα-separated ⇔A* and B* are ωα*-separated,(3)
A,B∈ωα(Ω)∩ℑ, then A and B are ωα-separated ⇔A and B are ωα*-separated ⇔A[α]∩B[α]=∅,(4)
ωα(Ω)~αℑ⇒A-A* and A* are ωα*-separated.Proof.
(1) Sinceωα*(A)[α]⊂ωα(A)[α] for every A∈LX; easy.
(2) By Theorem16 (4) and (5), we have ωα(A*)[α]=A[α]*⊂ωα*(A)[α] and ωα*(A*)[α]=(A*)[α]*∪A[α]*=A[α]*=ωα*(A)[α]; obvious.
(3) SinceA∈ωα(Ω)∩ℑ, A[α]*=∅ and ωα(A)[α]=A[α]=ωα*(A)[α]. The result follows.
(4) Sinceωα(Ω)~αℑ, A-A*∈ℑ. So ωα*(A-A*)[α]∩A[α]*=(A-A*)[α]∩A[α]*=ωα*(A*)[α]∩(A-A*)[α]=∅.Example 42.
In Example18, take B=(0,1/2),D=(1/2,0),E=(1/2,1/2). Then, ωα*(B)(𝔍)[α]=ωα(B)[α]=B[α], ωα*(D)(𝔍)[α]={x}. ωα(D)[α]={x,y}. Thus, B and D are ωα*(𝔍)-separated, but not ωα-separated. Therefore, E=(1/2,1/2) is ωα-connected, but not ωα*(𝔍)-connected. Moreover, B,E∈𝔍 implies B*(𝔍)[α]=∅, E*(𝔍)[α]=∅. So A* and B* are ωα-separated. However, since ωα*(𝔍)(B)[α]=B[α],ωα*(𝔍)(E)[α]=E[α], A and B are not ωα*-separated.Theorem 43.
Let(LX,ωα*(Ω)) be an ωα-opos, ℑ,𝔍  two α-ideals. A,B∈LX. Then,(1)
ifℑ⊂𝔍, ωα*(ℑ)-separated ⇒ωα*(𝔍)-separated,(2)
ωα*(ℑ∩𝔍)-separated if and only if  both ωα*(ℑ)-separated and ωα*(𝔍)-separated,(3)
ωα*(ℑ∨𝔍)-separated if and only if    ωα*(ωα*(ℑ))(𝔍)-separated if and only if    ωα*(ωα*(𝔍))(ℑ)-separated,(4)
ωα*(ωα*(ℑ))(ℑ)-separated ⇔ωα*(ℑ)-separated.Proof.
By Theorems19(1), 4.4 and Corollary 24, easy.Corollary 44.
Let(LX,ωα*(Ω)) be an ωα-opos, ℑ an α-ideals. Then,(1)
ωα*-connected ⇒ωα-connected,(2)
A* and B* are ωα-connected ⇔A* and B* are ωα*-connected ⇒A and B are ωα*-connected,(3)
A∈ωα(Ω)∩ℑ, then A is ωα-connected ⇔A is ωα*-connected.Corollary 45.
Let(LX,ωα*(Ω)) be an ωα-opos, ℑ,𝔍  two α-ideals. A,B∈LX. Then,(1)
ifℑ⊂𝔍, ωα*(𝔍)-connected ⇒ωα*(ℑ)-connected,(2)
ωα*(ℑ∩𝔍)-connected if and only if  both ωα*(ℑ)-connected and ωα*(𝔍)-connected,(3)
ωα*(ℑ∨𝔍)-connected if and only if  ωα*(ωα*(ℑ))(𝔍)-connected if and only if  ωα*(ωα*(𝔍))(ℑ)-connected.



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*Source: 582105-2012-08-29.xml*