# L-Fuzzy Connectedness in Stratified Order-Preserving Operator Spaces

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

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## Abstract

The concepts of stratified order-preserving
operators and stratified continuity are introduced inL-fuzzy topological spaces. Their basic properties are discussed, and their characteristic
properties are observed. The relationship between induced stratified
order-preserving topological spaces and general order-preserving operator topological spaces is studied. Finally, stratified connectedness is
introduced, and its properties are studied systematically.


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## Body

## 1. Introduction

Since the concept ofL-fuzzy order-preserving operator was introduced by Professor Chen in 2002 (see [1]), a lot of papers have being devoted to it (see [2–5]). In general topology, similar researches were established (see [5]). However, it has not yet been found in stratified space. As we know, it is quite different between L-fuzzy stratified topological spaces and L-fuzzy topological spaces. Thus, it is important and meaningful to introduce the concept.


## 2. Preliminaries

In this paper, a latticeL is called a completely distributive lattice with an order-reserving involution′, if the following conditions hold.(1)
L is a completely lattice. That is, the largest element 1∈L, and smallest elements 0∈L.            Besides, foralla,b∈L, a∨b∈L, a∧b∈L.(2)
Distributive law holds inL. That is, foralli∈I≠∅, forallj∈Ji≠∅, ai,j∈L, then
(1)⋀i∈I(⋁j∈Jiai,j)=⋁f∈∏i∈IJi(⋀i∈Iai,f(i)),⋁i∈I(⋀j∈Jiai,j)=⋀f∈∏i∈IJi(⋁i∈Iai,f(i)).(3)
There is a mapping′:L→L satisfies: foralla∈L, then (a′)′=a, and foralla,b∈L, a≤b implies b′≤a′ (see [6]).In this paper,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 [6]). 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) (see [7]). 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} (see [8]).The following are the concepts ofL-fuzzy order-preserving operator in L-fuzzy topological space and order-preserving operator in general topological space.LetX be an nonempty set. An operator ω:LX→LX is called an L-fuzzy order preserving operator in LX, if it satisfies: (1) ω(1X)=1X, (2) forallA,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 forallP∈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 (see [1]).LetX be an nonempty set, 𝒫(X) be the family of all subsets of X. An operator σ:𝒫(X)→𝒫(X) is called an order preserving operator in X, if it satisfies: (1) σ(X)=X, (2) forallA⊂B⊂X implies σ(A)⊂σ(B). A set A⊂X is called an σ-set, if σ(A)=A. The set of all σ-sets in X is denoted by Δ. And (X,Δ) is called an order-preserving operator space on X (briefly, opos). Let x∈X,P∈Δ, P is called an 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 an σ-adherent point of A, if forallP∈ση(x),  A⊄P. The union of all σ-adherent points of A is called the σ-closure of A, denoted by Aσ-. An 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 (see [5]).Let(X,Δ) be an opos, L be an fuzzy lattice, forallα∈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 topology in LX. The space (LX,ωL(Δ)) is called the induced L-fopos by (X,Δ) (see [5]).Considering the above definitions, if we say there is anL-fopos(LX,Ω) or an opos(X,Δ), we mean there is an order-preserving operator ω on LX, or σ on X. The two spaces are generated by them, respectively.Let(LX,δ) and (LY,τ) be two L-fuzzy topological spaces, f→:LX→LY be induced by simple mapping f:X→Y. If f→ and its reverse mapping f← satisfies:(1)
forallA∈LX,y∈Y, f→(A)(y)=∨{A(x)∣x∈X,f(x)=y};(2)
forallB∈LY,x∈X, f←(B)(x)=B(f(x)).Thenf→ is called an L-fuzzy mapping (see [6]).AnL-fuzzy mapping f→LX→LY is called homomorphism, if it satisfies:(1)
f→ is union preserving. That is, for {Ai∈LX,(i∈I)}⊂LX, f→(⋁i∈IAi)=⋁i∈If→(Ai);(2)
f→ is reserving involution preserving. That is, for B∈LY, f←(B′)=(f←(B))′ (see [6]).


## 3.ωα-Closed Set

Definition 1.
Let(LX,Ω) be an L-fopos, α∈M(L). An operator ωα:LX→LX is defined by: forallA∈LX,
(2)ωα(A)=∧{G∈Ωω-∣G[α]⊃A[α]}.
Since the operatorωα is related to α∈M(L), such kind of operator is generally called stratified order-preserving operator. Clearly, if ω is a closure operator, then ωα is Dα (see [9]).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).Proof.
(1), (2) are easy.
(3) by (1), we haveωα(A∨B)≥ωα(A)∨ωα(B). On the other hand,
(3)ωα(A)∨ωα(B)=(∧{G∈Ωω-∣G[α]⊃A[α]})∨(∧{H∈Ωω-∣H[α]⊃B[α]})=∧{G∨H∈Ωω-∣G[α]⊃A[α],H[α]⊃B[α]}≥∧{E∈Ωω-∣E[α]⊃A[α]∪B[α]}=ωα(A∨B).
(4)ForallA∈LX, we have
(4)ωα(ωα(A))=∧{G∈Ωω-∣G[α]⊃ωα(A)[α]}≥∧{G∈Ωω-∣G[α]⊃A[α]}=ωα(A).
On the other hand, as ωα(A)∈Ωω-, then by Definition 1, we have ωα(A)≤ωα(ωα(A)). Hence, (4) holds.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 ωα-order-preserving operator space, (briefly, ωα-opos).
By Definition3, if ω is the closure operator, then an ωα-closed set is a Dα-closed set (see [9]).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).Proof.
(1)  As(1X)[α]=X, it is easy to check ωα(1X)[α]=X=(1X)[α]. Thus, 1X∈ωα(Ω).
(2)  By Theorem2, if {Ai∣i=1,2,…,n}⊂ωα(Ω). We have ωα(⋁i=1nAi)=⋁i=1nωα(Ai). Then
(5)ωα(⋁i=1nAi)[α]=(⋁i=1nωα(Ai))[α]=⋃i=1nωα(Ai)[α]=⋃i=1n(Ai)[α]=(⋁i=1nAi)[α].
Hence, (2) holds.
(3)  If{Ai∣i∈I}⊂ωα(Ω), so
(6)ωα(⋀i∈IAi)[α]⊂(⋀i∈Iωα(Ai))[α]=⋂i∈Iωα(Ai)[α]=⋂i∈I(Ai)[α]=(⋀i∈IAi)[α].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). If there is A∈Ωω-, such that A[α]=∅. Then ωα(Ω) consists an L-fuzzy co-topology on LX.Proof.
By Theorem4, we only need to prove 0X∈ωα(Ω). In fact, as there is A∈Ωω-, such that A[α]=∅. Then
(7)ωα(0X)[α]=∧{G∈Ωω-∣G[α]⊃∅}=∅=(0X)[α].
Hence, 0X∈ωα(Ω).Remark 7.
In fact, the smallest fuzzy set0X is ω-closed. That is, (0X)ω-=0X. Otherwise, ∃xλ≤(0X)ω-, xλ≰0X. It means xλ is an ω-adherent point of 0X. However, for all P∈ωη(xλ)≠∅, 0X≤P. This is a contradiction with the hypothesis that xλ is an ω-adherent point of 0X.
Taking the fact above, we can conclude that given anL-fopos (LX,Ω), (LX,ωα(Ω)) is always an L-fuzzy co-topology.Corollary 8.
Let(LX,ωα(Ω)) be an ωα-opos. ωα(Ω) is an L-fuzzy co-topology on LX. Then ωα(Ω)[α] is a co-topology on X. So (X,ωα(Ω)[α]) is an opos on X. Moreover, one has forallA∈LX, (A[α])σω-=ω(A)[α].Proof.
We only prove(A[α])σω-=ω(A)[α]. By Theorem 2 (1), (A[α])σω-⊂(ω(A)[α])σω-. Again, by Theorem 2 (4), we have ω(A)[α]∈ωα(Ω)[α]. So (A[α])σω-⊂ω(A)[α].
Conversely, if there isx∈ω(A)[α], but x∉(A[α])σω-. Since(A[α])σω-=∩{H[α]∣H∈ωα(Ω),H[α]⊃A[α]}, there is H∈ωα(Ω), such that H[α]⊃A[α] and x∉H[α]. So we have H[α]=ωα(H)[α]⊃ωα(A)[α]. This is a contradiction, because x∈ω(A)[α]. Therefore, (A[α])σω-⊃ω(A)[α].Lemma 9.
Let(X,Δ) be an opos, (LX,ωL(Δ)) be an L-fopos induced by (X,Δ). E⊂X. Then E∈Δσ- if and only if χE∈(ωL(Δ))ω-.Proof.
For anyα∈M(L). If α≠1, then {x∈X∣χE≤α}=E′. If α=1, then {x∈X∣χE≤α}=X. Thus, by the definition of lower continuous function, we have E∈Δσ- if and only if χE∈(ωL(Δ))ω-.Theorem 10.
Let(X,Δ) be an opos, (LX,ωL(Δ)) be an L-fopos induced by (X,Δ). A∈LX, α∈M(L). Then A∈ωα(ωL(Δ)) if and only if A[α]∈Δσ-.Proof.
For anyG∈LX, by Lemma 9, we have G∈(ωL(Δ))ω- iff G[α]∈Δσ-. Let A∈ωα(ωL(Δ)). Then
(8)A[α]=ωα(A)[α]=(∧{G∈(ωL(Δ))ω-∣G[α]⊃A[α]})[α]=∩{G[α]∣G∈(ωL(Δ))ω-,G[α]⊃A[α]}=∩{G[α]∈Δω-∣G[α]⊃A[α]}=(A[α])σ-.
So A[α]∈Δσ-.
Conversely, ifA[α]∈Δσ-, by Lemma 9, we have χA[α]∈(ωL(Δ))ω-. And (χA[α])[α]=A[α], so
(9)ωα(A)[α]=∩{G[α]∣G∈(ωL(Δ))ω-,G[α]⊃A[α]}⊂(χA[α])[α]=A[α].
Hence, ωα(A)[α]=A[α]. This implies A∈ωα(ωL(Δ)).Theorem 11.
Let(X,Δ) be an opos, (LX,ωL(Δ)) is an L-fopos induced by (X,Δ). Then A∈ωL(Δ) iff forallα∈M(L),  A∈ωα(ωL(Δ)).Proof.
By Theorem10, we have
(10)A∈(ωL(Δ))′⟺A[α]∈Δσ-⟺A∈ωα(ωL(Δ)).Definition 12.
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 forallB∈ωα2(Ω2), then f←(B)∈ωα1(Ω1).Theorem 13.
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)
forallB∈LY, ωα1(f←(B))[α]⊂f←(ωα2(B))[α];(3)
forallA∈LX, f→(ωα1(A))[α]⊂ωα2(f→(A))[α].Proof.
(1)⇒(2) forallB∈LY, so ωα2(B)[α]⊃B[α], then f←(ωα2(B))[α]⊃f←(B)[α]. Therefore, ωα1(f←(ωα2(B)))[α]⊃ωα1(f←(B))[α].
On the other hand, by Theorem2, we get that ωα2(B)∈ωα2(Ω2). And by hypothesis, f→ is (ωα1(Ω1),ωα2(Ω2))-continuous, then, f←(ωα2(B))∈ωα1(Ω1). Hence, ωα1(f←(ωα2(B)))[α]=f←(ωα2(B))[α]. Therefore, we have
(11)ωα1(f←(B))[α]⊂ωα1(f←(ωα2(B)))[α]=f←(ωα2(B))[α].
(2)⇒(3)  forallA∈LX, so f→(A)∈LY. By (2), we get ωα1(f←(f→(A)))[α]⊂f←(ωα2(f→(A)))[α]. So,
(12)(f→f←(ωα2(f→(A))))[α]⊃f→(ωα1(f←(f→(A))))[α]⊃f→(ωα1(A))[α].
And obviously, (ωα2(f→(A)))[α]=(f→f←(ωα2(f→(A))))[α]. Thus, (2) holds.
(3)⇒(1). forallB∈ωα2(Ω2). By (3), we have
(13)f→(ωα1(f←(B)))[α]⊂ωα2(f→(f←(B)))[α]⊂ωα2(B)[α]=B[α].
So, ωα1(f←(B))[α]⊂f←(B)[α].
On the other hand, obviously,f←(B)[α]⊂ωα1(f←(B))[α]. Therefore, f←(B)[α]=ωα1(f←(B))[α]. This shows that f←(B)∈ωα1(Ω1).


## 4.ωα-Connectedness

Definition 14.
Let(LX,Ω) be an L-fopos, α∈M(L). A,B∈LX are called ωα-separated, if ωα(A)[α]∩B[α]=A[α]∩ωα(B)[α]=∅. Otherwise, A,B∈LX are called ωα-connected.
Clearly, ifω is the closure operator in LX, then ωα-connected is Dα-connected [8].Definition 15.
Let(LX,Ω) be an L-fopos, α∈M(L). A∈LX is called an ωα-connected set, if there not exist ωα-separated sets B,C∈LX, such that, B[α]≠∅,C[α]≠∅, and A[α]=B[α]∪B[α]. Specially, if 1X is ωα-connected, then, (LX,Ω) is called ωα-connected.Theorem 16.
Let(LX,Ω) be an L-fopos, α∈M(L). Then (LX,Ω) is not ωα-connected if and only if there are A,B∈ωα(Ω), such that, A[α]≠∅,B[α]≠∅, and A[α]∩B[α]=∅, A[α]∪B[α]=X.Proof.
Necessity. Let(LX,Ω) be not ωα-connected. That is, 1X is not ωα-connected. By Definition 15, there are two ωα-separated sets A,B∈LX, such that, A[α]≠∅,B[α]≠∅, and A[α]∪B[α]=(1X)[α]=X.
Since,A,B∈LX are ωα-separated, we have
(14)ωα(A)[α]=ωα(A)[α]∩X=ωα(A)[α]∩(A[α]∪B[α])=(ωα(A)[α]∩A[α])∪(ωα(A)[α]∩B[α])=A[α]∪∅=A[α].
Therefore,A[α]∩B[α]⊂ωα(A)[α]∩B[α]=∅. Hence, the necessity holds.
Sufficiency. TakeA,B∈ωα(Ω) satisfies the conditions stated in the theorem. So (1X)[α]=A[α]∪B[α]=X. As A[α]∩B[α]=∅, and ωα(A)[α]=A[α], we have ωα(A)[α]∩B[α]=A[α]∩B[α]=∅. Similarly, ωα(B)[α]∩A[α]=B[α]∩A[α]=∅. Hence, A,B are ωα-separated. By Definition 15, we have (LX,Ω) is not ωα-connected.Theorem 17.
Let(LX,Ω) is an L-fopos, α∈M(L). A be ωα-connected, then A[α]⊂B[α]⊂ωα(A)[α] implies B is ωα-connected.Proof.
SinceA is ωα-connected. A[α] is connected in (X,ωα(Ω)). By Theorem 20 in reference [10] (page 54) and Corollary 8, ωα(A)[α] is connected. So B[α] is connected. Therefore, B is ωα-connected.Theorem 18.
Let(LX,ωα(Ω)) be an ωα-opos. {At}t∈T⊂LX, At is ωα-connected, and there is s∈T, such that for any t∈T, At and As are ωα-connected. Then A=⋁t∈TAt is ωα-connected.Proof.
SinceAt is ωα-connected, At[α] is connected in (X,ωα(Ω)). As[α] and At[α] are collected for each s∈T. Then by Theorem 21 in reference [10] (page 54) and Corollary 8, A[α]=(⋁t∈TAt)[α]=⋃t∈TAt[α] is connected. Therefore, ⋁t∈TAt is ωα-connected.Theorem 19.
Let(X,Δ) be an opos, and (LX,ωL(Δ)) be the induced L-opos. Then A∈ωα(ωL(Δ)) is ωα-connected if and only if A[α]⊂X is connected.Proof.
Necessity. LetA∈ωα(ωL(Δ)) be ωα-connected. Suppose that  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 10, ωα(1U)[α]=Uσ-,  ωα(1V)[α]=Vσ-. So
(15)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]. A contradiction.
Sufficiency. SupposeA∈ωα(ωL(Δ)) 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 10, we have ωα(B)[α]∈Δσ-. Thus, (B[α])σ-⊂(ωα(B)[α])σ-=ωα(B)[α]. Therefore,
(16)(B[α])σ-∩C[α]⊂ωα(B)[α]∩C[α]=∅.
Similarly, we get that (B[α])σ-∩C[α]=∅. This means A[α]⊂X is not connected. A contradiction.Theorem 20.
Let(LX,Ωi)(i=1,2) be two L-oposs, f→:LX→LX be (ω1,ω2)-continuous, A∈LX be ωα1-connected. Then f→(A) is ωα2-connected.Proof.
Letf→(A)[α]=B[α]∪C[α], ωα2(B)[α]∩C[α]=ωα2(C)[α]∩B[α]=∅. Since f-1(B[α])=f←(B)[α]. Take
(17)E[α]=f-1(B[α]),F[α]=f-1(C[α]),
we have
(18)A[α]⊂(f←f→(A))[α]=f←(B)[α]∪f←(C)[α]=E[α]∪F[α].
Besides, by
(19)ωα1(E)[α]=ωα1(f←(B))[α]⊂f←(ωα2(B))[α],ωα1(F)[α]=ωα1(f←(C))[α]⊂f←(ωα2(C))[α]
we get
(20)ωα1(E)[α]∩F[α]⊂f←(ωα2(B))[α]∩f←(ωα2(C))[α]=f-1(ωα2(B)[α]∩C[α])=f-1(∅)=∅,ωα1(F)[α]∩E[α]⊂f←(ωα2(C))[α]∩f←(ωα2(B))[α]=f-1(ωα2(C)[α]∩B[α])=f-1(∅)=∅.
DenoteG[α]=A[α]∪E[α], H[α]=A[α]∪F[α]. Then A[α]=G[α]∪H[α], and
(21)ωα1(G)[α]∩H[α]=ωα1(H)[α]∩G[α]=∅.
SinceA is ωα1-connected, then G[α]=∅, or H[α]=∅. Take the former, for example, A[α]=H[α]⊂F[α]. Thus,
(22)f→(A)[α]⊂f→(F)[α]=(f→f←(C))[α]⊂C[α].
Thus,
(23)B[α]=B[α]∩f→(A)[α]⊂B[α]∩C[α]⊂B[α]∩ωα2(C)[α]=∅.Theorem 21.
Let(LX,Ω) be an L-opos, A∈LX, for allα∈M(L), M*(A)={a∈M*(LX)∣a[α]∈A[α]}. ThenA is ωα-connected if and only if for all a,b∈M*(A),  for all P:M*(A)→∪{ηωα(x)∣x∈M*(A)}, there are finite molecular x1=a,x2,…,xn=b∈M*(A), such that A[α]⊄P(xj)[α]∪P(xj)[α],(j=1,2,…,n-1).Proof.
Necessity. Suppose the result does not holds. That is, there isa,b∈M*(A), a≠b. And there is a mapping
(24)P:M*(A)⟶∪{ηωα(x)x∈M*(A)},
such that for any finite x1=a,x2,…,xn=b∈M*(A), A[α]⊄P(xj)[α]∪P(xj)[α] does not holds.
For the convenient sake,a,b∈M*(A), a and b are called joined, if there are finite x1=a,x2,…,xn=b∈M*(A), such that A[α]⊄P(xj)[α]∪P(xj)[α], denoted by a~b. Otherwise, a and b are called disjoined, denoted by a≁b. Let
(25)μ={x∈M*(A)a~x},ν={x∈M*(A)a≁x},
and B=∨μ,C=∨ν. Obviously, a~a, so a∈μ,a≤B. and a≁b, so b∈ν,b≤C. Hence, B[α]≠∅,C[α]≠∅. Since foralla∈M*(A), a∈μ, or a∈ν. So A[α]=B[α]∪C[α]. Next, let us prove B,C are not ωα-connected.
Suppose thatωα(B)[α]∩C[α]≠∅, then there is x[α]∈ωα(B)[α]∩C[α]. By x[α]∈ωα(B)[α], we have B[α]⊄P(x)[α]. So there is y∈μ, such that y[α]∉P(y)[α], and y[α]∈B[α]⊂A[α], hence A[α]⊄P(x)[α]∪P(y)[α]. This means that y~a, and x~a.
On the other hand, byx[α]∈C[α], then C[α]⊄P(x)[α], so there is z∈ν, such that z[α]∉P(x)[α]. Therefore, z[α]∉P(x)[α]∪P(x)[α]. And by z[α]∈C[α]⊂A[α], we have A[α]⊄P(x)[α]∪P(z)[α]. This means x~a and a~z. This is a contradiction with z∈ν. Therefore ωα(B)[α]∩C[α]=∅. Similarly, we get that ωα(C)[α]∩B[α]=∅.
Sufficiency. SupposeA is not ωα-connected. Then there are B,C∈LX, such that B,C are ωα-separated, and B[α]≠∅,C[α]≠∅, A[α]=B[α]∪C[α]. Define a mapping
(26)P:M*(A)⟶∪{ηωα(x)x∈M*(A)},P(x)={ωα(C),x[α]∈B[α],ωα(B),x[α]∈C[α].
Then forallx∈M*(A), we have x≰P(x), P(x)∈ωα(Ω).
Takea[α]∈B[α],b[α]∈C[α]. Then for any finite x1=a,x2,…,xn=b∈M*(A), xi[α]∈B[α], or xi[α]∈C[α]. Thus, P(xi)[α]∈ωα(C), or P(xi)[α]∈ωα(B), (i=1,2,…,n). However, since P(x1)[α]∈ωα(C),P(xn)[α]∈ωα(B), there is 1≤j≤n-1, such that P(xj)=ωα(C),P(xj+1)=ωα(B). Therefore, A[α]=B[α]∪C[α]⊂P(xj)[α]∪P(xj+1)[α]. A contradiction.



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*Source: 925948-2012-02-20.xml*