On 𝜶-Layer Ideal Topologies

Abstract

The purpose of this paper is to study theory of two different kinds of 𝛼-layer order-preserving operator space, namely, 𝜔_𝛼-opos and 𝜔^∗𝛼(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 𝜔_𝛼-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 𝜔^∗𝛼(ℑ)-oposs.

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 utilizing q-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 in 𝐿-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 of 𝜔_𝛼-opos. We introduce the concept of 𝛼-layer function in 𝐿-fuzzy order-preserving operator space. Then, based on its basic properties, we form 𝜔_𝛼-opos. We also prove it preserves many good properties. In the second part of the paper, we establish the theory of 𝜔^∗𝛼(ℑ)-opos. It is a 𝛼-layer space with local topological properties. We introduce the concept of local 𝛼-remote neighborhood function via an 𝜔_𝛼-opos and an 𝛼-ideal. On the basis of its basic properties, we form 𝜔^∗𝛼(ℑ)-opos. It is finer than the old one. We observe the structures of local 𝛼-remote neighborhood functions under different 𝛼-ideals and 𝜔_𝛼-opos as well as the relations of the correspondent 𝜔^∗𝛼-opos. We also obtain some equivalent conditions of 𝜔^∗𝛼-opos under compatibility of 𝛼-ideal and 𝜔_𝛼-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 lattice 𝐿 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.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 [1]). 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). 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} [8].

An 𝐿-fuzzy order-preserving operator space and some related conceptions are given in the following.

Let 𝑋 be an nonempty set. An operator 𝜔∶𝐿^𝑋→𝐿^𝑋 is called a 𝐿-fuzzy order preserving operator in 𝐿^𝑋, if it satisfies (1) 𝜔(1_𝑋)=1_𝑋, (2) for all 𝐴,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 for all 𝑃∈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 [9].

Similar concepts in a general topology are defined as follows.

Let 𝑋 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 𝐴^−𝜎. A 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.

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}∈Δ^−𝜎. Then, the set of all the lower continuous functions, denoted by 𝜔_𝐿(Δ), consists an 𝐿-fuzzy cotopology in 𝐿^𝑋. The space (𝐿^𝑋,𝜔_𝐿(Δ)) is called the induced 𝐿-𝑓opos by (𝑋,Δ).

A nonempty subfamily ℑ of 𝐿^𝑋 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 if 𝐵∈𝐿^𝑋, 𝐴∈ℑ, such that 𝐵_[𝛼]=𝐴_[𝛼], then 𝐵∈ℑ. Moreover, ℑ of 𝐿^𝑋 is an 𝛼-idea if and only if ℑ_[𝛼]={𝐴_[𝛼]∶𝐴∈ℑ} is an ideal on 𝑋. If 𝐵∈𝐿^𝑋, 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, if 𝔉⊂𝐿^𝑋, we denote 𝔉_[𝛼]={𝐴∶𝐴∈𝔉} and 𝔍^={𝐴^∶𝐴∈𝔉}.

3. 𝜔_𝛼-Closed Set and 𝜔_𝛼-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 (𝐿^𝑋,Ω) be an 𝐿-𝑓opos, 𝛼∈𝑀(𝐿). An operator 𝜔_𝛼∶𝐿^𝑋→𝐿^𝑋 is defined by, for all 𝐴∈𝐿^𝑋,𝜔_𝛼(𝐴)=𝐺∈Ω^−𝜔∶𝐺_[𝛼]⊃𝐴_[𝛼].(1)

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).

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 an 𝜔_𝛼-order-preserving operator space (briefly, 𝜔_𝛼-opos).

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 ⋁^𝑛𝑖=1𝐴_𝑖∈𝜔_𝛼(Ω);(3)if {𝐴_𝑖∶𝑖∈𝐼}⊂𝜔_𝛼(Ω), then ∧_𝑖∈𝐼𝐴_𝑖∈𝜔_𝛼(Ω).

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). Then, 𝜔_𝛼(Ω) consists an 𝐿-fuzzy cotopology on 𝐿^𝑋.

Lemma 7. Let (𝑋,Δ) be an opos, (𝐿^𝑋,𝜔_𝐿(Δ)) be an 𝐿-𝑓opos induced by (𝑋,Δ) and let 𝐸⊂𝑋. Then, 𝐸∈Δ^−𝜎 if and only if 𝜒_𝐸∈(𝜔_𝐿(Δ))^−𝜔.

Theorem 8. Let (𝑋,Δ) be an opos, (𝐿^𝑋,𝜔_𝐿(Δ)) 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.

Theorem 9. Let (𝑋,Δ) be an opos, (𝐿^𝑋,𝜔_𝐿(Δ)) is an 𝐿-𝑓opos induced by (𝑋,Δ). Then, 𝐴∈𝜔_𝐿(Δ) if and only if for all 𝛼∈𝑀(𝐿), 𝐴∈𝜔_𝛼(𝜔_𝐿(Δ)).

Definition 10. Let (𝐿^𝑋,𝜔^1𝛼(Ω₁)), (𝐿^𝑌,𝜔^2𝛼(Ω₂)) be 𝜔^1𝛼-opos, 𝜔^2𝛼-opos, respectively. An 𝐿-fuzzy homomorphism 𝑓^→∶𝐿^𝑋→𝐿^𝑌 is called (𝜔^1𝛼(Ω₁),𝜔^2𝛼(Ω₂))-continuous, if for all 𝐵∈𝜔^2𝛼(Ω₂), then 𝑓^←(𝐵)∈𝜔^1𝛼(Ω₁).

Theorem 11. Let (𝐿^𝑋,𝜔^1𝛼(Ω₁)), (𝐿^𝑌,𝜔^2𝛼(Ω₂)) be 𝜔^1𝛼-opos, 𝜔^2𝛼-opos, respectively. 𝑓^→∶𝐿^𝑋→𝐿^𝑌 is an 𝐿-fuzzy mapping. Then, the following statements are equivalent:(1)𝑓^→ is (𝜔^1𝛼(Ω₁),𝜔^2𝛼(Ω₂))-continuous;(2)for all 𝐵∈𝐿^𝑌, 𝜔^1𝛼(𝑓^←(𝐵))_[𝛼]⊂𝑓^←(𝜔^2𝛼(𝐵))_[𝛼];(3)for all 𝐴∈𝐿^𝑋, 𝑓^→(𝜔^1𝛼(𝐴))_[𝛼]⊂𝜔^2𝛼(𝑓^→(𝐴))_[𝛼].

Definition 12. Let (𝐿^𝑋,𝜔_𝛼(Ω)) be an 𝜔_𝛼-opos, 𝑥_𝛼∈𝑀^∗(𝐿^𝑋),𝐴∈𝜔_𝛼(Ω). 𝐴 is called an 𝜔_𝛼-closed remote neighborhood of 𝑎, if 𝑥∉𝐴_[𝛼]. The set of all 𝜔_𝛼-closed remote neighborhood of 𝑥_𝛼 will be denoted by 𝜂_𝜔𝛼(𝑥_𝛼). An fuzzy point 𝑥_𝛼 is called 𝜔_𝛼-adherent point of 𝐵∈𝐿^𝑋, if for every 𝑃∈𝜂_𝜔𝛼(𝑥_𝛼), 𝐵_([𝛼])⊄𝑃_([𝛼]).

Remark 13. 𝑃∈𝜂_𝜔𝛼(𝑥_𝛼)⇔(𝑃_[𝛼])′∈𝒩(𝑥), where 𝒩(𝑥) is the neighborhood system of 𝑥 in (𝑋,𝜔_𝛼(Ω)_[𝛼]). Hence, (𝜂_𝜔𝛼(𝑥_𝛼)_[𝛼])′=𝒩(𝑥).

Theorem 14. Let (𝐿^𝑋,𝜔_𝛼(Ω)) be an 𝜔_𝛼-opos, 𝑥_𝛼∈𝑀^∗(𝐿^𝑋),𝐴∈𝐿^𝑋, then(1)𝑥_𝛼 is an 𝜔_𝛼-adherent point of 𝐴∈𝐿^𝑋 if and only if 𝑥∈𝜔_𝛼(𝐴)_[𝛼],(2)𝜔_𝛼(𝐴) is the union of all 𝜔_𝛼-adherent points of 𝐴.

4. 𝜔_𝛼-ℑ-Closed Sets

Let (𝐿^𝑋,𝜔_𝛼(Ω)) be an 𝜔_𝛼-opos. ℑ⊂𝐿^𝑋 be an 𝛼-ideal. 𝑥_𝛼∈𝑀^∗(𝐿^𝑋), 𝜂_𝜔𝛼(𝑥_𝛼) be the 𝜔_𝛼-remote family of 𝑥_𝛼. For any 𝐴∈𝐿^𝑋, take 𝐴^∗ℑ=,𝜔_𝛼𝑥_𝛼∶𝐴_[𝛼]⊄𝑃_[𝛼]∪𝐼_[𝛼],∀𝑃∈𝜂_𝜔𝛼𝑥_𝛼,𝐼∈ℑ.(2) Then, 𝐴^∗(ℑ,𝜔_𝛼) is called the local 𝜔_𝛼-remote function of 𝐴 with respect to ℑ and 𝜔_𝛼(Ω), simply denoted by 𝐴^∗(ℑ) or 𝐴^∗.

Remark 15. If 𝐿=[0,1], 𝜔 is the closure operator, then (𝐴^∗[𝛼])′ is the local function of 𝐴_[𝛼] in (𝑋,𝜔_𝛼(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 (𝐿^𝑋,𝜔_𝛼(Ω)) be an 𝜔_𝛼-opos. ℑ,𝔍⊂𝐿^𝑋 be two 𝛼-ideals. 𝐴,𝐵∈𝐿^𝑋. Then, the following statements hold.(1)ℑ_[𝛼]={∅}⇒𝐴^∗(ℑ)_[𝛼]=𝜔_𝛼(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]=𝜔_𝛼(𝐴^∗(ℑ))_[𝛼]⊂𝜔_𝛼(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∈𝜔_𝛼(Ω)⇒𝑃^∗[𝛼]⊂𝑃_[𝛼].

Proof. (1) Suppose ℑ_[𝛼]={∅}, by Theorem 14, we have 𝑥_𝛼≤⟺𝐴^∗𝑥_𝛼ℑ,𝜔_𝛼⟺𝑥_𝛼≤𝑥_𝛼∶𝐴_[𝛼]⊄𝑃_[𝛼],∀𝑃∈𝜂_𝜔𝛼𝑥_𝛼≤𝜔_𝛼(𝐴).(3)
If 𝑋∈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∉𝜔_𝛼(𝐴)_[𝛼] implies 𝜔_𝛼(𝐴)∈𝜂_𝛼(𝑥_𝛼). By 𝐴_[𝛼]⊂𝜔_𝛼(𝐴)_[𝛼], we have 𝑥∉𝐴^∗[𝛼]. So 𝐴^∗[𝛼]⊂𝜔_𝛼(𝐴)_[𝛼]. Thus, 𝜔_𝛼(𝐴)^∗[𝛼]⊂𝜔_𝛼(𝜔_𝛼(𝐴))_[𝛼]=𝜔_𝛼(𝐴)_[𝛼]. Besides, 𝑥∈𝜔_𝛼(𝐴^∗)_[𝛼] implies for all 𝑃∈𝜂_𝜔𝛼(𝑥_𝛼), 𝐴^∗[𝛼]⊄𝑃_[𝛼]. Take 𝑦∈𝐴^∗[𝛼]−𝑃_[𝛼], so 𝑃∈𝜂_𝜔𝛼(𝑦_𝛼) and for all 𝑄∈𝜂_𝜔𝛼(𝑦_𝛼),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, 𝜔_𝛼(𝐴^∗)_[𝛼]⊂𝐴^∗[𝛼]. Therefore, 𝜔_𝛼(𝐴^∗)_[𝛼]=𝐴^∗[𝛼].
(5) By (4), (𝐴^∗(ℑ))^∗(ℑ)_[𝛼]=(𝜔_𝛼(𝐴^∗(ℑ)))^∗(ℑ)_[𝛼]=𝜔_𝛼(𝜔_𝛼(𝐴^∗(ℑ)))_[𝛼]=(𝜔_𝛼(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∈𝜂_𝜔𝛼(𝑥_𝛼),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 each 𝑥∈(𝐴^∗−𝐵^∗)_[𝛼], we may remark 𝑥∈(𝐴−𝐵)^∗[𝛼]. If not, there are 𝑃∈𝜂_𝜔𝛼(𝑥_𝛼),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∈𝜂_𝜔𝛼(𝑥_𝛼),𝐽∈ℑ, 𝐵_[𝛼]⊂𝑄_[𝛼]∪𝐽_[𝛼]. This shows 𝐴_[𝛼]⊂𝑃_[𝛼]∪𝐼_[𝛼]∪𝐵_[𝛼]⊂𝑃_[𝛼]∪𝐼_[𝛼]∪𝑄_[𝛼]∪𝐽_[𝛼]=(𝑃∨𝑄)_[𝛼]∪(𝐼∨𝐽)_[𝛼].(4) 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∈𝜂_𝜔𝛼(𝑥_𝛼),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 is 𝑥∈𝑃^∗[𝛼]−𝑃_[𝛼]. So 𝑃∈𝜂_𝜔𝛼(𝑥_𝛼). By 𝑃_[𝛼]⊂(𝑃∨𝐼)_[𝛼] for each 𝐼∈ℑ, we have 𝑥∉𝑃_[𝛼], a contradiction.

Theorem 17. Let (𝐿^𝑋,𝜔_𝛼(Ω)) be an 𝜔_𝛼-opos. Let ℑ⊂𝐿^𝑋 be an ideal. 𝐴∈𝐿^𝑋. Take 𝜔^∗𝛼(𝐴)=𝐴∨𝐴^∗, then(1)𝜔^∗𝛼(𝐴)_[𝛼]=∅, if and only if 𝐴_[𝛼]=∅,(2)𝜔^∗𝛼(𝐴∨𝐵)_[𝛼]=𝜔^∗𝛼(𝐴)_[𝛼]∪𝜔^∗𝛼(𝐵)_[𝛼], (3)𝜔^∗𝛼(𝜔^∗𝛼(𝐴))_[𝛼]=𝜔^∗𝛼(𝐴)_[𝛼],(4)(𝜔^∗𝛼(𝐴^∗))^∗[𝛼]=𝜔^∗𝛼(𝐴^∗)_[𝛼]=(𝜔^∗𝛼(𝐴))^∗[𝛼]=(𝜔_𝛼(𝐴^∗))^∗[𝛼]=𝐴^∗[𝛼].

Proof. According to (5), (7) in Theorem 16, the proof is trivial.
By Theorem 17, we know if and only if we take 𝜔^∗𝛼(ℑ,Ω)={𝐴∈𝐿^𝑋∶𝜔^∗𝛼(𝐴)_[𝛼]=𝐴_[𝛼]}, then 𝜔^∗𝛼(ℑ,Ω) (simply denoted by 𝜔^∗𝛼(ℑ)) consists an 𝐿-fuzzy cotopology on 𝐿^𝑋. So it is called 𝜔_𝛼-cotopology formed by the ideal ℑ and 𝜔_𝛼(Ω). The pair (𝐿^𝑋,𝜔^∗𝛼(ℑ,Ω)), simply denoted by (𝐿^𝑋,𝜔^∗𝛼(ℑ)), is called 𝜔^∗𝛼(ℑ)-𝑜𝑝𝑜𝑠. By Theorem 16 (1), we have ℑ_[𝛼]={∅}⇒𝜔^∗𝛼(𝐴)_[𝛼]=𝜔_𝛼(𝐴)_[𝛼], thus 𝜔^∗𝛼(ℑ,Ω)=𝜔_𝛼(Ω), and ℑ_[𝛼]=𝒫(𝑋)⇒𝜔^∗𝛼(𝐴)_[𝛼]=𝐴_[𝛼], thus 𝜔^∗𝛼(ℑ,Ω)=𝐿^𝑋. But in general, {∅}⊂ℑ⊂𝒫(𝑋), 𝐴_[𝛼]⊂𝜔^∗𝛼(𝐴)_[𝛼]⊂𝜔_𝛼(𝐴)_[𝛼], consequently, 𝜔_𝛼(Ω)⊂𝜔^∗𝛼(ℑ,Ω)⊂𝐿^𝑋. Moreover, if and only if 𝜔_𝛼(Ω)_[𝛼]={∅,𝑋}, then 𝜔^∗𝛼(Ω)={𝐴∶𝐴_[𝛼]∈ℑ_[𝛼]}.

Example 18. Let 𝑋={𝑥,𝑦},𝐿={0,1/2,1}. An 𝐿-fuzzy set 𝐺∈𝐿^𝑋 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 𝐿^𝑋. 𝛼=1/2. Put 𝜔_𝛼(𝐴)=⋀{𝐸∈Ω∶𝐸_[𝛼]⊃𝐴_[𝛼]}. Then, 𝜔_𝛼(Ω)={𝐴∶𝜔_𝛼(𝐴)_[𝛼]=𝐴_[𝛼]} consists an 𝜔_𝛼-cotopology, and 𝜂_𝜔𝛼(𝑥_𝛼)={(0,0),(0,1/2),(0,1)},𝜂_𝜔𝛼(𝑦_𝛼)={(0,0)}. Totally, there are four kinds of 𝛼-ideals in 𝐿^𝑋, namely, 𝔏={(0,0)}, 𝔐=𝐿^𝑋, ℑ={(0,0),(0,1/2),(0,1)} and 𝔍={(0,0),(1/2,0),(1,0)}. 𝔏 and 𝔐=𝐿^𝑋 are the trivial 𝛼-ideals. Let us study ℑ and 𝔍.
Denote 𝐵=(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∈𝜔^∗𝛼(ℑ), But 𝐷∉𝜔^∗𝛼(ℑ). In addition, 𝜔^∗𝛼(ℑ,Ω)=𝜔_𝛼(Ω). However, 𝐷^∗(𝔍)=(0,0), which means 𝐷∈𝜔^∗𝛼(𝔍). It is easy to check 𝜔^∗𝛼(𝔍,Ω)=𝐿^𝑋.
In this example, we see (𝜔_𝛼(𝐵))^∗(ℑ)_[𝛼]=𝐵^∗(ℑ)_[𝛼]=∅. However, 𝜔^∗𝛼(𝐵)_[𝛼]=(𝐵∨𝐵^∗(ℑ))_[𝛼]={𝑦}. Thus, (𝜔_𝛼(𝐵))^∗(ℑ)_[𝛼]≠𝜔^∗𝛼(𝐵)(ℑ)_[𝛼]. Furthermore, (𝜔_𝛼(Ω))^∗(ℑ)={𝐴^∗(ℑ)∶𝐴∈𝜔_𝛼(Ω)}={(0,0),(1/2,1/2)} is not an 𝜔_𝛼-cotopology.

Theorem 19. Let (𝐿^𝑋,𝜔_𝛼(Ω_𝑖))(𝑖=1,2) be an 𝜔_𝛼-opos. 𝜔_𝛼(Ω₁)⊂𝜔_𝛼(Ω₂). ℑ is an 𝛼-ideal. 𝐴∈𝐿^𝑋, then(1)𝐴^∗(ℑ,𝜔_𝛼(Ω₂))_[𝛼]⊂𝐴^∗(ℑ,𝜔_𝛼(Ω₁))_[𝛼],(2)𝐴^∗(ℑ,𝜔^∗𝛼(ℑ,Ω₁))_[𝛼]⊂𝐴^∗(ℑ,𝜔_𝛼(Ω₁))_[𝛼],(3)𝜔^∗𝛼(ℑ,Ω₁)⊂𝜔^∗𝛼(ℑ,Ω₂).

Theorem 20. Let (𝐿^𝑋,𝜔_𝛼(Ω)) be an 𝜔_𝛼-𝑜𝑝𝑜𝑠, an 𝛼-ideal ℑ⊂𝜔_𝛼(Ω). Then, 𝜔^∗𝛼(ℑ)=𝜔_𝛼(Ω).

Proof. Obviously, 𝜔^∗𝛼(ℑ)⊃𝜔_𝛼(Ω). If 𝐴∉𝜔_𝛼(Ω), that is, 𝐴_[𝛼]≠𝜔_𝛼(𝐴)_[𝛼]. Take 𝑦∈𝜔_𝛼(𝐴)_[𝛼]−𝐴_[𝛼]. Let us prove 𝑦∈𝐴^∗(ℑ)_[𝛼]. In fact, if 𝑦∉𝐴^∗(ℑ), there are 𝑄∈𝜂_𝜔𝛼(𝑦_𝛼),𝐼∈ℑ, such that 𝐴_[𝛼]⊂(𝑄∨𝐼)_[𝛼]. So 𝑦∈𝜔_𝛼(𝐴)_[𝛼]⊂𝜔_𝛼(𝑄∨𝐼)_[𝛼]=𝜔_𝛼(𝑄)∨𝜔_𝛼(𝐼)_[𝛼]=(𝑄∨𝐼)_[𝛼].(5)
It means 𝑦∈𝐼_[𝛼]. On the other hand, as 𝐴_[𝛼]⊂(𝑄∨𝐼)_[𝛼], we have (𝐴∧𝐼)_[𝛼]⊂𝑄_[𝛼]. Then, 𝜔_𝛼(𝐴∧𝐼)_[𝛼]=𝜔_𝛼(𝐴)∧𝜔_𝛼(𝐼)_[𝛼]=𝜔_𝛼(𝐴)_[𝛼]∩𝐼_[𝛼]⊂𝜔_𝛼(𝑄)_[𝛼]=𝑄_[𝛼].(6) Notice that 𝑦∈𝜔_𝛼(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∉𝜔^∗𝛼(ℑ). Therefore, 𝜔^∗𝛼(ℑ)⊂𝜔_𝛼(Ω). The proof is completed.

Theorem 21. Let (𝐿^𝑋,𝜔_𝛼(Ω)) be an 𝜔_𝛼-opos, and let ℑ be an 𝛼-ideal. Take 𝜔_𝛼(Ω)∨ℑ as the supremum 𝜔_𝛼- cotopology generated by 𝜔_𝛼(Ω)∪ℑ. Then, 𝜔^∗𝛼(ℑ)=𝜔_𝛼(Ω)∨ℑ.

Proof. Put 𝔅={𝑃∨𝐼∶𝑃∈𝜔_𝛼(Ω),𝐼∈ℑ}. It is easy to prove 𝔅⊂𝜔^∗𝛼(ℑ) is a base of 𝜔^∗𝛼(ℑ).

Lemma 22. Let (𝐿^𝑋,𝜔_𝛼(Ω)) be an 𝜔_𝛼-opos. Let ℑ,𝔍⊂𝐿^𝑋 be two ideals. 𝐼∈ℑ, 𝐴∈𝐿^𝑋. Then, (𝐴−𝐼)^∗(𝔍,𝜔_𝛼(Ω))=𝐴^∗(𝔍,𝜔^∗𝛼(ℑ)).

Proof. By Theorem 21, 𝑥∉(𝐴−𝐼)^∗𝔍,𝜔_𝛼(Ω)⟺∃(𝐴𝑃−∈𝐼𝜂)_{𝜔𝛼[𝛼]}𝑥_𝛼,𝐽∈𝔍,⊂(𝑃∨𝐽)_[𝛼]⟺∃𝐴𝑃∈𝜂_𝜔𝛼𝑥_𝛼,𝐽∈𝔍,_[𝛼]⊂(𝑃∨𝐽∨𝐼)_[𝛼]⟺∃𝐴𝑄∈𝜂_𝜔^∗𝛼_(ℑ)𝑥_𝛼,𝐽∈𝔍,_[𝛼]⊂(𝑄∨𝐽)_[𝛼]⟺𝑥∉𝐴^∗𝔍,𝜔^∗𝛼(ℑ).(7)

Theorem 23. Let (𝐿^𝑋,𝜔_𝛼(Ω)) be an 𝜔_𝛼-opos. Let ℑ,𝔍⊂𝐿^𝑋 be two 𝛼-ideals. 𝐴∈𝐿^𝑋, then(1)𝐴^∗(ℑ∩𝔍)_[𝛼]=𝐴^∗(ℑ)_[𝛼]∪𝐴^∗(𝔍)_[𝛼],(2)𝐴^∗(ℑ∨𝔍,𝜔_𝛼(Ω))_[𝛼]=𝐴^∗(ℑ,𝜔^∗𝛼(𝔍))_[𝛼]∩𝐴^∗(𝔍,𝜔^∗𝛼(ℑ))_[𝛼].

Proof. (1) Suppose 𝑥∉𝐴^∗(ℑ)_[𝛼]∪𝐴^∗(𝔍)_[𝛼], there are 𝑃,𝑄∈𝜂_𝜔𝛼(𝑥_𝛼), 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)∈𝜂_𝜔𝛼(𝑥_𝛼), 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) Suppose 𝑥∉𝐴^∗(ℑ∨𝔍,𝜔_𝛼(Ω))_[𝛼]. There are 𝑃∈𝜂_𝜔𝛼(𝑥_𝛼), 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,𝜔_𝛼(Ω))_[𝛼]=𝐴^∗(ℑ,𝜔^∗𝛼(𝔍))_[𝛼], and 𝑥∉(𝐴−𝐼)^∗(𝔍,𝜔_𝛼(Ω))_[𝛼]=𝐴^∗(𝔍,𝜔^∗𝛼(ℑ))_[𝛼]. This shows that 𝑥∉𝐴^∗(ℑ,𝜔^∗𝛼(𝔍))_[𝛼]∩𝐴^∗(𝔍,𝜔^∗𝛼(ℑ))_[𝛼]. Therefore, 𝐴^∗(ℑ∨𝔍,𝜔_𝛼(Ω))_[𝛼]⊃𝐴^∗(ℑ,𝜔^∗𝛼(𝔍))_[𝛼]∩𝐴^∗(𝔍,𝜔^∗𝛼(ℑ))_[𝛼].
Conversely. suppose 𝑥∉𝐴^∗(ℑ,𝜔^∗𝛼(𝔍))_[𝛼]. Then, there exists 𝑃∈𝜂_𝜔^∗𝛼_(𝔍)(𝑥_𝛼), and 𝐼∈ℑ, such that 𝐴_[𝛼]⊂(𝑃∨𝐼)_[𝛼]. Because of 𝑃∈𝜔^∗𝛼(𝔍,Ω)=𝜔_𝛼(Ω)∨𝔍, there is 𝐽∈𝔍, such that 𝜔_𝛼(𝑃−𝐽)∈𝜂_𝜔𝛼(𝑥_𝛼). Hence, 𝐴_[𝛼]⊂(𝑃∨𝐼)_[𝛼]=(𝑃−𝐽)_[𝛼]∪(𝐼∨𝐽)_[𝛼]. This shows 𝑥∉𝐴^∗(ℑ∨𝔍,𝜔_𝛼(Ω))_[𝛼]. Therefore, 𝐴^∗(ℑ∨𝔍,𝜔_𝛼(Ω))_[𝛼]⊂𝐴^∗(ℑ,𝜔^∗𝛼(𝔍))_[𝛼]. Similarly, we can prove 𝐴^∗(ℑ∨𝔍,𝜔_𝛼(Ω))_[𝛼]⊂𝐴^∗(𝔍,𝜔^∗𝛼(ℑ))_[𝛼]. So 𝐴^∗(ℑ∨𝔍,𝜔_𝛼(Ω))_[𝛼]⊂𝐴^∗(ℑ,𝜔^∗𝛼(𝔍))_[𝛼]∩𝐴^∗(𝔍,𝜔^∗𝛼(ℑ))_[𝛼]. The proof is completed.

By Theorem 20, we get two important results.

Corollary 24. Let (𝐿^𝑋,𝜔_𝛼(Ω)) be an 𝜔_𝛼-opos. Let ℑ be an 𝛼-ideal, then 𝐴^∗(ℑ,𝜔_𝛼(Ω))_[𝛼]=𝐴^∗(ℑ,𝜔^∗𝛼(ℑ))_[𝛼], and as a result 𝜔^∗𝛼(ℑ)=𝜔^∗𝛼(𝜔^∗𝛼(ℑ))(ℑ).

Proof. By (2) in Theorem 23, take ℑ=𝔍, clearly.

Corollary 25. Let (𝐿^𝑋,𝜔_𝛼(Ω)) be an 𝜔_𝛼-opos. Let ℑ,𝔍⊂𝐿^𝑋 be two 𝛼-ideals. Then,(1)𝜔^∗𝛼(ℑ∨𝔍)=𝜔^∗𝛼(𝜔^∗𝛼(𝔍))(ℑ)=𝜔^∗𝛼(𝜔^∗𝛼(ℑ))(𝔍),(2)𝜔^∗𝛼(ℑ∨𝔍)=𝜔^∗𝛼(𝔍)∨𝜔^∗𝛼(ℑ),(3)𝜔^∗𝛼(ℑ∩𝔍)=𝜔^∗𝛼(𝔍)∩𝜔^∗𝛼(ℑ).

Proof. (1) By (2) in Theorem 23, easy.
(2) Since 𝜔_𝛼(Ω)⊂𝜔^∗𝛼(𝔍) for every 𝛼-ideal 𝔍, by (1) and Theorem 20, we have 𝜔^∗𝛼(ℑ∨𝔍)=𝜔^∗𝛼(𝜔^∗𝛼(𝔍))(ℑ)=𝜔^∗𝛼(𝔍)∨ℑ=𝜔_𝛼(Ω)∨𝜔^∗𝛼(𝔍)∨ℑ=𝜔^∗𝛼(𝔍)∨𝜔^∗𝛼(ℑ).
(3) Clearly, 𝜔^∗𝛼(ℑ∩𝔍)⊂𝜔^∗𝛼(𝔍)∩𝜔^∗𝛼(ℑ). Conversely, 𝑃∈𝜔^∗𝛼(𝔍)∩𝜔^∗𝛼(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∈𝜔^∗𝛼(ℑ∩𝔍).

Theorem 26. Let (𝐿^𝑋,𝜔_𝛼(Ω)) be an 𝜔_𝛼-opos, and let ℑ be an 𝛼-ideal. 𝐴∈𝐿^𝑋. Then, 𝜔^∗𝛼(𝐴)_[𝛼]=𝜔_𝛼(𝐴)_[𝛼] if and only if ℑ⊂𝜔_𝛼(Ω).

Proof. Sufficiency. according to (4) in Theorem 16, 𝜔^∗𝛼(𝐴)_[𝛼]⊂𝜔_𝛼(𝐴)_[𝛼]. On the other hand, if 𝑥∉𝜔^∗𝛼(𝐴)_[𝛼], then 𝑦∉𝐴^∗[𝛼] and 𝑦∉𝐴_[𝛼]. So there are 𝑃∈𝜂_𝜔𝛼(𝑦_𝛼), 𝐼∈ℑ, such that 𝐴_[𝛼]⊂(𝑃∨𝐼)_[𝛼]. Here, we may assume 𝑦∉𝐼_[𝛼], since ℑ is an 𝛼-ideal. Therefore, 𝜔_𝛼(𝐴)_[𝛼]⊂𝜔_𝛼(𝑃∨𝐼)_[𝛼]=(𝑃∨𝐼)_[𝛼]. This implies 𝑦∉𝜔_𝛼(𝐴)_[𝛼]. 𝜔^∗𝛼(𝐴)_[𝛼]⊃𝜔_𝛼(𝐴)_[𝛼].
Necessary. if 𝐴∈ℑ, then 𝐴^∗[𝛼]=∅. Since 𝐴_[𝛼]=(𝐴∨𝐴^∗)_[𝛼]=𝜔^∗𝛼(𝐴)_[𝛼]=𝜔_𝛼(𝐴)_[𝛼], 𝐴∈𝜔_𝛼(Ω).
Since (𝑋,𝜔_𝛼(Ω)_[𝛼]) is an opos on 𝑋. ℑ_[𝛼] is an ideal on 𝑋. We have the following results.

Theorem 27. Let (𝐿^𝑋,𝜔_𝛼(Ω)) be an 𝜔_𝛼-𝑜𝑝𝑜𝑠, and let ℑ be an 𝛼-ideal. Then, for every 𝐴∈𝐿^𝑋, (𝐴_[𝛼])^∗=𝐴^∗[𝛼], and therefore 𝜔^∗𝛼(𝐴)_[𝛼]=𝑐𝑙^∗(𝐴_[𝛼]).

Proof. Since 𝑥∉𝐴^∗[𝛼]⟺∃𝑃∈𝜂_𝜔𝛼𝑥_𝛼,𝐼∈ℑ,𝐴_[𝛼]⊂(𝑃∨𝐼)_[𝛼]⟺∃𝐼𝑃∈ℑ_[𝛼]∈,𝐴𝜔_𝛼[𝛼](Ω)_[𝛼],𝑥∈𝑃_[𝛼]^,∩𝑃_[𝛼]^=𝐼_[𝛼]⟺𝑥∉𝐴_[𝛼]^∗,(8) then (𝐴_[𝛼])^∗=𝐴^∗[𝛼], 𝜔^∗𝛼(𝐴)_[𝛼]=(𝐴∨𝐴^∗)_[𝛼]=𝐴_[𝛼]∪(𝐴_[𝛼])^∗=𝑐𝑙^∗(𝐴_[𝛼]).

Corollary 28. Let (𝐿^𝑋,𝜔_𝛼(Ω)) be an 𝜔_𝛼-opos, and let ℑ be an 𝛼-ideal. Then, for every 𝐴∈𝐿^𝑋, 𝒳^∗𝐴_[𝛼]=𝒳_𝐴^∗[𝛼]=𝒳_(𝐴[𝛼])^∗.

Proof. 𝑥∉(𝒳^∗𝐴_[𝛼])_[𝛼]⇔thereexists𝑃∈𝜂_𝜔𝛼(𝑥_𝛼),𝐼∈ℑ,𝐴_[𝛼]⊂(𝑃∨𝐼)_[𝛼]⇔𝑥∉𝐴^∗[𝛼]⇔𝑥∉(𝒳_𝐴^∗[𝛼])_[𝛼]. So 𝒳^∗𝐴_[𝛼]=𝒳_𝐴^∗[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 𝜔_𝛼-opos and 𝛼-ideal are denoted, respectively, by (𝐿^𝑋,𝜔_𝛼(𝜔_𝐿(Δ))) and ℑ={𝒳_𝐼∶𝐼∈ℐ}. Then, 𝜔^∗𝛼(ℑ,𝜔_𝛼(𝜔_𝐿(Δ)))_[𝛼]=Δ^∗.

Proof. By Theorem 27 and Corollary 28, 𝜔^∗𝛼(ℑ,𝜔_𝛼(𝜔_𝐿(Δ)))_[𝛼]⊂Δ^∗. On other hand, again by Corollary 28, for every 𝑈⊂𝑋, 𝜔^∗𝛼(𝒳_𝑈)_[𝛼]=(𝒳_𝑈∨𝒳^∗𝑈)_[𝛼]=𝑈∪(𝒳_𝑈^∗)_[𝛼]=𝑈∪𝑈^∗=𝑐𝑙^∗(𝑈).

5. Compatibility of 𝜔_𝛼(Ω) with ℑ

Definition 30. Let (𝐿^𝑋,𝜔_𝛼(Ω)) be an 𝜔_𝛼-opos. ℜ⊂𝜔_𝛼(Ω) is called an 𝛼-𝜔_𝛼-remote neighborhood family of 𝐴 (briefly, 𝛼-𝜔_𝛼-RF of 𝐴, if for all 𝑥∈𝐴_[𝛼], there is 𝑃∈ℜ, such that 𝑃∈𝜂_𝜔𝛼(𝑥_𝛼)).

Definition 31. Let (𝐿^𝑋,𝜔_𝛼(Ω)) be an 𝜔_𝛼-opos. Let ℑ be an 𝛼-ideal. 𝜔_𝛼(Ω) is said to be compatible with ℑ, denoted by 𝜔_𝛼(Ω)∼_𝛼ℑ, if for any 𝐴∈𝐿^𝑋 and 𝑥∈𝐴_[𝛼], there are 𝑃∈𝜂_𝜔𝛼(𝑥_𝛼) and 𝐼∈ℑ, such that 𝐴_[𝛼]⊂(𝑃∨𝐼)_[𝛼], then 𝐴∈ℑ.

Example 32. In Example 18, we know 𝜔_𝛼={(0,0),(0,1/2),(0,1),(1/2,1/2),(1/2,1),(1,1/2),(1,1)}, 𝜂_𝜔𝛼(𝑥_𝛼)={(0,0),(0,1/2),(0,1)},𝜂_𝜔𝛼(𝑦_𝛼)={(0,0)}. Take ℑ₁=𝐿^𝑋, ℑ₂={(0,0)}, ℑ₃={(0,0),(1/2,0),(1,0)}, ℑ₄={(0,0),(0,1/2),(0,1)}.
It is easy to check ℑ_𝑖,(𝑖≤4) are all 𝛼-ideals, 𝜔_𝛼(Ω)∼_𝛼ℑ₁ and 𝜔_𝛼(Ω)∼_𝛼ℑ₂. Take 𝐺=(1/2,1/2), then 𝑥∈𝐺_[𝛼] and 𝑅=(0,1/2)∈𝜂_𝜔𝛼(𝑥_𝛼),𝐽=(1/2,0)∈ℑ₃, such that 𝐺_[𝛼]⊂(𝑅∨𝐽)_[𝛼]. But 𝐺∉ℑ₃, therefore 𝜔_𝛼(Ω)≁_𝛼ℑ₃. Furthermore, for all 𝐴∈𝐿^𝑋, if 𝐴_[𝛼]=∅ or 𝐴_[𝛼]={𝑦}, then 𝐴∈ℑ₄. If 𝐴_[𝛼]={𝑥} or 𝐴_[𝛼]={𝑥,𝑦}, there does not exist 𝐸∈𝜂_𝜔𝛼(𝑥_𝛼),𝐼∈ℑ₄, such that 𝐴_[𝛼]⊂(𝑅∨𝐽)_[𝛼]. Therefore, 𝜔_𝛼(Ω)∼_𝛼ℑ₄.

Theorem 33. Let (𝐿^𝑋,𝜔_𝛼(Ω)) be an 𝜔_𝛼-opos, ℑ an ideal, 𝜔_𝛼(Ω)∼_𝛼ℑ. Then, 𝔅=𝜔^∗𝛼(ℑ).

Proof. For every 𝐵∈𝜔^∗𝛼(ℑ), 𝐵^∗[𝛼]⊂𝐵_[𝛼]. Put 𝐴=𝐵−𝐵^∗(ℑ). If 𝑥∈𝐴_[𝛼], then 𝑥∉𝐵^∗[𝛼]. So there are 𝑃∈𝜂_𝜔𝛼(𝑥_𝛼) and 𝐼∈ℑ, such that 𝐵_[𝛼]⊂(𝑃∨𝐼)_[𝛼]. Thus, 𝐴_[𝛼]⊂𝐵_[𝛼]⊂(𝑃∨𝐼)_[𝛼]. Since 𝜔_𝛼(Ω)∼_𝛼ℑ, we get 𝐴∈ℑ. Besides, by (4) in Theorem 16, 𝐵^∗∈𝜔_𝛼(Ω). Notice that 𝐵_[𝛼]=(𝐴∨𝐵^∗)_[𝛼]. Thus, 𝐵∈𝔅, and 𝜔^∗𝛼(ℑ)⊂𝔅. The reverse inclusion is obvious. Therefore, 𝔅=𝜔^∗𝛼(ℑ).

Lemma 34. Let (𝐿^𝑋,𝜔_𝛼(Ω)) be an 𝜔_𝛼-opos, ℑ an 𝛼-ideal. Then, for every 𝐴∈𝐿^𝑋, (𝐴−𝐴^∗(ℑ))_[𝛼]∩(𝐴−𝐴^∗(ℑ))^∗(ℑ)_[𝛼]=∅.

Proof. If 𝑥∈(𝐴−𝐴^∗(ℑ))_[𝛼], so 𝑥∉𝐴^∗(ℑ)_[𝛼]. That is, there are 𝑃∈𝜂_𝜔𝛼(𝑥_𝛼),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 (𝐿^𝑋,𝜔_𝛼(Ω)) be an 𝜔_𝛼-opos, ℑ an ideal. Then, the following statements are equivalent.(1)𝜔_𝛼(Ω)∼_𝛼ℑ.(2)If 𝐴∈𝐿^𝑋 has an 𝛼-𝜔_𝛼-RF ℜ, satisfying for all 𝑃∈ℜ, there is 𝐼∈ℑ, such that 𝐴_[𝛼]⊂(𝑃∨𝐼)_[𝛼], then 𝐴∈ℑ.(3)For all 𝐴∈𝐿^𝑋, (𝐴∧𝐴^∗(ℑ))_[𝛼]=∅⇒𝐴∈ℑ.(4)For all 𝐴∈𝐿^𝑋, 𝐴−𝐴^∗(ℑ)∈ℑ.(5)For all 𝐴∈𝜔^∗𝛼(ℑ), 𝐴−𝐴^∗(ℑ)∈ℑ.(6)For all 𝐴∈𝐿^𝑋, if ∄∅≠𝐵_[𝛼]⊂𝐵^∗(ℑ)_[𝛼], 𝐵_[𝛼]⊂𝐴_[𝛼], then 𝐴∈ℑ.

Proof. (1) ⇒ (2) If 𝐴∈𝐿^𝑋 has a 𝛼-𝜔_𝛼-RF ℜ, satisfying the condition in (2), then for every 𝑥∈𝐴_[𝛼], there are 𝑃∈ℜ and 𝐼∈ℑ such that 𝑃∈𝜂_𝜔𝛼(𝑥_𝛼) and 𝐴_[𝛼]⊂(𝑃∨𝐼)_[𝛼]. As 𝜔_𝛼(Ω)∼_𝛼ℑ, we have 𝐴∈ℑ.
(2) ⇒ (1) By Definitions 30 and 31, clearly.
(1) ⇒ (3) For all 𝐴∈𝐿^𝑋, (𝐴∧𝐴^∗(ℑ))_[𝛼]=∅. If 𝐴_[𝛼]=∅, then 𝐴∈ℑ. If 𝑥∈𝐴_[𝛼], then 𝑥∉𝐴^∗(ℑ)_[𝛼], there are 𝑃∈𝜂_𝜔𝛼(𝑥_𝛼) and 𝐼∈ℑ, such that 𝐴_[𝛼]⊂(𝑃∨𝐼)_[𝛼]. Because 𝜔_𝛼(Ω)∼_𝛼ℑ, we have 𝐴∈ℑ.
(3) ⇒ (1) if 𝐴∈𝐿^𝑋 satisfying, for every 𝑥∈𝐴_[𝛼], there are 𝑃∈𝜂_𝜔𝛼(𝑥_𝛼) and 𝐼∈ℑ, such that 𝐴_[𝛼]⊂(𝑃∨𝐼)_[𝛼]. Then, 𝑥∉𝐴^∗(ℑ)_[𝛼]. By (3), we have 𝐴∈ℑ. Hence, 𝜔_𝛼(Ω)∼_𝛼ℑ.
(3) ⇒ (4) By Lemma 34, obviously.
(4) ⇒ (5) Directly.
(5) ⇒ (1) If for every 𝐴∈𝐿^𝑋 and 𝑥∈𝐴_[𝛼], there are 𝑃∈𝜂_𝜔𝛼(𝑥_𝛼) 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 𝜔^∗𝛼(𝐵)_[𝛼]=(𝐵∨𝐵^∗(ℑ))_[𝛼]=𝐵_[𝛼]. This means 𝐵∈𝜔^∗𝛼(ℑ). By (5), we have 𝐵−𝐵^∗∈ℑ. Since 𝐵−𝐵^∗_[𝛼]=𝐴∨𝐴^∗−𝐴∨𝐴^∗^∗_[𝛼]=𝐴∨𝐴^∗_[𝛼]−𝐴^∗[𝛼]=𝐴_[𝛼],(9) we draw a conclusion: for every 𝑥∈𝐴_[𝛼], 𝑥∉𝐵^∗[𝛼]=𝐴^∗[𝛼]. Therefore, 𝐴∈ℑ.
(4) ⇒ (6) For any 𝐴∈𝐿^𝑋, 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∈𝐿^𝑋, (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 (𝐿^𝑋,𝜔_𝛼(Ω)) be an 𝜔_𝛼-opos, ℑ be an 𝛼-ideal. Then the following statements are equivalent, and implied by 𝜔_𝛼(Ω)∼_𝛼ℑ.(1)For every 𝐴∈𝐿^𝑋, (𝐴∧𝐴^∗)_[𝛼]=∅⇒𝐴^∗[𝛼]=∅.(2)For every 𝐴∈𝐿^𝑋, (𝐴−𝐴^∗)^∗[𝛼]=∅.(3)For every 𝐴∈𝐿^𝑋, (𝐴∧𝐴^∗)^∗[𝛼]=𝐴^∗[𝛼].

Proof. 𝜔_𝛼(Ω)∼_𝛼ℑ⇒(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 (𝐿^𝑋,𝜔_𝛼(Ω)) be an 𝜔_𝛼-opos, ℑ be an 𝛼-ideal. 𝜔_𝛼(Ω)∼_𝛼ℑ. Then for every 𝐴∈𝜔^∗𝛼(ℑ) if and only if there are 𝐵∈𝜔_𝛼(Ω) and 𝐼∈ℑ, such that 𝐴_[𝛼]=(𝐵∨𝐼)_[𝛼].

Proof. Necessity. suppose 𝐴∈𝜔^∗𝛼(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^∗∈𝜔_𝛼(Ω) and 𝐴−𝐴^∗∈ℑ.
Sufficiency. suppose 𝐴_[𝛼]=(𝐵∨𝐼)_[𝛼], 𝜔_𝛼(𝐵)_[𝛼]=𝐵_[𝛼] and 𝐼∈ℑ. Then 𝐴^∗[𝛼]=𝐵^∗[𝛼]∪𝐼^∗[𝛼]=𝐵^∗[𝛼]⊂𝜔_𝛼(𝐵)_[𝛼]=𝐵_[𝛼]⊂𝐴_[𝛼]. Therefore, 𝐴∈𝜔^∗𝛼(ℑ).

6. 𝜔_𝛼-Connectivity and 𝜔^∗𝛼-Connectivity

In this section, we shall introduce four kinds of connectedness and discuss their relations.

Concepts of 𝜔^∗𝛼(ℑ)-separated sets and 𝜔^∗𝛼-connected are defined as:

Definition 38. Let (𝐿^𝑋,𝜔_𝛼) and (𝐿^𝑋,𝜔^∗𝛼(ℑ)) be 𝜔_𝛼-opos and 𝜔^∗𝛼(ℑ)-opos, respectively.(1)𝐴,𝐵∈𝐿^𝑋 are called 𝜔_𝛼-separated sets (resp. 𝜔^∗𝛼(ℑ)-separated set), if 𝜔_𝛼(𝐴)_[𝛼]∩𝐵_[𝛼]=𝜔_𝛼(𝐵)_[𝛼]∩𝐴_[𝛼]=∅, (resp. 𝜔^∗𝛼(𝐴)_[𝛼]∩𝐵_[𝛼]=𝜔^∗𝛼(𝐵)_[𝛼]∩𝐴_[𝛼]=∅).(2)𝐺∈𝐿^𝑋 is called an 𝜔_𝛼-connected set (resp. 𝜔^∗𝛼-connected set), if there not exist 𝜔_𝛼-separated sets 𝐵,𝐶∈𝐿^𝑋, such that, 𝐵_[𝛼]≠∅,𝐶_[𝛼]≠∅, and 𝐺_[𝛼]=𝐵_[𝛼]∪𝐶_[𝛼], (resp. there not exist 𝜔^∗𝛼(ℑ)-separated sets 𝐵,𝐶∈𝐿^𝑋, such that, 𝐵_[𝛼]≠∅,𝐶_[𝛼]≠∅, and 𝐺_[𝛼]=𝐵_[𝛼]∪𝐶_[𝛼]).

Theorem 39. Let (𝐿^𝑋,𝜔^∗𝛼(Ω)) be an 𝜔^∗𝛼-opos. 𝐴 is 𝜔^∗𝛼-connected and 𝐴^∗[𝛼]⊂𝐵^∗[𝛼]⊂𝜔^∗𝛼(𝐴)_[𝛼]. Then, 𝐵 is 𝜔_𝛼-connected.

Proof. Let 𝐵_[𝛼]=𝐸_[𝛼]∪𝐹_[𝛼], and 𝐸,𝐹 are 𝜔^∗𝛼-separated. Take 𝐸^Δ[𝛼]=𝐴_[𝛼]∩𝐸_[𝛼],𝐹^Δ[𝛼]=𝐴_[𝛼]∩𝐹_[𝛼].(10) Then, 𝐴_[𝛼]=𝐸^Δ[𝛼]∪𝐹^Δ[𝛼], and 𝜔_𝛼𝐸^Δ_[𝛼]∩𝐹^Δ[𝛼]=𝐸^Δ^∗∨𝐸^Δ_[𝛼]∩𝐹^Δ[𝛼]⊂𝜔^∗𝛼(𝐸)_[𝛼]∩𝐹_[𝛼]=∅.(11) Similarly, we have 𝜔_𝛼(𝐹^Δ)_[𝛼]∩𝐸^Δ[𝛼]=∅. Thus, 𝐸^Δ,𝐹^Δ are 𝜔^∗𝛼-separated.
As 𝐴 is 𝜔^∗[𝛼]-connected, so 𝐸^Δ[𝛼]=∅, or 𝐹^Δ[𝛼]=∅. Take 𝐸^Δ[𝛼]=∅, for example. Then, 𝐴_[𝛼]=𝐹^Δ[𝛼]. This means 𝐴_[𝛼]⊂𝐹_[𝛼]. As 𝜔^∗𝛼𝐴_[𝛼]⊂𝜔^∗𝛼𝐹_[𝛼], and 𝐸_[𝛼]⊂𝐸_[𝛼]∩𝜔^∗𝛼(𝐴)_[𝛼]⊂𝐸_[𝛼]∩𝜔^∗𝛼(𝐹)_[𝛼]=∅.(12) Then, 𝐵 is 𝜔^∗𝛼-connected.

Theorem 40. Let (𝑋,Δ) be an opos, ℐ⊂𝑋 an ideal. Take ℑ={𝒳_𝐼∶𝐼∈ℐ}. The induced 𝜔_𝛼-opos and 𝜔^∗𝛼(ℑ)-opos are denoted, respectively, by (𝐿^𝑋,𝜔_𝛼(𝜔_𝐿(Δ))) and (𝐿^𝑋,𝜔^∗𝛼(ℑ,𝜔_𝛼(𝜔_𝐿(Δ)))). Then, for every 𝐴∈𝐿^𝑋,(1)𝐴 is 𝜔_𝛼-connected in (𝐿^𝑋,𝜔_𝛼(𝜔_𝐿(Δ))) if and only if 𝐴_[𝛼]⊂𝑋 is connected in (𝑋,Δ);(2)𝐴 is 𝜔^∗𝛼-connected in (𝐿^𝑋,𝜔^∗𝛼(ℑ,𝜔_𝛼(𝜔_𝐿(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 𝑈^−𝜎∩𝑉=𝑉^−𝜎∩𝑈=∅. By Theorem 8, 𝜔_𝛼(1_𝑈)_[𝛼]=𝑈^−𝜎, 𝜔_𝛼(1_𝑉)_[𝛼]=𝑉^−𝜎. So 𝜔_𝛼1_𝑈_[𝛼]∩1_𝑉_[𝛼]=𝜔_𝛼1_𝑉_[𝛼]∩1_𝑈_[𝛼]=∅.(13) This means 1_𝑈,1_𝑉 are 𝜔_𝛼-separated sets, and 𝐴_[𝛼]=(1_𝑈)_[𝛼]∪(1_𝑉)_[𝛼]. This is a contradiction with 𝐴 is 𝜔_𝛼-connected.
Sufficiency. suppose 𝐴 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 8, 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]=∅.(14) Similarly, we get (𝐶_[𝛼])^−𝜎∩𝐵_[𝛼]=∅. This means 𝐴_[𝛼]⊂𝑋 is not connected. It is another contradiction.
(2) By Theorem 27, similarly.
(3) Notice that (𝐴_[𝛼])^−𝜎⊃𝑐𝑙^∗(𝐴_[𝛼]); easily.

Theorem 41. Let (𝐿^𝑋,𝜔^∗𝛼(ℑ)) be an 𝜔^∗𝛼(ℑ)-opos. 𝐴,𝐵∈𝐿^𝑋. Then,(1)𝜔_𝛼-separated ⇒𝜔^∗𝛼-separated,(2)𝐴 and 𝐵 are 𝜔^∗𝛼-separated ⇒𝐴^∗ and 𝐵^∗ are 𝜔_𝛼-separated ⇔𝐴^∗ and 𝐵^∗ are 𝜔^∗𝛼-separated,(3)𝐴,𝐵∈𝜔_𝛼(Ω)∩ℑ, then 𝐴 and 𝐵 are 𝜔_𝛼-separated ⇔𝐴 and 𝐵 are 𝜔^∗𝛼-separated ⇔𝐴_[𝛼]∩𝐵_[𝛼]=∅,(4)𝜔_𝛼(Ω)∼_𝛼ℑ⇒𝐴−𝐴^∗ and 𝐴^∗ are 𝜔^∗𝛼-separated.

Proof. (1) Since 𝜔^∗𝛼(𝐴)_[𝛼]⊂𝜔_𝛼(𝐴)_[𝛼] for every 𝐴∈𝐿^𝑋; easy.
(2) By Theorem 16 (4) and (5), we have 𝜔_𝛼(𝐴^∗)_[𝛼]=𝐴^∗[𝛼]⊂𝜔^∗𝛼(𝐴)_[𝛼] and 𝜔^∗𝛼(𝐴^∗)_[𝛼]=(𝐴^∗)^∗[𝛼]∪𝐴^∗[𝛼]=𝐴^∗[𝛼]=𝜔^∗𝛼(𝐴)_[𝛼]; obvious.
(3) Since 𝐴∈𝜔_𝛼(Ω)∩ℑ, 𝐴^∗[𝛼]=∅ and 𝜔_𝛼(𝐴)_[𝛼]=𝐴_[𝛼]=𝜔^∗𝛼(𝐴)_[𝛼]. The result follows.
(4) Since 𝜔_𝛼(Ω)∼_𝛼ℑ, 𝐴−𝐴^∗∈ℑ. So 𝜔^∗𝛼(𝐴−𝐴^∗)_[𝛼]∩𝐴^∗[𝛼]=(𝐴−𝐴^∗)_[𝛼]∩𝐴^∗[𝛼]=𝜔^∗𝛼(𝐴^∗)_[𝛼]∩(𝐴−𝐴^∗)_[𝛼]=∅.

Example 42. In Example 18, take 𝐵=(0,1/2),𝐷=(1/2,0),𝐸=(1/2,1/2). Then, 𝜔^∗𝛼(𝐵)(𝔍)_[𝛼]=𝜔_𝛼(𝐵)_[𝛼]=𝐵_[𝛼], 𝜔^∗𝛼(𝐷)(𝔍)_[𝛼]={𝑥}. 𝜔_𝛼(𝐷)_[𝛼]={𝑥,𝑦}. Thus, 𝐵 and 𝐷 are 𝜔^∗𝛼(𝔍)-separated, but not 𝜔_𝛼-separated. Therefore, 𝐸=(1/2,1/2) is 𝜔_𝛼-connected, but not 𝜔^∗𝛼(𝔍)-connected. Moreover, 𝐵,𝐸∈𝔍 implies 𝐵^∗(𝔍)_[𝛼]=∅, 𝐸^∗(𝔍)_[𝛼]=∅. So 𝐴^∗ and 𝐵^∗ are 𝜔_𝛼-separated. However, since 𝜔^∗𝛼(𝔍)(𝐵)_[𝛼]=𝐵_[𝛼],𝜔^∗𝛼(𝔍)(𝐸)_[𝛼]=𝐸_[𝛼], 𝐴 and 𝐵 are not 𝜔^∗𝛼-separated.

Theorem 43. Let (𝐿^𝑋,𝜔^∗𝛼(Ω)) be an 𝜔_𝛼-opos, ℑ,𝔍 two 𝛼-ideals. 𝐴,𝐵∈𝐿^𝑋. Then,(1)if ℑ⊂𝔍, 𝜔^∗𝛼(ℑ)-separated ⇒𝜔^∗𝛼(𝔍)-separated,(2)𝜔^∗𝛼(ℑ∩𝔍)-separated if and only if both 𝜔^∗𝛼(ℑ)-separated and 𝜔^∗𝛼(𝔍)-separated,(3)𝜔^∗𝛼(ℑ∨𝔍)-separated if and only if 𝜔^∗𝛼(𝜔^∗𝛼(ℑ))(𝔍)-separated if and only if 𝜔^∗𝛼(𝜔^∗𝛼(𝔍))(ℑ)-separated,(4)𝜔^∗𝛼(𝜔^∗𝛼(ℑ))(ℑ)-separated ⇔𝜔^∗𝛼(ℑ)-separated.

Proof. By Theorems 19(1), 4.4 and Corollary 24, easy.

Corollary 44. Let (𝐿^𝑋,𝜔^∗𝛼(Ω)) be an 𝜔_𝛼-opos, ℑ an 𝛼-ideals. Then,(1)𝜔^∗𝛼-connected ⇒𝜔_𝛼-connected,(2)𝐴^∗ and 𝐵^∗ are 𝜔_𝛼-connected ⇔𝐴^∗ and 𝐵^∗ are 𝜔^∗𝛼-connected ⇒𝐴 and 𝐵 are 𝜔^∗𝛼-connected,(3)𝐴∈𝜔_𝛼(Ω)∩ℑ, then 𝐴 is 𝜔_𝛼-connected ⇔𝐴 is 𝜔^∗𝛼-connected.

Corollary 45. Let (𝐿^𝑋,𝜔^∗𝛼(Ω)) be an 𝜔_𝛼-opos, ℑ,𝔍 two 𝛼-ideals. 𝐴,𝐵∈𝐿^𝑋. Then,(1)if ℑ⊂𝔍, 𝜔^∗𝛼(𝔍)-connected ⇒𝜔^∗𝛼(ℑ)-connected,(2)𝜔^∗𝛼(ℑ∩𝔍)-connected if and only if both 𝜔^∗𝛼(ℑ)-connected and 𝜔^∗𝛼(𝔍)-connected,(3)𝜔^∗𝛼(ℑ∨𝔍)-connected if and only if 𝜔^∗𝛼(𝜔^∗𝛼(ℑ))(𝔍)-connected if and only if 𝜔^∗𝛼(𝜔^∗𝛼(𝔍))(ℑ)-connected.

Acknowledgments

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