𝐿-Fuzzy Connectedness in Stratified Order-Preserving Operator Spaces

Abstract

The concepts of stratified order-preserving operators and stratified continuity are introduced in 𝐿-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.

1. Introduction

Since the concept of 𝐿-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 lattice 𝐿 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 in 𝐿. That is, forall𝑖∈𝐼≠∅, forall𝑗∈𝐽_𝑖≠∅, 𝑎_𝑖,𝑗∈𝐿, then _{𝑖∈𝐼𝑗∈𝐽𝑖}𝑎_𝑖,𝑗=_{𝑓∈∏𝑖∈𝐼𝐽𝑖𝑖∈𝐼}𝑎_{𝑖,𝑓(𝑖)},_{𝑖∈𝐼𝑗∈𝐽𝑖}𝑎_𝑖,𝑗=_{𝑓∈∏𝑖∈𝐼𝐽𝑖𝑖∈𝐼}𝑎_{𝑖,𝑓(𝑖)}. (1)(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. 𝐿^𝑋 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 𝑥_𝛼 is called a molecule in 𝐿^𝑋. The set of all molecules in 𝐿^𝑋 is denoted by 𝑀^∗(𝐿^𝑋) (see [7]). If 𝐴∈𝐿^𝑋, 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 of 𝐿-fuzzy order-preserving operator in 𝐿-fuzzy topological space and order-preserving operator in general topological space.

Let 𝑋 be an nonempty set. An operator 𝜔∶𝐿^𝑋→𝐿^𝑋 is called an 𝐿-fuzzy order preserving operator in 𝐿^𝑋, if it satisfies: (1) 𝜔(1_𝑋)=1_𝑋, (2) forall𝐴,𝐵∈𝐿^𝑋 and 𝐴≤𝐵 implies 𝜔(𝐴)≤𝜔(𝐵). A set 𝐴∈𝐿^𝑋 is called an 𝜔-set, if 𝜔(𝐴)=𝐴. The set of all 𝜔-sets in 𝐿^𝑋 is denoted by Ω. And (𝐿^𝑋,Ω) is called an order-preserving operator space (briefly, 𝐿-𝑓opos). A molecule 𝑥_𝛼∈𝑀^∗(𝐿^𝑋),𝑃∈Ω, 𝑃 is called an 𝜔-remote neighborhood of 𝑥_𝛼, if 𝑥_𝛼∉𝑃. The set of all 𝜔-remote neighborhood of 𝑥_𝛼 is denoted by 𝜔𝜂(𝑥_𝛼). Let 𝑥_𝛼∈𝑀^∗(𝐿^𝑋),𝐴∈𝐿^𝑋, 𝑥_𝛼 is called an 𝜔-adherent point of 𝐴, if forall𝑃∈𝜔𝜂(𝑥_𝛼), 𝐴∉𝑃. The union of all 𝜔-adherent points of 𝐴 is called the 𝜔-closure of 𝐴, denoted by 𝐴⁻_𝜔. A set 𝐴∈𝐿^𝑋 is called 𝜔-closed, if 𝐴⁻_𝜔=𝐴. The set of all 𝜔-closed sets in 𝐿^𝑋 is denoted by Ω⁻_𝜔. Ω⁻_𝜔 is finite union and infinite intersection preserving (see [1]).

Let 𝑋 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 𝐴⁻_𝜎. An set 𝐴⊂𝑋 is called 𝜎-closed, if 𝐴⁻_𝜎=𝐴. The set of all 𝜎-closed sets in 𝑋 is denoted by Δ⁻_𝜎. Δ⁻_𝜎 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}∈Δ⁻_𝜎. Then, the set of all the lower continuous functions, denoted by 𝜔_𝐿(Δ) consists an 𝐿-fuzzy topology in 𝐿^𝑋. The space (𝐿^𝑋,𝜔_𝐿(Δ)) is called the induced 𝐿-𝑓opos by (𝑋,Δ) (see [5]).

Considering the above definitions, if we say there is an 𝐿-𝑓opos(𝐿^𝑋,Ω) or an opos(𝑋,Δ), we mean there is an order-preserving operator 𝜔 on 𝐿^𝑋, or 𝜎 on 𝑋. The two spaces are generated by them, respectively.

Let (𝐿^𝑋,𝛿) and (𝐿^𝑌,𝜏) be two 𝐿-fuzzy topological spaces, 𝑓^→∶𝐿^𝑋→𝐿^𝑌 be induced by simple mapping 𝑓∶𝑋→𝑌. If 𝑓^→ and its reverse mapping 𝑓^← satisfies:(1)forall𝐴∈𝐿^𝑋,𝑦∈𝑌, 𝑓^→(𝐴)(𝑦)=∨{𝐴(𝑥)∣𝑥∈𝑋,𝑓(𝑥)=𝑦};(2)forall𝐵∈𝐿^𝑌,𝑥∈𝑋, 𝑓^←(𝐵)(𝑥)=𝐵(𝑓(𝑥)).

Then 𝑓^→ is called an 𝐿-fuzzy mapping (see [6]).

An 𝐿-fuzzy mapping 𝑓^→𝐿^𝑋→𝐿^𝑌 is called homomorphism, if it satisfies:(1)𝑓^→ is union preserving. That is, for {𝐴_𝑖∈𝐿^𝑋,(𝑖∈𝐼)}⊂𝐿^𝑋, 𝑓^→(⋁_𝑖∈𝐼𝐴_𝑖)=⋁_𝑖∈𝐼𝑓^→(𝐴_𝑖);(2)𝑓^→ is reserving involution preserving. That is, for 𝐵∈𝐿^𝑌, 𝑓^←(𝐵^′)=(𝑓^←(𝐵))′ (see [6]).

3. 𝜔_𝛼-Closed Set

Definition 1. Let (𝐿^𝑋,Ω) be an 𝐿-𝑓opos, 𝛼∈𝑀(𝐿). An operator 𝜔_𝛼∶𝐿^𝑋→𝐿^𝑋 is defined by: forall𝐴∈𝐿^𝑋, 𝜔_𝛼(𝐴)=∧𝐺∈Ω⁻_𝜔∣𝐺_[𝛼]⊃𝐴_[𝛼]. (2)
Since the operator 𝜔_𝛼 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 𝜔_𝛼 is 𝐷_𝛼 (see [9]).

Theorem 2. Let (𝐿^𝑋,Ω) be an 𝐿-𝑓𝑜𝑝𝑜𝑠, 𝛼∈𝑀(𝐿). 𝐴,𝐵∈𝐿^𝑋, then the following statements hold:(1)𝐴_[𝛼]⊂𝐵_[𝛼] implies 𝜔_𝛼(𝐴)≤𝜔_𝛼(𝐵);(2)𝐴_[𝛼]⊂𝜔_𝛼(𝐴)_[𝛼];(3)𝜔_𝛼(𝐴∨𝐵)=𝜔_𝛼(𝐴)∨𝜔_𝛼(𝐵);(4)𝜔_𝛼(𝜔_𝛼(𝐴))=𝜔_𝛼(𝐴).

Proof. (1), (2) are easy.
(3) by (1), we have 𝜔_𝛼(𝐴∨𝐵)≥𝜔_𝛼(𝐴)∨𝜔_𝛼(𝐵). On the other hand, 𝜔_𝛼(𝐴)∨𝜔_𝛼(𝐵)=∧𝐺∈Ω⁻_𝜔∣𝐺_[𝛼]⊃𝐴_[𝛼]∨∧𝐻∈Ω⁻_𝜔∣𝐻_[𝛼]⊃𝐵_[𝛼]=∧𝐺∨𝐻∈Ω⁻_𝜔∣𝐺_[𝛼]⊃𝐴_[𝛼],𝐻_[𝛼]⊃𝐵_[𝛼]≥∧𝐸∈Ω⁻_𝜔∣𝐸_[𝛼]⊃𝐴_[𝛼]∪𝐵_[𝛼]=𝜔_𝛼(𝐴∨𝐵). (3)
(4) Forall𝐴∈𝐿^𝑋, we have 𝜔_𝛼𝜔_𝛼(𝐴)=∧𝐺∈Ω⁻_𝜔∣G_[𝛼]⊃𝜔_𝛼(𝐴)_[𝛼]≥∧𝐺∈Ω⁻_𝜔∣𝐺_[𝛼]⊃𝐴_[𝛼]=𝜔_𝛼(𝐴). (4) On the other hand, as 𝜔_𝛼(𝐴)∈Ω⁻_𝜔, then by Definition 1, we have 𝜔_𝛼(𝐴)≤𝜔_𝛼(𝜔_𝛼(𝐴)). Hence, (4) holds.

Definition 3. Let (𝐿^𝑋,Ω) be an 𝐿-𝑓opos, 𝛼∈𝑀(𝐿). A set 𝐴∈𝐿^𝑋 is called 𝜔_𝛼-closed, if 𝜔_𝛼(𝐴)_[𝛼]=𝐴_[𝛼]. The set of all 𝜔_𝛼-closed sets in 𝐿^𝑋 is denoted by 𝜔_𝛼(Ω). (𝐿^𝑋,𝜔_𝛼(Ω)) is called 𝜔_𝛼-order-preserving operator space, (briefly, 𝜔_𝛼-opos).
By Definition 3, if 𝜔 is the closure operator, then an 𝜔_𝛼-closed set is a 𝐷_𝛼-closed set (see [9]).

Theorem 4. Let (𝐿^𝑋,Ω) be an 𝐿-𝑓opos, 𝛼∈𝑀(𝐿). Then(1)1_𝑋∈𝜔_𝛼(Ω);(2)If {𝐴_𝑖∣𝑖=1,2,…,𝑛}⊂𝜔_𝛼(Ω), then ⋁^𝑛_𝑖=1𝐴_𝑖∈𝜔_𝛼(Ω);(3)If {𝐴_𝑖∣𝑖∈𝐼}⊂𝜔_𝛼(Ω), then ⋀_𝑖∈𝐼𝐴_𝑖∈𝜔_𝛼(Ω).

Proof. (1) As (1_𝑋)_[𝛼]=𝑋, it is easy to check 𝜔_𝛼(1_𝑋)_[𝛼]=𝑋=(1_𝑋)_[𝛼]. Thus, 1_𝑋∈𝜔_𝛼(Ω).
(2) By Theorem 2, if {𝐴_𝑖∣𝑖=1,2,…,𝑛}⊂𝜔_𝛼(Ω). We have 𝜔_𝛼(⋁^𝑛_𝑖=1𝐴_𝑖)=⋁^𝑛_𝑖=1𝜔_𝛼(𝐴_𝑖). Then 𝜔_𝛼^𝑛_𝑖=1𝐴_𝑖[𝛼]=^𝑛_𝑖=1𝜔_𝛼𝐴_𝑖[𝛼]=^𝑛_𝑖=1𝜔_𝛼𝐴_𝑖[𝛼]=^𝑛_𝑖=1𝐴_𝑖[𝛼]=^𝑛_𝑖=1𝐴_𝑖[𝛼]. (5) Hence, (2) holds.
(3) If {𝐴_𝑖∣𝑖∈𝐼}⊂𝜔_𝛼(Ω), so 𝜔_{𝛼𝑖∈𝐼}𝐴_𝑖[𝛼]⊂_𝑖∈𝐼𝜔_𝛼𝐴_𝑖[𝛼]=_𝑖∈𝐼𝜔_𝛼𝐴_𝑖[𝛼]=_𝑖∈𝐼𝐴_𝑖[𝛼]=_𝑖∈𝐼𝐴_𝑖[𝛼]. (6)

Remark 5. The theorem shows that 𝜔_𝛼 is finite union and infinite intersection preserving.

Corollary 6. Let (𝐿^𝑋,Ω) be an 𝐿-𝑓𝑜𝑝𝑜𝑠, 𝛼∈𝑀(𝐿). If there is 𝐴∈Ω⁻_𝜔, such that 𝐴_[𝛼]=∅. Then 𝜔_𝛼(Ω) consists an 𝐿-fuzzy co-topology on 𝐿^𝑋.

Proof. By Theorem 4, we only need to prove 0_𝑋∈𝜔_𝛼(Ω). In fact, as there is 𝐴∈Ω⁻_𝜔, such that 𝐴_[𝛼]=∅. Then 𝜔_𝛼0_𝑋[𝛼]=∧𝐺∈Ω⁻_𝜔∣𝐺_[𝛼]⊃∅=∅=0_𝑋[𝛼]. (7) Hence, 0_𝑋∈𝜔_𝛼(Ω).

Remark 7. In fact, the smallest fuzzy set 0_𝑋 is 𝜔-closed. That is, (0_𝑋)⁻_𝜔=0_𝑋. Otherwise, ∃𝑥_𝜆≤(0_𝑋)⁻_𝜔, 𝑥_𝜆≰0_𝑋. It means 𝑥_𝜆 is an 𝜔-adherent point of 0_𝑋. However, for all 𝑃∈𝜔𝜂(𝑥_𝜆)≠∅, 0_𝑋≤𝑃. This is a contradiction with the hypothesis that 𝑥_𝜆 is an 𝜔-adherent point of 0_𝑋.
Taking the fact above, we can conclude that given an 𝐿-fopos (𝐿^𝑋,Ω), (𝐿^𝑋,𝜔_𝛼(Ω)) is always an 𝐿-fuzzy co-topology.

Corollary 8. Let (𝐿^𝑋,𝜔_𝛼(Ω)) be an 𝜔_𝛼-𝑜𝑝𝑜𝑠. 𝜔_𝛼(Ω) is an 𝐿-fuzzy co-topology on 𝐿^𝑋. Then 𝜔_𝛼(Ω)_[𝛼] is a co-topology on 𝑋. So (𝑋,𝜔_𝛼(Ω)_[𝛼]) is an 𝑜𝑝𝑜𝑠 on 𝑋. Moreover, one has forall𝐴∈𝐿^𝑋, (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]∈𝜔_𝛼(Ω)_[𝛼]. So (𝐴_[𝛼])⁻_𝜎𝜔⊂𝜔(𝐴)_[𝛼].
Conversely, if there is 𝑥∈𝜔(𝐴)_[𝛼], but 𝑥∉(𝐴_[𝛼])⁻_𝜎𝜔. Since(𝐴_[𝛼])⁻_𝜎𝜔=∩{𝐻_[𝛼]∣𝐻∈𝜔_𝛼(Ω),𝐻_[𝛼]⊃𝐴_[𝛼]}, there is 𝐻∈𝜔_𝛼(Ω), such that 𝐻_[𝛼]⊃𝐴_[𝛼] and 𝑥∉𝐻_[𝛼]. So we have 𝐻_[𝛼]=𝜔_𝛼(𝐻)_[𝛼]⊃𝜔_𝛼(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, (𝐿^𝑋,𝜔_𝐿(Δ)) be an 𝐿-𝑓𝑜𝑝𝑜𝑠 induced by (𝑋,Δ). 𝐸⊂𝑋. Then 𝐸∈Δ⁻_𝜎 if and only if 𝜒_𝐸∈(𝜔_𝐿(Δ))⁻_𝜔.

Proof. For any 𝛼∈𝑀(𝐿). If 𝛼≠1, then {𝑥∈𝑋∣𝜒_𝐸≤𝛼}=𝐸′. If 𝛼=1, then {𝑥∈𝑋∣𝜒_𝐸≤𝛼}=𝑋. Thus, by the definition of lower continuous function, we have 𝐸∈Δ⁻_𝜎 if and only if 𝜒_𝐸∈(𝜔_𝐿(Δ))⁻_𝜔.

Theorem 10. Let (𝑋,Δ) be an 𝑜𝑝𝑜𝑠, (𝐿^𝑋,𝜔_𝐿(Δ)) be an 𝐿-𝑓𝑜𝑝𝑜𝑠 induced by (𝑋,Δ). 𝐴∈𝐿^𝑋, 𝛼∈𝑀(𝐿). Then 𝐴∈𝜔_𝛼(𝜔_𝐿(Δ)) if and only if 𝐴_[𝛼]∈Δ⁻_𝜎.

Proof. For any 𝐺∈𝐿^𝑋, by Lemma 9, we have 𝐺∈(𝜔_𝐿(Δ))⁻_𝜔 iff 𝐺_[𝛼]∈Δ⁻_𝜎. Let 𝐴∈𝜔_𝛼(𝜔_𝐿(Δ)). Then 𝐴_[𝛼]=𝜔_𝛼(𝐴)_[𝛼]=∧𝐺∈𝜔_𝐿(Δ)⁻_𝜔∣𝐺_[𝛼]⊃𝐴_[𝛼][𝛼]=∩𝐺_[𝛼]∣𝐺∈𝜔_𝐿(Δ)⁻_𝜔,𝐺_[𝛼]⊃𝐴_[𝛼]=∩𝐺_[𝛼]∈Δ⁻_𝜔∣𝐺_[𝛼]⊃𝐴_[𝛼]=𝐴_[𝛼]⁻_𝜎. (8) So 𝐴_[𝛼]∈Δ⁻_𝜎.
Conversely, if 𝐴_[𝛼]∈Δ⁻_𝜎, by Lemma 9, we have 𝜒_𝐴[𝛼]∈(𝜔_𝐿(Δ))⁻_𝜔. And (𝜒_𝐴[𝛼])_[𝛼]=𝐴_[𝛼], so 𝜔_𝛼(𝐴)_[𝛼]=∩𝐺_[𝛼]∣𝐺∈𝜔_𝐿(Δ)⁻_𝜔,𝐺_[𝛼]⊃𝐴_[𝛼]⊂𝜒_{𝐴[𝛼][𝛼]}=𝐴_[𝛼]. (9) Hence, 𝜔_𝛼(𝐴)_[𝛼]=𝐴_[𝛼]. This implies 𝐴∈𝜔_𝛼(𝜔_𝐿(Δ)).

Theorem 11. Let (𝑋,Δ) be an 𝑜𝑝𝑜𝑠, (𝐿^𝑋,𝜔_𝐿(Δ)) is an 𝐿-𝑓𝑜𝑝𝑜𝑠 induced by (𝑋,Δ). Then 𝐴∈𝜔_𝐿(Δ) iff forall𝛼∈𝑀(𝐿), 𝐴∈𝜔_𝛼(𝜔_𝐿(Δ)).

Proof. By Theorem 10, we have 𝐴∈𝜔_𝐿(Δ)′⟺𝐴_[𝛼]∈Δ⁻_𝜎⟺𝐴∈𝜔_𝛼𝜔_𝐿(Δ). (10)

Definition 12. Let (𝐿^𝑋,𝜔¹_𝛼(Ω₁)), (𝐿^𝑌,𝜔²_𝛼(Ω₂)) be 𝜔¹_𝛼-𝑜𝑝𝑜𝑠, 𝜔²_𝛼-𝑜𝑝𝑜𝑠, respectively. An 𝐿-fuzzy homomorphism 𝑓^→∶𝐿^𝑋→𝐿^𝑌 is called (𝜔¹_𝛼(Ω₁),𝜔²_𝛼(Ω₂))-continuous, if forall𝐵∈𝜔²_𝛼(Ω₂), then 𝑓^←(𝐵)∈𝜔¹_𝛼(Ω₁).

Theorem 13. Let (𝐿^𝑋,𝜔¹_𝛼(Ω₁)), (𝐿^𝑌,𝜔²_𝛼(Ω₂)) be 𝜔¹_𝛼-𝑜𝑝𝑜𝑠, 𝜔²_𝛼-𝑜𝑝𝑜𝑠, respectively. 𝑓^→∶𝐿^𝑋→𝐿^𝑌 is an 𝐿-fuzzy mapping. Then the following statements are equivalent.(1)𝑓^→ is (𝜔¹_𝛼(Ω₁),𝜔²_𝛼(Ω₂))-continuous;(2)forall𝐵∈𝐿^𝑌, 𝜔¹_𝛼(𝑓^←(𝐵))_[𝛼]⊂𝑓^←(𝜔²_𝛼(𝐵))_[𝛼];(3)forall𝐴∈𝐿^𝑋, 𝑓^→(𝜔¹_𝛼(𝐴))_[𝛼]⊂𝜔²_𝛼(𝑓^→(𝐴))_[𝛼].

Proof. (1)⇒(2) forall𝐵∈𝐿^𝑌, so 𝜔²_𝛼(𝐵)_[𝛼]⊃𝐵_[𝛼], then 𝑓^←(𝜔²_𝛼(𝐵))_[𝛼]⊃𝑓^←(𝐵)_[𝛼]. Therefore, 𝜔¹_𝛼(𝑓^←(𝜔²_𝛼(𝐵)))_[𝛼]⊃𝜔¹_𝛼(𝑓^←(𝐵))_[𝛼].
On the other hand, by Theorem 2, we get that 𝜔²_𝛼(𝐵)∈𝜔²_𝛼(Ω₂). And by hypothesis, 𝑓^→ is (𝜔¹_𝛼(Ω₁),𝜔²_𝛼(Ω₂))-continuous, then, 𝑓^←(𝜔²_𝛼(𝐵))∈𝜔¹_𝛼(Ω₁). Hence, 𝜔¹_𝛼(𝑓^←(𝜔²_𝛼(𝐵)))_[𝛼]=𝑓^←(𝜔²_𝛼(𝐵))_[𝛼]. Therefore, we have 𝜔¹_𝛼𝑓^←(𝐵)_[𝛼]⊂𝜔¹_𝛼𝑓^←𝜔²_𝛼(𝐵)_[𝛼]=𝑓^←𝜔²_𝛼(𝐵)_[𝛼]. (11)
(2)⇒(3) forall𝐴∈𝐿^𝑋, so 𝑓^→(𝐴)∈𝐿^𝑌. By (2), we get 𝜔¹_𝛼(𝑓^←(𝑓^→(𝐴)))_[𝛼]⊂𝑓^←(𝜔²_𝛼(𝑓^→(𝐴)))_[𝛼]. So, 𝑓^→𝑓^←𝜔²_𝛼𝑓^→(𝐴)_[𝛼]⊃𝑓^→𝜔¹_𝛼𝑓^←𝑓^→(𝐴)_[𝛼]⊃𝑓^→𝜔¹_𝛼(𝐴)_[𝛼]. (12) And obviously, (𝜔²_𝛼(𝑓^→(𝐴)))_[𝛼]=(𝑓^→𝑓^←(𝜔²_𝛼(𝑓^→(𝐴))))_[𝛼]. Thus, (2) holds.
(3)⇒(1). forall𝐵∈𝜔²_𝛼(Ω₂). By (3), we have 𝑓^→𝜔¹_𝛼𝑓^←(𝐵)_[𝛼]⊂𝜔²_𝛼𝑓^→𝑓^←(𝐵)_[𝛼]⊂𝜔²_𝛼(𝐵)_[𝛼]=𝐵_[𝛼]. (13) So, 𝜔¹_𝛼(𝑓^←(𝐵))_[𝛼]⊂𝑓^←(𝐵)_[𝛼].
On the other hand, obviously, 𝑓^←(𝐵)_[𝛼]⊂𝜔¹_𝛼(𝑓^←(𝐵))_[𝛼]. Therefore, 𝑓^←(𝐵)_[𝛼]=𝜔¹_𝛼(𝑓^←(𝐵))_[𝛼]. This shows that 𝑓^←(𝐵)∈𝜔¹_𝛼(Ω₁).

4. 𝜔_𝛼-Connectedness

Definition 14. Let (𝐿^𝑋,Ω) be an 𝐿-𝑓opos, 𝛼∈𝑀(𝐿). 𝐴,𝐵∈𝐿^𝑋 are called 𝜔_𝛼-separated, if 𝜔_𝛼(𝐴)_[𝛼]∩𝐵_[𝛼]=𝐴_[𝛼]∩𝜔_𝛼(𝐵)_[𝛼]=∅. Otherwise, 𝐴,𝐵∈𝐿^𝑋 are called 𝜔_𝛼-connected.
Clearly, if 𝜔 is the closure operator in 𝐿^𝑋, then 𝜔_𝛼-connected is 𝐷_𝛼-connected [8].

Definition 15. Let (𝐿^𝑋,Ω) be an 𝐿-𝑓opos, 𝛼∈𝑀(𝐿). 𝐴∈𝐿^𝑋 is called an 𝜔_𝛼-connected set, if there not exist 𝜔_𝛼-separated sets 𝐵,𝐶∈𝐿^𝑋, such that, 𝐵_[𝛼]≠∅,𝐶_[𝛼]≠∅, and 𝐴_[𝛼]=𝐵_[𝛼]∪𝐵_[𝛼]. Specially, if 1_𝑋 is 𝜔_𝛼-connected, then, (𝐿^𝑋,Ω) is called 𝜔_𝛼-connected.

Theorem 16. Let (𝐿^𝑋,Ω) be an 𝐿-𝑓𝑜𝑝𝑜𝑠, 𝛼∈𝑀(𝐿). Then (𝐿^𝑋,Ω) is not 𝜔_𝛼-connected if and only if there are 𝐴,𝐵∈𝜔_𝛼(Ω), such that, 𝐴_[𝛼]≠∅,𝐵_[𝛼]≠∅, and 𝐴_[𝛼]∩𝐵_[𝛼]=∅, 𝐴_[𝛼]∪𝐵_[𝛼]=𝑋.

Proof. Necessity. Let (𝐿^𝑋,Ω) be not 𝜔_𝛼-connected. That is, 1_𝑋 is not 𝜔_𝛼-connected. By Definition 15, there are two 𝜔_𝛼-separated sets 𝐴,𝐵∈𝐿^𝑋, such that, 𝐴_[𝛼]≠∅,𝐵_[𝛼]≠∅, and 𝐴_[𝛼]∪𝐵_[𝛼]=(1_𝑋)_[𝛼]=𝑋.
Since, 𝐴,𝐵∈𝐿^𝑋 are 𝜔_𝛼-separated, we have 𝜔_𝛼(𝐴)_[𝛼]=𝜔_𝛼(𝐴)_[𝛼]∩𝑋=𝜔_𝛼(𝐴)_[𝛼]∩𝐴_[𝛼]∪𝐵_[𝛼]=𝜔_𝛼(𝐴)_[𝛼]∩𝐴_[𝛼]∪𝜔_𝛼(𝐴)_[𝛼]∩𝐵_[𝛼]=𝐴_[𝛼]∪∅=𝐴_[𝛼]. (14)
Therefore, 𝐴_[𝛼]∩𝐵_[𝛼]⊂𝜔_𝛼(𝐴)_[𝛼]∩𝐵_[𝛼]=∅. Hence, the necessity holds.
Sufficiency. Take 𝐴,𝐵∈𝜔_𝛼(Ω) satisfies the conditions stated in the theorem. So (1_𝑋)_[𝛼]=𝐴_[𝛼]∪𝐵_[𝛼]=𝑋. As 𝐴_[𝛼]∩𝐵_[𝛼]=∅, and 𝜔_𝛼(𝐴)_[𝛼]=𝐴_[𝛼], we have 𝜔_𝛼(𝐴)_[𝛼]∩𝐵_[𝛼]=𝐴_[𝛼]∩𝐵_[𝛼]=∅. Similarly, 𝜔_𝛼(𝐵)_[𝛼]∩𝐴_[𝛼]=𝐵_[𝛼]∩𝐴_[𝛼]=∅. Hence, 𝐴,𝐵 are 𝜔_𝛼-separated. By Definition 15, we have (𝐿^𝑋,Ω) is not 𝜔_𝛼-connected.

Theorem 17. Let (𝐿^𝑋,Ω) is an 𝐿-𝑓𝑜𝑝𝑜𝑠, 𝛼∈𝑀(𝐿). 𝐴 be 𝜔_𝛼-connected, then 𝐴_[𝛼]⊂𝐵_[𝛼]⊂𝜔_𝛼(𝐴)_[𝛼] implies 𝐵 is 𝜔_𝛼-connected.

Proof. Since 𝐴 is 𝜔_𝛼-connected. 𝐴_[𝛼] is connected in (𝑋,𝜔_𝛼(Ω)). By Theorem 20 in reference [10] (page 54) and Corollary 8, 𝜔_𝛼(𝐴)_[𝛼] is connected. So 𝐵_[𝛼] is connected. Therefore, 𝐵 is 𝜔_𝛼-connected.

Theorem 18. Let (𝐿^𝑋,𝜔_𝛼(Ω)) be an 𝜔_𝛼-𝑜𝑝𝑜𝑠. {𝐴_𝑡}_𝑡∈𝑇⊂𝐿^𝑋, 𝐴_𝑡 is 𝜔_𝛼-connected, and there is 𝑠∈𝑇, such that for any 𝑡∈𝑇, 𝐴_𝑡 and 𝐴_𝑠 are 𝜔_𝛼-connected. Then 𝐴=⋁_𝑡∈𝑇𝐴_𝑡 is 𝜔_𝛼-connected.

Proof. Since 𝐴_𝑡 is 𝜔_𝛼-connected, 𝐴_𝑡[𝛼] is connected in (𝑋,𝜔_𝛼(Ω)). 𝐴_𝑠[𝛼] and 𝐴_𝑡[𝛼] are collected for each 𝑠∈𝑇. Then by Theorem 21 in reference [10] (page 54) and Corollary 8, 𝐴_[𝛼]=(⋁_𝑡∈𝑇𝐴_𝑡)_[𝛼]=⋃_𝑡∈𝑇𝐴_𝑡[𝛼] is connected. Therefore, ⋁_𝑡∈𝑇𝐴_𝑡 is 𝜔_𝛼-connected.

Theorem 19. Let (𝑋,Δ) be an 𝑜𝑝𝑜𝑠, and (𝐿^𝑋,𝜔_𝐿(Δ)) be the induced 𝐿-𝑜𝑝𝑜𝑠. Then 𝐴∈𝜔_𝛼(𝜔_𝐿(Δ)) is 𝜔_𝛼-connected if and only if 𝐴_[𝛼]⊂𝑋 is connected.

Proof. Necessity. Let 𝐴∈𝜔_𝛼(𝜔_𝐿(Δ)) be 𝜔_𝛼-connected. Suppose that 𝐴_[𝛼] is not connected, then there are nonempty 𝑈,𝑉⊂𝑋, such that 𝐴_[𝛼]=𝑈∪𝑉, and 𝑈⁻_𝜎∩𝑉=𝑉⁻_𝜎∩𝑈=∅. By Theorem 10, 𝜔_𝛼(1_𝑈)_[𝛼]=𝑈⁻_𝜎, 𝜔_𝛼(1_𝑉)_[𝛼]=𝑉⁻_𝜎. So 𝜔_𝛼(1_𝑈)_[𝛼]∩1_𝑉[𝛼]=𝜔_𝛼1_𝑉[𝛼]∩1_𝑈[𝛼]=∅. (15) This means 1_𝑈,1_𝑉 are 𝜔_𝛼-separated sets, and 𝐴_[𝛼]=(1_𝑈)_[𝛼]∪(1_𝑉)_[𝛼]. A contradiction.
Sufficiency. Suppose 𝐴∈𝜔_𝛼(𝜔_𝐿(Δ)) is not 𝜔_𝛼-connected. Then There are 𝜔_𝛼-separated sets 𝐵,𝐶∈𝐿^𝑋, such that 𝐵_[𝛼],𝐶_[𝛼]≠∅, and 𝐴_[𝛼]=𝐵_[𝛼]∪𝐵_[𝛼]. Since 𝜔_𝛼(𝐵)∈𝜔_𝛼(𝜔_𝐿(Δ)), by Theorem 10, we have 𝜔_𝛼(𝐵)_[𝛼]∈Δ⁻_𝜎. Thus, (𝐵_[𝛼])⁻_𝜎⊂(𝜔_𝛼(𝐵)_[𝛼])⁻_𝜎=𝜔_𝛼(𝐵)_[𝛼]. Therefore, 𝐵_[𝛼]⁻_𝜎∩𝐶_[𝛼]⊂𝜔_𝛼(𝐵)_[𝛼]∩𝐶_[𝛼]=∅. (16) Similarly, we get that (𝐵_[𝛼])⁻_𝜎∩𝐶_[𝛼]=∅. This means 𝐴_[𝛼]⊂𝑋 is not connected. A contradiction.

Theorem 20. Let (𝐿^𝑋,Ω_𝑖)(𝑖=1,2) be two 𝐿-𝑜𝑝𝑜𝑠s, 𝑓^→∶𝐿^𝑋→𝐿^𝑋 be (𝜔¹,𝜔²)-continuous, 𝐴∈𝐿^𝑋 be 𝜔¹_𝛼-connected. Then 𝑓^→(𝐴) is 𝜔²_𝛼-connected.

Proof. Let 𝑓^→(𝐴)_[𝛼]=𝐵_[𝛼]∪𝐶_[𝛼], 𝜔²_𝛼(𝐵)_[𝛼]∩𝐶_[𝛼]=𝜔²_𝛼(𝐶)_[𝛼]∩𝐵_[𝛼]=∅. Since 𝑓⁻¹(𝐵_[𝛼])=𝑓^←(𝐵)_[𝛼]. Take 𝐸_[𝛼]=𝑓⁻¹𝐵_[𝛼],𝐹_[𝛼]=𝑓⁻¹𝐶_[𝛼], (17) we have 𝐴_[𝛼]⊂𝑓^←𝑓^→(𝐴)_[𝛼]=𝑓^←(𝐵)_[𝛼]∪𝑓^←(𝐶)_[𝛼]=𝐸_[𝛼]∪𝐹_[𝛼]. (18) Besides, by 𝜔¹_𝛼(𝐸)_[𝛼]=𝜔¹_𝛼𝑓^←(𝐵)_[𝛼]⊂𝑓^←𝜔²_𝛼(𝐵)_[𝛼],𝜔¹_𝛼(𝐹)_[𝛼]=𝜔¹_𝛼𝑓^←(𝐶)_[𝛼]⊂𝑓^←𝜔²_𝛼(𝐶)_[𝛼] (19) we get 𝜔¹_𝛼(𝐸)_[𝛼]∩𝐹_[𝛼]⊂𝑓^←𝜔²_𝛼(𝐵)_[𝛼]∩𝑓^←𝜔²_𝛼(𝐶)_[𝛼]=𝑓⁻¹𝜔²_𝛼(𝐵)_[𝛼]∩𝐶_[𝛼]=𝑓⁻¹(∅)𝜔=∅,_1𝛼(𝐹)_[𝛼]∩𝐸_[𝛼]⊂𝑓^←𝜔²_𝛼(𝐶)_[𝛼]∩𝑓^←𝜔²_𝛼(𝐵)_[𝛼]=𝑓⁻¹𝜔²_𝛼(𝐶)_[𝛼]∩𝐵_[𝛼]=𝑓⁻¹(∅)=∅. (20)
Denote 𝐺_[𝛼]=𝐴_[𝛼]∪𝐸_[𝛼], 𝐻_[𝛼]=𝐴_[𝛼]∪𝐹_[𝛼]. Then 𝐴_[𝛼]=𝐺_[𝛼]∪𝐻_[𝛼], and 𝜔¹_𝛼(𝐺)_[𝛼]∩𝐻_[𝛼]=𝜔¹_𝛼(𝐻)_[𝛼]∩𝐺_[𝛼]=∅. (21)
Since 𝐴 is 𝜔¹_𝛼-connected, then 𝐺_[𝛼]=∅, or 𝐻_[𝛼]=∅. Take the former, for example, 𝐴_[𝛼]=𝐻_[𝛼]⊂𝐹_[𝛼]. Thus, 𝑓^→(𝐴)_[𝛼]⊂𝑓^→(𝐹)_[𝛼]=𝑓^→𝑓^←(𝐶)_[𝛼]⊂𝐶_[𝛼]. (22) Thus, 𝐵_[𝛼]=𝐵_[𝛼]∩𝑓^→(𝐴)_[𝛼]⊂𝐵_[𝛼]∩𝐶_[𝛼]⊂𝐵_[𝛼]∩𝜔²_𝛼(𝐶)_[𝛼]=∅. (23)

Theorem 21. Let (𝐿^𝑋,Ω) be an 𝐿-𝑜𝑝𝑜𝑠, 𝐴∈𝐿^𝑋, for all𝛼∈𝑀(𝐿), 𝑀^∗(𝐴)={𝑎∈𝑀^∗(𝐿^𝑋)∣𝑎_[𝛼]∈A_[𝛼]}. Then𝐴 is 𝜔_𝛼-connected if and only if for all 𝑎,𝑏∈𝑀^∗(𝐴), for all 𝑃∶𝑀^∗(𝐴)→∪{𝜂_𝜔𝛼(𝑥)∣𝑥∈𝑀^∗(𝐴)}, there are finite molecular 𝑥₁=𝑎,𝑥₂,…,𝑥_𝑛=𝑏∈𝑀^∗(𝐴), such that 𝐴_[𝛼]⊄𝑃(𝑥_𝑗)_[𝛼]∪𝑃(𝑥_𝑗)_[𝛼],(𝑗=1,2,…,𝑛−1).

Proof. Necessity. Suppose the result does not holds. That is, there is 𝑎,𝑏∈𝑀^∗(𝐴), 𝑎≠𝑏. And there is a mapping 𝑃∶𝑀^∗(𝐴)⟶∪𝜂_𝜔𝛼(𝑥)𝑥∈𝑀^∗(𝐴), (24) such that for any finite 𝑥₁=𝑎,𝑥₂,…,𝑥_𝑛=𝑏∈𝑀^∗(𝐴), 𝐴_[𝛼]⊄𝑃(𝑥_𝑗)_[𝛼]∪𝑃(𝑥_𝑗)_[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,…,𝑥_𝑛=𝑏∈𝑀^∗(𝐴), such that 𝐴_[𝛼]⊄𝑃(𝑥_𝑗)_[𝛼]∪𝑃(𝑥_𝑗)_[𝛼], denoted by 𝑎∼𝑏. Otherwise, 𝑎 and 𝑏 are called disjoined, denoted by 𝑎≁𝑏. Let 𝜇=𝑥∈𝑀^∗(𝐴)𝑎∼𝑥,𝜈=𝑥∈𝑀^∗(𝐴)𝑎≁𝑥, (25) 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 𝜔_𝛼-connected.
Suppose that 𝜔_𝛼(𝐵)_[𝛼]∩𝐶_[𝛼]≠∅, then there is 𝑥_[𝛼]∈𝜔_𝛼(𝐵)_[𝛼]∩𝐶_[𝛼]. By 𝑥_[𝛼]∈𝜔_𝛼(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, by 𝑥_[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 𝜔_𝛼(𝐵)_[𝛼]∩𝐶_[𝛼]=∅. Similarly, we get that 𝜔_𝛼(𝐶)_[𝛼]∩𝐵_[𝛼]=∅.
Sufficiency. Suppose 𝐴 is not 𝜔_𝛼-connected. Then there are 𝐵,𝐶∈𝐿^𝑋, such that 𝐵,𝐶 are 𝜔_𝛼-separated, and 𝐵_[𝛼]≠∅,𝐶_[𝛼]≠∅, 𝐴_[𝛼]=𝐵_[𝛼]∪𝐶_[𝛼]. Define a mapping 𝑃∶𝑀^∗(𝐴)⟶∪𝜂_𝜔𝛼(𝑥)𝑥∈𝑀^∗(𝐴,𝑃(𝑥)=𝜔_𝛼)(𝐶),𝑥_[𝛼]∈𝐵_[𝛼],𝜔_𝛼(𝐵),𝑥_[𝛼]∈𝐶_[𝛼]. (26) Then forall𝑥∈𝑀^∗(𝐴), we have 𝑥≰𝑃(𝑥), 𝑃(𝑥)∈𝜔_𝛼(Ω).
Take 𝑎_[𝛼]∈𝐵_[𝛼],𝑏_[𝛼]∈𝐶_[𝛼]. Then for any finite 𝑥₁=𝑎,𝑥₂,…,𝑥_𝑛=𝑏∈𝑀^∗(𝐴), 𝑥_𝑖[𝛼]∈𝐵_[𝛼], or 𝑥_𝑖[𝛼]∈𝐶_[𝛼]. Thus, 𝑃(𝑥_𝑖)_[𝛼]∈𝜔_𝛼(𝐶), or 𝑃(𝑥_𝑖)_[𝛼]∈𝜔_𝛼(𝐵), (𝑖=1,2,…,𝑛). However, since 𝑃(𝑥₁)_[𝛼]∈𝜔_𝛼(𝐶),𝑃(𝑥_𝑛)_[𝛼]∈𝜔_𝛼(𝐵), there is 1≤𝑗≤𝑛−1, such that 𝑃(𝑥_𝑗)=𝜔_𝛼(𝐶),𝑃(𝑥_𝑗+1)=𝜔_𝛼(𝐵). Therefore, 𝐴_[𝛼]=𝐵_[𝛼]∪𝐶_[𝛼]⊂𝑃(𝑥_𝑗)_[𝛼]∪𝑃(𝑥_𝑗+1)_[𝛼]. A contradiction.

Acknowledgment

The work is supported by Youth Science Foundation of Hunan University of Science and Engineering (10XKYTB038).