Pairwise Weakly Regular-Lindelöf Spaces

Abstract

We will introduce and study the pairwise weakly regular-Lindelöf bitopological spaces and obtain some results. Furthermore, we study the pairwise weakly regular-Lindelöf subspaces and subsets, and investigate some of their characterizations. We also show that a pairwise weakly regular-Lindelöf property is not a hereditary property. Some counterexamples will be considered in order to establish some of their relations.

1. Introduction

The study of bitopological spaces was first initiated by Kelly [1] in 1963 and thereafter a large number of papers have been done to generalize the topological concepts to bitopological setting. In literature, there are several generalizations of the notion of Lindelöf spaces, and these are studied separately for different reasons and purposes. In 1959, Frolík [2] introduced the notion of weakly Lindelöf spaces and in 1996, Cammaroto and Santoro [3] studied and gave further new results about these spaces followed by K𝚤l𝚤çman and Fawakhreh [4]. In the same paper, Cammaroto and Santoro introduced the notion of weakly regular-Lindelöf spaces by using regular covers and leave open the study of this new concept. In 2001, Fawakhreh and K𝚤l𝚤çman [5] studied this new generalization of Lindelöf spaces and obtained some results. Then, K𝚤l𝚤çman and Fawakhreh [6] studied subspaces of this spaces and obtained some results.

Recently, the authors studied pairwise Lindelöfness in [7] and introduced and studied the notion of pairwise weakly Lindelöf spaces in bitopological spaces, see [8], where the authors extended some results that were due to Cammaroto and Santoro [3], K𝚤l𝚤çman and Fawakhreh [4], and Fawakhreh [9]. In [10], the authors also studied the mappings and pairwise continuity on pairwise Lindelöf bitopological spaces. The purpose of this paper is to define the notion of weakly regular-Lindelöf property in bitopological spaces, which we will call pairwise weakly regular- spaces and investigate some of their characterizations. Moreover, we study the pairwise weakly regular-Lindelöf subspaces and subsets and also investigate some of their characterizations.

In Section 3, we will introduce the concept of pairwise weakly regular-Lindelöf bitopological spaces by using pairwise regular cover. This study begin by investigating the 𝑖𝑗-weakly regular-Lindelöf property and some results obtained. Furthermore, we study the relation between 𝑖𝑗-nearly Lindelöf, 𝑖𝑗-almost Lindelöf, 𝑖𝑗-weakly Lindelöf, 𝑖𝑗-almost regular-Lindelöf, 𝑖𝑗-nearly regular-Lindelöf, and 𝑖𝑗-weakly regular-Lindelöf spaces, where 𝑖,𝑗=1 or 2, 𝑖≠𝑗.

In Section 4, we will define the concept of pairwise weakly regular-Lindelöf subspaces and subsets. We will define the concept of pairwise weakly regular-Lindelöf relative to a bitopological space by investigating the 𝑖𝑗-weakly regular-Lindelöf property and obtain some results. The main result obtained is pairwise, and weakly regular-Lindelöf property is not a hereditary property by a counterexample given.

2. Preliminaries

Throughout this paper, all spaces (𝑋,𝜏) and (𝑋,𝜏₁,𝜏₂) (or simply 𝑋) are always mean topological spaces and bitopological spaces, respectively, unless explicitly stated. We always use 𝑖𝑗- to denote the certain properties with respect to topology 𝜏_𝑖 and 𝜏_𝑗, where 𝑖,𝑗∈{1,2} and 𝑖≠𝑗. By 𝑖-int(𝐴) and 𝑖-cl(𝐴), we will mean the interior and the closure of a subset 𝐴 of 𝑋 with respect to topology 𝜏_𝑖, respectively. We denote by int(𝐴) and cl(𝐴) for the interior and the closure of a subset 𝐴 of 𝑋 with respect to topology 𝜏_𝑖 for each 𝑖=1,2, respectively.

If 𝑆⊆𝐴⊆𝑋, then 𝑖-int_𝐴(𝑆) and 𝑖-cl_𝐴(𝑆) will be used to denote the interior and closure of 𝑆 with respect to topology 𝜏_𝑖 in the subspace 𝐴, respectively. By 𝑖-open cover of 𝑋, we mean that the cover of 𝑋 by 𝑖-open sets in 𝑋; similar for the 𝑖𝑗-regular open cover of 𝑋 and so forth. We will use the notation, 𝑋 is 𝑖-Lindelöf space which mean that (𝑋,𝜏_𝑖) is a Lindelöf space, where 𝑖∈{1,2}.

Definition 2.1 (See [11, 12]). A subset 𝑆 of a bitopological space (𝑋,𝜏₁,𝜏₂) is said to be 𝑖𝑗-regular open (resp., 𝑖𝑗-regular closed) if 𝑖-int(𝑗-cl(𝑆))=𝑆 (resp., 𝑖-cl(𝑗-int(𝑆))=𝑆), and 𝑆 is said pairwise regular open (resp., pairwise regular closed) if it is both 𝑖𝑗-regular open and 𝑗𝑖-regular open (resp., 𝑖𝑗-regular closed and 𝑗𝑖-regular closed).

Definition 2.2. Let (𝑋,𝜏₁,𝜏₂) be a bitopological space. A subset 𝐹 of 𝑋 is said to be
(i)𝑖-open if 𝐹 is open with respect to 𝜏_𝑖 in 𝑋, 𝐹 is said open in 𝑋 if it is both 1-open and 2-open in 𝑋, or equivalently, 𝐹∈𝜏₁∩𝜏₂;(ii)𝑖-closed if 𝐹 is closed with respect 𝜏_𝑖 in 𝑋, 𝐹 is said closed in 𝑋 if it is both 1-closed and 2-closed in 𝑋, or equivalently, 𝑋⧵𝐹∈𝜏₁∩𝜏₂;(iii)𝑖-clopen if 𝐹 is both 𝑖-closed and 𝑖-open set in 𝑋, 𝐹 is said clopen in 𝑋 if it is both 1-clopen and 2-clopen in 𝑋;(iv)𝑖𝑗-clopen if 𝐹 is 𝑖-closed and 𝑗-open set in 𝑋, 𝐹 is said clopen if it is both 𝑖𝑗-clopen and 𝑗𝑖-clopen in 𝑋.

Definition 2.3 (See [13]). A bitopological space (𝑋,𝜏₁,𝜏₂) is said to be Lindelöf if the topological space (𝑋,𝜏₁) and (𝑋,𝜏₂) are both Lindelöf. Equivalently, (𝑋,𝜏₁,𝜏₂) is Lindelöf if every 𝑖-open cover of 𝑋 has a countable subcover for each 𝑖=1,2.

Definition 2.4 (See [1, 11]). A bitopological space (𝑋,𝜏₁,𝜏₂) is said to be 𝑖𝑗-regular if for each point 𝑥∈𝑋 and for each 𝑖-open set 𝑉 of 𝑋 containing 𝑥 there exists an 𝑖-open set 𝑈 such that 𝑥∈𝑈⊆𝑗-cl(𝑈)⊆𝑉, and 𝑋 is said to be pairwise regular if it is both 𝑖𝑗-regular and 𝑗𝑖-regular.

Definition 2.5 (See [11, 14]). A bitopological space 𝑋 is said to be 𝑖𝑗-almost regular if for each 𝑥∈𝑋 and for each 𝑖𝑗-regular open set 𝑉 of 𝑋 containing 𝑥 there is an 𝑖𝑗-regular open set 𝑈 such that 𝑥∈𝑈⊆𝑗-cl(𝑈)⊆𝑉, then 𝑋 is said to be pairwise almost regular if it is both 𝑖𝑗-almost regular and 𝑗𝑖-almost regular.

Definition 2.6 (See [11, 12]). A bitopological space 𝑋 is said to be 𝑖𝑗-semiregular if for each 𝑥∈𝑋 and for each 𝑖-open set 𝑉 of 𝑋 containing 𝑥 there is an 𝑖-open set 𝑈 such that 𝑥∈𝑈⊆𝑖-int(𝑗-cl(𝑈))⊆𝑉, and 𝑋 is said pairwise semiregular if it is both 𝑖𝑗-semiregular and 𝑗𝑖-semiregular.

Definition 2.7. A bitopological space 𝑋 is said to be 𝑖𝑗-nearly Lindelöf [15] (resp., 𝑖𝑗-almost Lindelöf [16], 𝑖𝑗-weakly Lindelöf [8]) if for every 𝑖-open cover {𝑈_𝛼∶𝛼∈Δ} of 𝑋 there exists a countable subset {𝛼_𝑛∶𝑛∈ℕ} of Δ such that 𝑋=_𝑛∈ℕ𝑖-int(𝑗-cl(𝑈_𝛼𝑛))(resp.,𝑋=_𝑛∈ℕ𝑗-cl(𝑈_𝛼𝑛),𝑋=𝑗-cl(_𝑛∈ℕ𝑈_𝛼𝑛)),(2.1)and 𝑋 is said pairwise nearly Lindelöf (resp., pairwise almost Lindelöf, pairwise weakly Lindelöf) if it is both 𝑖𝑗-nearly Lindelöf (resp., 𝑖𝑗-almost Lindelöf, 𝑖𝑗-weakly Lindelöf) and 𝑗𝑖-nearly Lindelöf (resp., 𝑗𝑖-almost Lindelöf, 𝑗𝑖-weakly Lindelöf).

Definition 2.8 (See [8]). A subset 𝑆 of a bitopological space 𝑋 is said to be 𝑖𝑗-weakly Lindelöf relative to 𝑋 if for every cover {𝑈_𝛼∶𝛼∈Δ} of 𝑆 by 𝑖-open subsets of 𝑋 such that 𝑆⊆⋃_𝛼∈Δ𝑈_𝛼 there exists a countable subset {𝛼_𝑛∶𝑛∈ℕ} of Δ such that 𝑆⊆𝑗-cl(⋃_𝑛∈ℕ𝑈_𝛼𝑛). 𝑆 is said pairwise weakly Lindelöf relative to 𝑋 if it is both 𝑖𝑗-weakly Lindelöf relative to 𝑋 and 𝑗𝑖-weakly Lindelöf relative to 𝑋.

Definition 2.9 (See [8]). A bitopological space 𝑋 is said to be 𝑖𝑗-nearly paracompact if every cover of 𝑋 by 𝑖𝑗-regular open sets admits a locally finite refinement. 𝑋 is said pairwise nearly paracompact if it is both 𝑖𝑗-nearly paracompact and 𝑗𝑖-nearly paracompact.

3. Pairwise Weakly Regular-Lindelöf Spaces

Definition 3.1 (See [17]). An 𝑖-open cover {𝑈_𝛼∶𝛼∈Δ} of a bitopological space 𝑋 is said to be 𝑖𝑗-regular cover if for every 𝛼∈Δ there exists a nonempty 𝑗𝑖-regular closed subset 𝐶_𝛼 of 𝑋 such that 𝐶_𝛼⊆𝑈_𝛼 and 𝑋=⋃_𝛼∈Δ𝑖-int(𝐶_𝛼). {𝑈_𝛼∶𝛼∈Δ} is said pairwise regular cover if it is both 𝑖𝑗-regular cover and 𝑗𝑖-regular cover.

Definition 3.2. A bitopological space 𝑋 is said to be 𝑖𝑗-almost regular-Lindelöf [17] (resp., 𝑖𝑗-nearly regular-Lindelöf [18]) if for every 𝑖𝑗-regular cover {𝑈_𝛼∶𝛼∈Δ} of 𝑋 there exists a countable subset {𝛼_𝑛∶𝑛∈ℕ} of Δ such that 𝑋=_𝑛∈ℕ𝑗-cl(𝑈_𝛼𝑛)(resp.,𝑋=_𝑛∈ℕ𝑖-int(𝑗-cl(𝑈_𝛼𝑛))),(3.1)then 𝑋 is said pairwise almost regular-Lindelöf (resp., pairwise nearly regular-Lindelöf) if it is both 𝑖𝑗-almost regular-Lindelöf (resp., 𝑖𝑗-nearly regular-Lindelöf) and 𝑗𝑖-almost regular-Lindelöf (resp., 𝑗𝑖-nearly regular-Lindelöf).

Definition 3.3. A bitopological space 𝑋 is said to be 𝑖𝑗-weakly regular-Lindelöf if for every 𝑖𝑗-regular cover {𝑈_𝛼∶𝛼∈Δ} of 𝑋, there exists a countable subset {𝛼_𝑛∶𝑛∈ℕ} of Δ such that 𝑋=𝑗-cl(⋃_𝑛∈ℕ𝑈_𝛼𝑛). 𝑋 is said pairwise weakly regular-Lindelöf if it is both 𝑖𝑗-weakly regular-Lindelöf and 𝑗𝑖-weakly regular-Lindelöf.

Obviously, every 𝑖𝑗-weakly Lindelöf space is 𝑖𝑗-weakly regular-Lindelöf, and every 𝑖𝑗-almost regular-Lindelöf space is 𝑖𝑗-weakly regular-Lindelöf.

Question. Is 𝑖𝑗-weakly regular-Lindelöf spaces implies 𝑖𝑗-weakly Lindelöf?

Question. Is 𝑖𝑗-weakly regular-Lindelöf spaces implies 𝑖𝑗-almost regular-Lindelöf?

The authors expected that the answer of these questions is no. We can answer Question 1. by some restrictions on the space with the following proposition. First of all, we need the following lemmas.

Lemma 3.4 (See [17]). Let 𝑋 be an 𝑖𝑗-almost regular space. Then, for each 𝑥∈𝑋 and for each 𝑖𝑗-regular open subset 𝑊 of 𝑋 containing 𝑥 there exist two 𝑖𝑗-regular open subsets 𝑈 and 𝑉 of 𝑋 such that 𝑥∈𝑈⊆𝑗-cl(𝑈)⊆𝑉⊆𝑗-cl(𝑉)⊆𝑊.

Lemma 3.5 (See [17]). A space 𝑋 is 𝑖𝑗-regular if and only if it is 𝑖𝑗-almost regular and 𝑖𝑗-semiregular.

Proposition 3.6. An 𝑖𝑗-weakly regular-Lindelöf and 𝑖𝑗-regular space 𝑋 is 𝑖𝑗-weakly Lindelöf.

Proof. Let {𝑈_𝛼∶𝛼∈Δ} be an 𝑖𝑗-regular open cover of 𝑋. For each 𝑥∈𝑋, there exists 𝛼_𝑥∈Δ such that 𝑥∈𝑈_𝛼𝑥. Since 𝑋 is 𝑖𝑗-almost regular, there exist two 𝑖𝑗-regular open subsets 𝑉_𝛼𝑥 and 𝑊_𝛼𝑥 of 𝑋 such that 𝑥∈𝑉_𝛼𝑥⊆𝑗-cl(𝑉_𝛼𝑥)⊆𝑊_𝛼𝑥⊆𝑗-cl(𝑊_𝛼𝑥)⊆𝑈_𝛼𝑥 by Lemma 3.4. Since for each 𝛼∈Δ, there exists a 𝑗𝑖-regular closed set 𝑗-cl(𝑉_𝛼𝑥) in 𝑋 such that 𝑗-cl(𝑉_𝛼𝑥)⊆𝑊_𝛼𝑥 and 𝑋=⋃_𝛼∈Δ𝑉_𝛼𝑥=⋃_𝛼∈Δ𝑖-int(𝑗-cl(𝑉_𝛼𝑥)), the family {𝑊_𝛼𝑥∶𝑥∈𝑋} is an 𝑖𝑗-regular cover of 𝑋. Since 𝑋 is 𝑖𝑗-weakly regular-Lindelöf, there exists a countable set of points {𝑥_𝑛∶𝑛∈ℕ} of 𝑋 such that 𝑋=𝑗-cl(⋃_𝑛∈ℕ𝑊_𝛼𝑥𝑛)⊆𝑗-cl(⋃_𝑛∈ℕ𝑈_𝛼𝑥𝑛). So, 𝑋=𝑗-cl(⋃_𝑛∈ℕ𝑈_𝛼𝑥𝑛) and since 𝑋 is 𝑖𝑗-semiregular, therefore 𝑋 is 𝑖𝑗-weakly Lindelöf.

Corollary 3.7. A pairwise weakly regular-Lindelöf and pairwise regular space 𝑋 is pairwise weakly Lindelöf.

Proposition 3.6 implies the following corollaries.

Corollary 3.8. Let 𝑋 be an 𝑖𝑗-regular space. Then, 𝑋 is 𝑖𝑗-weakly regular-Lindelöf if and only if it is 𝑖𝑗-weakly Lindelöf.

Corollary 3.9. Let 𝑋 be a pairwise regular space. Then, 𝑋 is pairwise weakly regular-Lindelöf if and only if it is pairwise weakly Lindelöf.

Definition 3.10 (See [8]). A bitopological space 𝑋 is called 𝑖𝑗-weak 𝑃-space if for each countable family {𝑈_𝑛∶𝑛∈ℕ} of 𝑖-open sets in 𝑋, we have 𝑗-cl(⋃_𝑛∈ℕ𝑈_𝛼𝑛)=⋃_𝑛∈ℕ𝑗-cl(𝑈_𝛼𝑛) then 𝑋 is called pairwise weak 𝑃-space if it is both 𝑖𝑗-weak 𝑃-space and 𝑗𝑖-weak 𝑃-space.

The following proposition shows that in 𝑖𝑗-weak 𝑃-spaces, 𝑖𝑗-almost regular-Lindelöf property equivalent to 𝑖𝑗-weakly regular-Lindelöf property.

Proposition 3.11. Let 𝑋 be an 𝑖𝑗-weak 𝑃-spaces. Then, 𝑋 is 𝑖𝑗-almost regular-Lindelöf if and only if 𝑋 is 𝑖𝑗-weakly regular-Lindelöf.

Proof. The proof follows immediately from the fact that in 𝑖𝑗-weak 𝑃-spaces, ⋃_𝑛∈ℕ𝑗-cl(𝑈_𝛼𝑛)=𝑗-cl(⋃_𝑛∈ℕ𝑈_𝛼𝑛) for any countable family {𝑈_𝑛∶𝑛∈ℕ} of 𝑖-open sets in 𝑋.

Corollary 3.12. Let 𝑋 be a pairwise weak 𝑃-spaces. Then, 𝑋 is pairwise almost regular-Lindelöf if and only if 𝑋 is pairwise weakly regular-Lindelöf.

If 𝑋 is an 𝑖𝑗-almost regular space, then 𝑋 is 𝑖𝑗-almost regular-Lindelöf if and only if it is 𝑖𝑗-nearly Lindelöf (see [17]). Thus, we have the following corollary.

Corollary 3.13. In 𝑖𝑗-almost regular and 𝑖𝑗-weak 𝑃-spaces, 𝑖𝑗-weakly regular-Lindelöf property is equivalent to 𝑖𝑗-nearly Lindelöf property.

Proof. This is a direct consequence of Proposition 3.11 and the previous fact.

Corollary 3.14. In pairwise almost regular and pairwise weak 𝑃-spaces, pairwise weakly regular-Lindelöf property is equivalent to pairwise nearly Lindelöf property.

Lemma 3.15 (See [17]). An 𝑖𝑗-regular and 𝑖𝑗-almost regular-Lindelöf space 𝑋 is 𝑖-Lindelöf.

Corollary 3.16. In 𝑖𝑗-regular and 𝑖𝑗-weak 𝑃-spaces, 𝑖𝑗-weakly regular-Lindelöf property is equivalent to 𝑖-Lindelöf property.

Proof. This is a direct consequence of Proposition 3.11 and Lemma 3.15.

Corollary 3.17. In pairwise regular and pairwise weak 𝑃-spaces, pairwise weakly regular-Lindelöf property is equivalent to Lindelöf property.

Definition 3.18 (See [8]). A subset 𝐸 of a bitopological space 𝑋 is said to be 𝑖-dense in 𝑋 or is an 𝑖-dense subset of 𝑋 if 𝑖-cl(𝐸)=𝑋. 𝐸 is said dense in 𝑋 or is a dense subset of 𝑋 if it is 𝑖-dense in 𝑋 or is an 𝑖-dense subset of 𝑋 for each 𝑖=1,2.

Definition 3.19 (See [8]). A bitopological space 𝑋 is said to be 𝑖-separable if there exists a countable 𝑖-dense subset of 𝑋. 𝑋 is said separable if it is 𝑖-separable for each 𝑖=1,2.

Lemma 3.20 (See [8]). If the bitopological space 𝑋 is 𝑗-separable, then it is 𝑖𝑗-weakly Lindelöf.

Lemma 3.21 (See [18]). An 𝑖𝑗-regular and 𝑖𝑗-nearly regular-Lindelöf space 𝑋 is 𝑖-Lindelöf.

It is clear that every 𝑖𝑗-nearly regular-Lindelöf is 𝑖𝑗-weakly regular-Lindelöf and every 𝑖𝑗-almost regular-Lindelöf space is 𝑖𝑗-weakly regular-Lindelöf, but the converses are not true in general as the following example show.

Example 3.22. Let ℬ be the collection of closed-open intervals in the real line ℝ: ℬ={[𝑎,𝑏)∶𝑎,𝑏∈ℝ,𝑎<𝑏}.(3.2) Hence, ℬ is a base for the lower limit topology 𝜏₁ on ℝ. Choose usual topology as topology 𝜏₂ on ℝ. Thus, (ℝ,𝜏₁,𝜏₂) is a Lindelöf bitopological space (see [19]). Note that, sets of the form (-∞,𝑎),[𝑎,𝑏) or [𝑎,∞) are both 1-open and 1-closed in ℝ, and sets of the form (𝑎,𝑏) and (𝑎,∞) are 1-open in ℝ (see [19]). It is easy to check that (ℝ,𝜏₁,𝜏₂) is 12-regular since for each 𝑥∈ℝ and for each 1-open set of the form [𝑎,𝑏) in ℝ containing 𝑥, there exists a 1-open set [𝑎,𝑏-𝜖) with 𝜖>0 such that 𝑥∈[𝑎,𝑏-𝜖)⊆2-cl[𝑎,𝑏-𝜖)=[𝑎,𝑏-𝜖]⊆[𝑎,𝑏). We left to the reader to check for other forms of 1-open sets in ℝ. It is clear that ℝ is 2-separable since the rational numbers are a countable 2-dense subset of ℝ. So (ℝ×ℝ,𝜏₁×𝜏₁,𝜏₂×𝜏₂) is 12-regular and 2-separable. Thus, ℝ×ℝ is 12-weakly Lindelöf by Lemma 3.20, and so ℝ×ℝ is 12-weakly regular-Lindelöf. It is known that ℝ×ℝ is not 1-Lindelöf since the 1-closed subspace 𝐿={(𝑥,𝑦)∶𝑦=−𝑥} is not 1-Lindelöf for it is a discrete subspace (see [19]). Since ℝ×ℝ is 12-regular, but not 1-Lindelöf, then it is neither 12-almost regular-Lindelöf nor 12-nearly regular-Lindelöf by Lemmas 3.15 and 3.21.

It is clear that every 𝑖𝑗-almost Lindelöf is 𝑖𝑗-weakly Lindelöf, but the converse is not true as in the following example show.

Lemma 3.23 (See [16]). An 𝑖𝑗-regular space is 𝑖𝑗-almost Lindelöf if and only if it is 𝑖-Lindelöf.

Example 3.24. Let (ℝ,𝜏₁,𝜏₂) be a bitopological space defined as in Example 3.22 above. Example 3.22 shows that ℝ×ℝ is 12-weakly Lindelöf, but not 1-Lindelöf. Since ℝ×ℝ is 12-regular, but not 1-Lindelöf, then it is nor 12-almost Lindelöf by Lemma 3.23.

Remark 3.25. Example 3.24 solves the open problem in [8, Question 1].

Lemma 3.26 (See [8]). An 𝑖𝑗-weakly Lindelöf, 𝑖𝑗-regular, and 𝑖𝑗-nearly paracompact bitopological space 𝑋 is 𝑖-Lindelöf.

Proposition 3.27. Let 𝑋 be an 𝑖𝑗-regular and 𝑖𝑗-nearly paracompact spaces. Then, 𝑋 is 𝑖-Lindelöf if and only if 𝑋 is 𝑖𝑗-weakly regular-Lindelöf.

Proof. Let 𝑋 be an 𝑖𝑗-regular, 𝑖𝑗-nearly paracompact, and 𝑖𝑗-weakly regular-Lindelöf space. Then, 𝑋 is 𝑖𝑗-weakly Lindelöf by Proposition 3.6. So 𝑋 is 𝑖-Lindelöf by Lemma 3.26. The converse is obvious.

Corollary 3.28. Let 𝑋 be a pairwise regular and pairwise nearly paracompact spaces. Then, 𝑋 is Lindelöf if and only if 𝑋 is pairwise weakly regular-Lindelöf.

Now, we give a characterization of 𝑖𝑗-weakly regular-Lindelöf spaces.

Theorem 3.29. A bitopological spaces 𝑋 is 𝑖𝑗-weakly regular-Lindelöf if and only if for every family {𝐶_𝛼∶𝛼∈Δ} of 𝑖-closed subsets of 𝑋 such that for each 𝛼∈Δ, there exists a 𝑗-open subset 𝐴_𝛼 of 𝑋 with 𝐴_𝛼⊇𝐶_𝛼 and ⋂_𝛼∈Δ𝑖-cl(𝐴_𝛼)=∅, there exists a countable subfamily {𝐶_𝛼𝑛∶𝑛∈ℕ} such that 𝑗-int(⋂_𝑛∈ℕ𝐶_𝛼𝑛)=∅.

Proof. Let {𝐶_𝛼∶𝛼∈Δ} be a family of 𝑖-closed subsets of 𝑋 such that for each 𝛼∈Δ there exists a 𝑗-open subset 𝐴_𝛼 of 𝑋 with 𝐴_𝛼⊇𝐶_𝛼 and ⋂_𝛼∈Δ𝑖-cl(𝐴_𝛼)=∅. It follows that 𝑋=𝑋⧵(⋂_𝛼∈Δ𝑖-cl(𝐴_𝛼))=⋃_𝛼∈Δ(𝑋⧵𝑖-cl(𝐴_𝛼))=⋃_𝛼∈Δ𝑖-int(𝑋⧵𝐴_𝛼). Since 𝐶_𝛼⊆𝐴_𝛼⊆𝑗-int(𝑖-cl(𝐴_𝛼))⊆𝑖-cl(𝐴_𝛼), then 𝑋⧵𝑖-cl(𝐴_𝛼)⊆𝑋⧵𝑗-int(𝑖-cl(𝐴_𝛼))⊆𝑋⧵𝐶_𝛼, that is, 𝑖-int(𝑋⧵𝐴_𝛼)⊆𝑗-cl(𝑖-int(𝑋⧵𝐴_𝛼))⊆𝑋⧵𝐶_𝛼. Therefore,𝑋=_𝛼∈Δ𝑖-int(𝑋⧵𝐴_𝛼)⊆_𝛼∈Δ(𝑋⧵𝐶_𝛼).(3.3)So 𝑋=⋃_𝛼∈Δ(𝑋⧵𝐶_𝛼) and the family {𝑋⧵𝐶_𝛼∶𝛼∈Δ} is an 𝑖𝑗-regular cover of 𝑋. Since 𝑋 is 𝑖𝑗-weakly regular-Lindelöf, there exists a countable subfamily {𝑋⧵𝐶_𝛼𝑛∶𝑛∈ℕ} such that𝑋=𝑗-cl(_𝑛∈ℕ𝑋⧵𝐶_𝛼𝑛)=𝑗-cl(𝑋⧵(_𝑛∈ℕ𝐶_𝛼𝑛))=𝑋⧵(𝑗-int(_𝑛∈ℕ𝐶_𝛼𝑛)).(3.4)Therefore, 𝑗-int(⋂_𝑛∈ℕ𝐶_𝛼𝑛)=∅.
Conversely, let {𝑈_𝛼∶𝛼∈Δ} be an 𝑖𝑗-regular cover of 𝑋. By Definition 3.1, for each 𝛼∈Δ, 𝑈_𝛼 is 𝑖-open set in 𝑋 and there exists a 𝑗𝑖-regular closed subset 𝐶_𝛼 of 𝑋 such that 𝐶_𝛼⊆𝑈_𝛼 and 𝑋=⋃_𝛼∈Δ𝑖-int(𝐶_𝛼). The family {𝑋⧵𝑈_𝛼∶𝛼∈Δ} of 𝑖-closed subsets of 𝑋 is satisfying the condition, for each 𝛼∈Δ, there exists a 𝑗-open subset 𝑋⧵𝐶_𝛼 of 𝑋 such that 𝑋⧵𝐶_𝛼⊇𝑋⧵𝑈_𝛼 and _𝛼∈Δ𝑖-cl(𝑋⧵𝐶_𝛼)=_𝛼∈Δ(𝑋⧵𝑖-int(𝐶_𝛼))=𝑋⧵(_𝛼∈Δ𝑖-int(𝐶_𝛼))=𝑋⧵𝑋=∅.(3.5)By hypothesis, there exists a countable subset {𝛼_𝑛∶𝑛∈ℕ} of Δ such that 𝑗-int(⋂_𝑛∈ℕ(𝑋⧵𝑈_𝛼𝑛))=∅, that is, 𝑗-int(𝑋⧵⋃_𝑛∈ℕ𝑈_𝛼𝑛)=∅. So 𝑋⧵𝑗-cl(⋃_𝑛∈ℕ𝑈_𝛼𝑛)=∅ and, therefore, 𝑋=𝑗-cl(⋃_𝑛∈ℕ𝑈_𝛼𝑛). This completes the proof.

Corollary 3.30. A bitopological spaces 𝑋 is pairwise weakly regular-Lindelöf if and only if for every family {𝐶_𝛼∶𝛼∈Δ} of closed subsets of 𝑋 such that for each 𝛼∈Δ, there exists an open subset 𝐴_𝛼 of 𝑋 with 𝐴_𝛼⊇𝐶_𝛼 and ⋂_𝛼∈Δcl(𝐴_𝛼)=∅, there exists a countable subfamily {𝐶_𝛼𝑛∶𝑛∈ℕ} such that int(⋂_𝑛∈ℕ𝐶_𝛼𝑛)=∅.

The following diagram illustrates the relationship among the generalizations of pairwise Lindelöf spaces and the generalizations of pairwise regular-Lindelöf spaces in terms of 𝑖𝑗-: (3.6)

4. Pairwise Weakly Regular-Lindelöf Subspaces and Subsets

A subset 𝑆 of a bitopological space 𝑋 is said to be 𝑖𝑗-weakly regular-Lindelöf (resp., pairwise weakly regular-Lindelöf) if 𝑆 is 𝑖𝑗-weakly regular-Lindelöf (resp., pairwise weakly regular-Lindelöf) as a subspace of 𝑋, that is, 𝑆 is 𝑖𝑗-weakly regular-Lindelöf (resp., pairwise weakly regular-Lindelöf) with respect to the inducted bitopology from the bitopology of 𝑋.

Definition 4.1 (See [17]). Let 𝑆 be a subset of a bitopological space 𝑋. A cover {𝑈_𝛼∶𝛼∈Δ} of 𝑆 by 𝑖-open subsets of 𝑋 such that 𝑆⊆⋃_𝛼∈Δ𝑈_𝛼 is said to be 𝑖𝑗-regular cover of 𝑆 by 𝑖-open subsets of 𝑋 if for each 𝛼∈Δ, there exists a nonempty 𝑗𝑖-regular closed subset 𝐶_𝛼 of 𝑋 such that 𝐶_𝛼⊆𝑈_𝛼 and 𝑆⊆⋃_𝛼∈Δ𝑖-int(𝐶_𝛼). {𝑈_𝛼∶𝛼∈Δ} is said pairwise regular cover by open subsets of 𝑋 if it is both 𝑖𝑗-regular cover of 𝑆 by 𝑖-open subsets of 𝑋 and 𝑗𝑖-regular cover of 𝑆 by 𝑗-open subsets of 𝑋.

Definition 4.2 (See [17]). A subset 𝑆 of a bitopological space 𝑋 is said to be 𝑖𝑗-almost regular-Lindelöf relative to 𝑋 if for every 𝑖𝑗-regular cover {𝑈_𝛼∶𝛼∈Δ} of 𝑆 by 𝑖-open subsets of 𝑋 there exists a countable subset {𝛼_𝑛∶𝑛∈ℕ} of Δ such that 𝑆⊆⋃_𝑛∈ℕ𝑗-cl(𝑈_𝛼𝑛). 𝑆 is said pairwise almost regular-Lindelöf relative to 𝑋 if it is both 𝑖𝑗-almost regular-Lindelöf relative to 𝑋 and 𝑗𝑖-almost regular-Lindelöf relative to 𝑋.

Definition 4.3. A subset 𝑆 of a bitopological space 𝑋 is said to be 𝑖𝑗-weakly regular-Lindelöf relative to 𝑋 if for every 𝑖𝑗-regular cover {𝑈_𝛼∶𝛼∈Δ} of 𝑆 by 𝑖-open subsets of 𝑋 there exists a countable subset {𝛼_𝑛∶𝑛∈ℕ} of Δ such that 𝑆⊆𝑗-cl(⋃_𝑛∈ℕ𝑈_𝛼𝑛). 𝑆 is said pairwise weakly regular-Lindelöf relative to 𝑋 if it is both 𝑖𝑗-weakly regular-Lindelöf relative to 𝑋 and 𝑗𝑖-weakly regular-Lindelöf relative to 𝑋.

Obviously, every 𝑖𝑗-weakly Lindelöf relative to the space is 𝑖𝑗-weakly regular-Lindelöf relative to the space and every 𝑖𝑗-almost regular-Lindelöf relative to the space is 𝑖𝑗-weakly regular-Lindelöf relative to the space.

Question. Is 𝑖𝑗-weakly regular-Lindelöf relative to the space implies 𝑖𝑗-weakly Lindelöf relative to the space?

Question. Is 𝑖𝑗-weakly regular-Lindelöf relative to the space implies 𝑖𝑗-almost regular-Lindelöf relative to the space?

The authors expected that the answer of both questions is no.

Theorem 4.4. A subset 𝑆 of a bitopological spaces 𝑋 is 𝑖𝑗-weakly regular-Lindelöf relative to 𝑋 if and only if for every family {𝐶_𝛼∶𝛼∈Δ} of 𝑖-closed subsets of 𝑋 such that for each 𝛼∈Δ there exists a 𝑗-open subset 𝐴_𝛼 of 𝑋 with 𝐴_𝛼⊇𝐶_𝛼 and (⋂_𝛼∈Δ𝑖-cl(𝐴_𝛼))∩𝑆=∅ there exists a countable subfamily {𝐶_𝛼𝑛∶𝑛∈ℕ} such that (𝑗-int(⋂_𝑛∈ℕ𝐶_𝛼𝑛))∩𝑆=∅.

Proof. Let {𝐶_𝛼∶𝛼∈Δ} be a family of 𝑖-closed subsets of 𝑋 such that for each 𝛼∈Δ there exists a 𝑗-open subset 𝐴_𝛼 of 𝑋 with 𝐴_𝛼⊇𝐶_𝛼 and (⋂_𝛼∈Δ𝑖-cl(𝐴_𝛼))∩𝑆=∅. It follows that 𝑆⊆𝑋⧵(⋂_𝛼∈Δ𝑖-cl(𝐴_𝛼))=⋃_𝛼∈Δ(𝑋⧵𝑖-cl(𝐴_𝛼))=⋃_𝛼∈Δ𝑖-int(𝑋⧵𝐴_𝛼). Since 𝐶_𝛼⊆𝐴_𝛼⊆𝑗-int(𝑖-cl(𝐴_𝛼))⊆𝑖-cl(𝐴_𝛼), then 𝑋⧵𝑖-cl(𝐴_𝛼)⊆𝑋⧵𝑗-int(𝑖-cl(𝐴_𝛼))⊆𝑋⧵𝐶_𝛼, that is, 𝑖-int(𝑋⧵𝐴_𝛼)⊆𝑗-cl(𝑖-int(𝑋⧵𝐴_𝛼))⊆𝑋⧵𝐶_𝛼. Therefore, 𝑆⊆⋃_𝛼∈Δ𝑖-int(𝑋⧵𝐴_𝛼)⊆⋃_𝛼∈Δ(𝑋⧵𝐶_𝛼). So 𝑗-cl(𝑖-int(𝑋⧵𝐴_𝛼)) is a 𝑗𝑖-regular closed subset of 𝑋 satisfying the condition of Definition 4.1. Thus, the family {𝑋⧵𝐶_𝛼∶𝛼∈Δ} is an 𝑖𝑗-regular cover of 𝑆 by 𝑖-open subsets of 𝑋. Since 𝑋 is 𝑖𝑗-weakly regular-Lindelöf relative to 𝑋, there exists a countable subfamily {𝑋⧵𝐶_𝛼𝑛∶𝑛∈ℕ} such that𝑆⊆𝑗-cl(_𝑛∈ℕ(𝑋⧵𝐶_𝛼𝑛))=𝑗-cl(𝑋⧵_𝑛∈ℕ𝐶_𝛼𝑛)=𝑋⧵𝑗-int(_𝑛∈ℕ𝐶_𝛼𝑛).(4.1)Therefore, (𝑗-int(⋂_𝑛∈ℕ𝐶_𝛼𝑛))∩𝑆=∅.
Conversely, let {𝑈_𝛼∶𝛼∈Δ} be an 𝑖𝑗-regular cover of 𝑆 by 𝑖-open subsets of 𝑋. By Definition 4.1, for each 𝛼∈Δ, there exists a 𝑗𝑖-regular closed subset 𝐶_𝛼 of 𝑋 such that 𝐶_𝛼⊆𝑈_𝛼 and 𝑆⊆⋃_𝛼∈Δ𝑖-int(𝐶_𝛼). The family {𝑋⧵𝑈_𝛼∶𝛼∈Δ} of 𝑖-closed subsets of 𝑋 is satisfying the condition, for each 𝛼∈Δ, there exists a 𝑗-open set 𝑋⧵𝐶_𝛼⊇𝑋⧵𝑈_𝛼 with𝑆⊆_𝛼∈Δ𝑖-int(𝐶_𝛼)=𝑋⧵(_𝛼∈Δ𝑋⧵𝑖-int(𝐶_𝛼))=𝑋⧵(_𝛼∈Δ𝑖-cl(𝑋⧵𝐶_𝛼)),(4.2)then it follows that, (⋂_𝛼∈Δ𝑖-cl(𝑋⧵𝐶_𝛼))∩𝑆=∅. By hypothesis, there exists a countable subset {𝛼_𝑛∶𝑛∈ℕ} of Δ such that(𝑗-int(_𝑛∈ℕ(𝑋⧵𝑈_𝛼𝑛)))∩𝑆=∅,thatis,(𝑗-int(𝑋⧵_𝑛∈ℕ𝑈_𝛼𝑛))∩𝑆=∅.(4.3)Thus we have, (𝑋⧵𝑗-cl(⋃_𝑛∈ℕ𝑈_𝛼𝑛))∩𝑆=∅ and, therefore, 𝑆⊆𝑗-cl(⋃_𝑛∈ℕ𝑈_𝛼𝑛). This completes the proof.

Corollary 4.5. A subset 𝑆 of a bitopological spaces 𝑋 is pairwise weakly regular-Lindelöf relative to 𝑋 if and only if for every family {𝐶_𝛼∶𝛼∈Δ} of closed subsets of 𝑋 such that for each 𝛼∈Δ there exists an open subset 𝐴_𝛼 of 𝑋 with 𝐴_𝛼⊇𝐶_𝛼 and (⋂_𝛼∈Δcl(𝐴_𝛼))∩𝑆=∅, there exists a countable subfamily {𝐶_𝛼𝑛∶𝑛∈ℕ} such that (int(⋂_𝑛∈ℕ𝐶_𝛼𝑛))∩𝑆=∅.

Proposition 4.6. A subset 𝑆 of a space 𝑋 is 𝑖𝑗-weakly regular-Lindelöf relative to 𝑋 if and only if for every family {𝑈_𝛼∶𝛼∈Δ} of 𝑖𝑗-regular open subsets of 𝑋 satisfying the conditions 𝑆⊆⋃_𝛼∈Δ𝑈_𝛼 and for each 𝛼∈Δ there exists a nonempty 𝑗𝑖-regular closed subset 𝐶_𝛼 of 𝑋 such that 𝐶_𝛼⊆𝑈_𝛼 and 𝑆⊆⋃_𝛼∈Δ𝑖-int(𝐶_𝛼), then there exists a countable subset {𝛼_𝑛∶𝑛∈ℕ} of Δ such that 𝑆⊆𝑗-cl(⋃_𝑛∈ℕ𝑈_𝛼𝑛).

Proof. The necessity is obvious by the Definitions 4.1 and 4.2 since every 𝑖𝑗-regular open set in 𝑋 is 𝑖-open. For the sufficiency, let {𝑈_𝛼∶𝛼∈Δ} be a family of 𝑖-open sets in 𝑋 satisfying the conditions of Definition 4.1 above. Then {𝑖-int(𝑗-cl(𝑈_𝛼))∶𝛼∈Δ} is a family of 𝑖𝑗-regular open sets in 𝑋 satisfying the conditions of the theorem, since for each 𝛼∈Δ, we have 𝐶_𝛼⊆𝑈_𝛼⊆𝑖-int(𝑗-cl(𝑈_𝛼)). By hypothesis, there exists a countable subset {𝛼_𝑛∶𝑛∈ℕ} of Δ such that𝑆⊆⊆𝑗𝑗--ccll((𝑗-c_𝑛∈ℕl((𝑖-in_𝑛∈ℕt(𝑗-cl(𝑈_𝛼𝑛))))⊆𝑗-cl(_𝑛∈ℕ𝑗-cl(𝑈_𝛼𝑛))𝑈_𝛼𝑛))=𝑗-cl(_𝑛∈ℕ𝑈_𝛼𝑛).(4.4)This implies that 𝑆 is 𝑖𝑗-weakly regular-Lindelöf relative to 𝑋 and completes the proof.

Corollary 4.7. A subset 𝑆 of a space 𝑋 is pairwise weakly regular-Lindelöf relative to 𝑋 if and only if for every family {𝑈_𝛼∶𝛼∈Δ} of pairwise regular open subsets of 𝑋 satisfying the conditions 𝑆⊆⋃_𝛼∈Δ𝑈_𝛼 and for each 𝛼∈Δ there exists a nonempty pairwise regular closed subset 𝐶_𝛼 of 𝑋 such that 𝐶_𝛼⊆𝑈_𝛼 and 𝑆⊆⋃_𝛼∈Δint(𝐶_𝛼), then there exists a countable subset {𝛼_𝑛∶𝑛∈ℕ} of Δ such that 𝑆⊆cl(⋃_𝑛∈ℕ𝑈_𝛼𝑛).

Proposition 4.8. If {𝐴_𝑘∶𝑘∈ℕ} is a countable family of subsets of a space 𝑋 such that each 𝐴_𝑘 is 𝑖𝑗-weakly regular-Lindelöf relative to 𝑋, then ⋃{𝐴_𝑘∶𝑘∈ℕ} is 𝑖𝑗-weakly regular-Lindelöf relative to 𝑋.

Proof. Let {𝑈_𝛼∶𝛼∈Δ} be an 𝑖𝑗-regular cover of ⋃{𝐴_𝑘∶𝑘∈ℕ} by 𝑖-open subsets of 𝑋. Then for each 𝛼∈Δ, there exists a nonempty 𝑗𝑖-regular closed subset 𝐶_𝛼 of 𝑋 such that 𝐶_𝛼⊆𝑈_𝛼 and ⋃_𝑘∈ℕ𝐴_𝑘⊆⋃_𝛼∈Δ𝑖-int(𝐶_𝛼). Let Δ_𝑘={𝛼∈Δ∶𝑈_𝛼∩𝐴_𝑘≠∅}, then for each 𝛼_𝑘∈Δ_𝑘⊆Δ there exists a nonempty 𝑗𝑖-regular closed subset 𝐶_𝛼𝑘 of 𝑋 such that 𝐶_𝛼𝑘⊆𝑈_𝛼𝑘 and 𝐴_𝑘⊆⋃_{𝛼𝑘∈Δ𝑘}𝑖-int(𝐶_𝛼𝑘). So {𝑈_𝛼𝑘∶𝛼_𝑘∈Δ_𝑘} is an 𝑖𝑗-regular cover of 𝐴_𝑘 by 𝑖-open subsets of 𝑋. Since 𝐴_𝑘 is 𝑖𝑗-weakly regular-Lindelöf relative to 𝑋, there exists a countable subfamily {𝑈_𝛼𝑘𝑛∶𝑛∈ℕ} such that 𝐴_𝑘⊆𝑗-cl(⋃_𝑛∈ℕ𝑈_𝛼𝑘𝑛). But a countable union of countable sets is countable, so_𝑘∈ℕ𝐴_𝑘⊆_𝑘∈ℕ(𝑗-cl(_𝑛∈ℕ𝑈_𝛼𝑘𝑛))⊆𝑗-cl(_𝑘∈ℕ(_𝑛∈ℕ𝑈_𝛼𝑘𝑛))=𝑗-cl(_𝑛∈ℕ𝑈_𝛼𝑘𝑛).(4.5)This implies that ⋃{𝐴_𝑘∶𝑘∈ℕ} is 𝑖𝑗-weakly regular-Lindelöf relative to 𝑋 and completes the proof.

Corollary 4.9. If {𝐴_𝑘∶𝑘∈ℕ} is a countable family of subsets of a space 𝑋 such that each 𝐴_𝑘 is pairwise weakly regular-Lindelöf relative to 𝑋, then ⋃{𝐴_𝑘∶𝑘∈ℕ} is pairwise weakly regular-Lindelöf relative to 𝑋.

Proposition 4.10. If 𝑆 is an 𝑖𝑗-weakly regular-Lindelöf subspace of a bitopological space 𝑋, then 𝑆 is 𝑖𝑗-weakly regular-Lindelöf relative to 𝑋.

Proof. Let {𝑈_𝛼∶𝛼∈Δ} be an 𝑖𝑗-regular cover of 𝑆 by 𝑖-open subsets of 𝑋. Then, for each 𝛼∈Δ there exists a nonempty 𝑗𝑖-regular closed subset 𝐶_𝛼 of 𝑋 such that 𝐶_𝛼⊆𝑈_𝛼 and 𝑆⊆⋃_𝛼∈Δ𝑖-int_𝑋(𝐶_𝛼). For each 𝛼∈Δ, we have 𝑖-int_𝑋(𝐶_𝛼)∩𝑆 and 𝑈_𝛼∩𝑆 are 𝑖-open sets in 𝑆, and 𝐶_𝛼∩𝑆 is 𝑗-closed set in 𝑆. Since for each 𝛼∈Δ, there exists a 𝑗𝑖-regular closed set 𝑗-cl_𝑆(𝑖-int_𝑋(𝐶_𝛼)∩𝑆) in 𝑆 such that 𝑗-cl_𝑆(𝑖-int_𝑋(𝐶_𝛼)∩𝑆)⊆𝐶_𝛼∩𝑆⊆𝑈_𝛼∩𝑆 and𝑆=(_𝛼∈Δ𝑖-int_𝑋(𝐶_𝛼))∩𝑆=_𝛼∈Δ(𝑖-int_𝑋(𝐶_𝛼)∩𝑆)⊆_𝛼∈Δ𝑖-int_𝑆(𝑗-cl_𝑆(𝑖-int_𝑋(𝐶_𝛼)∩𝑆)),(4.6)that is, 𝑆=⋃_𝛼∈Δ𝑖-int_𝑆(𝑗-cl_𝑆(𝑖-int_𝑋(𝐶_𝛼)∩𝑆)), then the family {𝑈_𝛼∩𝑆∶𝛼∈Δ} is an 𝑖𝑗-regular cover of 𝑆. Since 𝑆 is an 𝑖𝑗-weakly regular-Lindelöf subspace of 𝑋, there exists a countable subset {𝛼_𝑛∶𝑛∈ℕ} of Δ such that𝑆=𝑗-cl_𝑆(_𝑛∈ℕ(𝑈_𝛼𝑛∩𝑆))=(𝑗-cl_𝑋(_𝑛∈ℕ(𝑈_𝛼𝑛∩𝑆)))∩𝑆⊆𝑗-cl_𝑋(_𝑛∈ℕ𝑈_𝛼𝑛).(4.7)This shows that 𝑆 is 𝑖𝑗-weakly regular-Lindelöf relative to 𝑋.

Corollary 4.11. If 𝑆 is a pairwise weakly regular-Lindelöf subspace of a bitopological space 𝑋, then 𝑆 is pairwise weakly regular-Lindelöf relative to 𝑋.

Question. Is the converse of Proposition 4.10 above true?

The authors expected that the answer is no.

Theorem 4.12. If every 𝑖𝑗-regular closed proper subset of a bitopological space 𝑋 is 𝑖𝑗-weakly regular-Lindelöf relative to 𝑋, then 𝑋 is 𝑖𝑗-weakly regular-Lindelöf.

Proof. Let {𝑈_𝛼∶𝛼∈Δ} be an 𝑖𝑗-regular cover of 𝑋. For each 𝛼∈Δ, there exists a nonempty 𝑗𝑖-regular closed subset 𝐶_𝛼 of 𝑋 such that 𝐶_𝛼⊆𝑈_𝛼 and 𝑋=⋃_𝛼∈Δ𝑖-int(𝐶_𝛼). Fix an arbitrary 𝛼₀∈Δ and let Δ^∗=Δ⧵{𝛼₀}. Put 𝐾=𝑋⧵(𝑖-int(𝐶_𝛼0)), then 𝐾 is an 𝑖𝑗-regular closed subset of 𝑋 and 𝐾⊆⋃_𝛼∈Δ∗𝑖-int(𝐶_𝛼). Therefore, {𝑈_𝛼∶𝛼∈Δ^∗} is an 𝑖𝑗-regular cover of 𝐾 by 𝑖-open subsets of 𝑋 by Definition 4.1. By hypothesis, 𝐾 is 𝑖𝑗-weakly regular-Lindelöf relative to 𝑋, hence there exists a countable subset {𝛼_𝑛∶𝑛∈ℕ^∗} of Δ^∗ such that 𝐾⊆𝑗-cl(⋃_{𝑛∈ℕ∗}𝑈_𝛼𝑛). So, we have𝑋==𝐾∪(𝑖-int(𝐶_𝛼0))⊆𝐾∪(𝑗-cl(𝑈_𝛼0))⊆(𝑗-cl(_{𝑛∈ℕ∗}𝑈_𝛼𝑛))∪(𝑗-cl(𝑈_𝛼0))_𝑛∈ℕ𝑗-cl(𝑈_𝛼𝑛).(4.8)So 𝑋=𝑗-cl(⋃_𝑛∈ℕ𝑈_𝛼𝑛) and this shows that 𝑋 is 𝑖𝑗-weakly regular-Lindelöf.

Corollary 4.13. If every pairwise regular closed proper subset of a bitopological space 𝑋 is pairwise weakly regular-Lindelöf relative to 𝑋, then 𝑋 is pairwise weakly regular-Lindelöf.

It is very clear that Theorem 4.12 implies the following corollaries.

Corollary 4.14. If every 𝑖𝑗-regular closed subset of a bitopological space 𝑋 is 𝑖𝑗-weakly regular-Lindelöf relative to 𝑋, then 𝑋 is 𝑖𝑗-weakly regular-Lindelöf.

Corollary 4.15. If every pairwise regular closed subset of a bitopological space 𝑋 is pairwise weakly regular-Lindelöf relative to 𝑋, then 𝑋 is pairwise weakly regular-Lindelöf.

Note that, the space 𝑋 in above propositions is any bitopological space. If we consider 𝑋 itself is an 𝑖𝑗-weakly regular-Lindelöf, we have the following results.

Theorem 4.16. Let 𝑋 be an 𝑖𝑗-weakly regular-Lindelöf space. If 𝐴 is a proper 𝑖𝑗-clopen subset of 𝑋, then 𝐴 is 𝑖𝑗-weakly regular-Lindelöf relative to 𝑋.

Proof. Let {𝑈_𝛼∶𝛼∈Δ} be an 𝑖𝑗-regular cover of 𝐴 by 𝑖-open subsets of 𝑋. Hence the family {𝑈_𝛼∶𝛼∈Δ}∪{𝑋⧵𝐴} is an 𝑖𝑗-regular cover of 𝑋 since 𝑋⧵𝐴 is a proper 𝑗𝑖-clopen subset of 𝑋 is also a 𝑗𝑖-regular closed subset of 𝑋. Since 𝑋 is 𝑖𝑗-weakly regular-Lindelöf, there exists a countable subfamily {𝑋⧵𝐴,𝑈_𝛼1,𝑈_𝛼2,…} such that𝑋=𝑗-cl(_𝑛∈ℕ𝑈_𝛼𝑛)∪𝑗-cl(𝑋⧵𝐴)=(𝑗-cl(_𝑛∈ℕ𝑈_𝛼𝑛))∪(𝑋⧵𝐴).(4.9)But 𝐴 and 𝑋⧵𝐴 are disjoint; therefore, we have 𝐴⊆𝑗-cl(⋃_𝑛∈ℕ𝑈_𝛼𝑛). This completes the proof.

Corollary 4.17. Let 𝑋 be a pairwise weakly regular-Lindelöf space. If 𝐴 is a proper clopen subset of 𝑋, then 𝐴 is pairwise weakly regular-Lindelöf relative to 𝑋.

It is very clear that Theorem 4.16 implies the following corollary.

Corollary 4.18. Let 𝑋 be an 𝑖𝑗-weakly regular-Lindelöf space. If 𝐴 is an 𝑖𝑗-clopen subset of 𝑋, then 𝐴 is 𝑖𝑗-weakly regular-Lindelöf relative to 𝑋.

Corollary 4.19. Let 𝑋 be a pairwise weakly regular-Lindelöf space. If 𝐴 is a clopen subset of 𝑋, then 𝐴 is pairwise weakly regular-Lindelöf relative to 𝑋.

Question 6.Is 𝑖-closed subspace of an 𝑖𝑗-weakly regular-Lindelöf space 𝑋𝑖𝑗-weakly regular-Lindelöf?

Question 7.Is 𝑖𝑗-regular closed subspace of an 𝑖𝑗-weakly regular-Lindelöf space 𝑋𝑖𝑗-weakly regular-Lindelöf?

The authors expected that the answer of both questions is no. Observe that the condition in Theorem 4.16 that a subset should be 𝑖𝑗-clopen is necessary and it is not sufficient to be only 𝑖-open or 𝑖𝑗-regular open as example below shows. Arbitrary subspaces of 𝑖𝑗-weakly regular-Lindelöf spaces need not be 𝑖𝑗-weakly regular-Lindelöf nor 𝑖𝑗-weakly regular-Lindelöf relative to the spaces. An 𝑖-open or 𝑖𝑗-regular open subset of an 𝑖𝑗-weakly regular-Lindelöf space is neither 𝑖𝑗-weakly regular-Lindelöf nor 𝑖𝑗-weakly regular-Lindelöf relative to the spaces as in the following example also show. We need the following lemma (see [20, page 11]).

Lemma 4.20. If 𝐴 is a countable subset of ordinals Ω not containing 𝜔₁, where 𝜔₁ being the first uncountable ordinal, then sup𝐴<𝜔₁.

Example 4.21. Let Ω denote the set of ordinals which are less than or equal to the first uncountable ordinal number 𝜔₁, that is, Ω=[1,𝜔₁]. This Ω is an uncountable well-ordered set with a largest element 𝜔₁, having the property that if 𝛼∈Ω with 𝛼<𝜔₁, then {𝛽∈Ω∶𝛽≤𝛼} is countable. Since Ω is a totally ordered space, it can be provided with its order topology. Let us denote this order topology by 𝜏₁. Choose discrete topology as another topology for Ω denoted by 𝜏₂. So (Ω,𝜏₁,𝜏₂) form a bitopological space. Now it is known that Ω is a 1-Lindelöf space [20], so it is 12-weakly Lindelöf and thus 12-weakly regular-Lindelöf. The subspace Ω₀=Ω⧵{𝜔₁}=[1,𝜔₁), however, is not 1-Lindelöf (see [20]). We notice that Ω₀ is 1-open subspace of Ω and also 12-regular open subset of Ω. Observe that Ω₀ is not 12-weakly regular-Lindelöf by Corollary 3.16 since it is 12-regular and 12-weak 𝑃-space. Moreover, Ω₀ is not 12-weakly regular-Lindelöf relative to Ω. In fact, the family {[1,𝛼)∶𝛼∈Ω₀} of 1-open sets in Ω is 12-regular cover of Ω₀ by 1-open subsets of Ω because Ω₀⊆⋃_𝛼∈Ω0[1,𝛼) and for each 𝛼∈Ω₀, there exists a nonempty 21-regular closed subset [1,𝛼) of Ω such that [1,𝛼)⊆[1,𝛼) and Ω₀⊆⋃_𝛼∈Ω0[1,𝛼)=⋃_𝛼∈Ω01-int([1,𝛼)). But the family {[1,𝛼)∶𝛼∈Ω₀} has no countable subfamily {[1,𝛼_𝑛)∶𝑛∈ℕ} such that Ω₀⊆2-cl(⋃_𝑛∈ℕ[1,𝛼_𝑛))=⋃_𝑛∈ℕ[1,𝛼_𝑛). For if {[1,𝛼₁),[1,𝛼₂),…} satisfy the condition: 2-closures of unions of it elements cover Ω₀, then sup{𝛼₁,𝛼₂,…}=𝜔₁ which is impossible by Lemma 4.20.

So we can conclude that an 𝑖𝑗-weakly regular-Lindelöf property is not hereditary property and, therefore, pairwise weakly regular-Lindelöf property is not so.

Acknowledgments

The authors gratefully acknowledge the Ministry of Higher Education, Malaysia, and University Putra Malaysia (UPM) that this research was partially supported under the Fundamental Grant Project 01-01-07-158FR.