Abstract and Applied Analysis
Volume 2008 (2008), Article ID 184243, 13 pages
doi:10.1155/2008/184243
Research Article

Pairwise Weakly Regular-Lindelöf Spaces

Adem Kılıçman1 and Zabidin Salleh2

1Department of Mathematics, University Malaysia Terengganu, 21030 Kuala Terengganu, Terengganu, Malaysia
2Institute for Mathematical Research, University Putra Malaysia, 43400 UPM Serdang, Selangor Darul Ehsan, Malaysia

Received 19 December 2007; Accepted 4 April 2008

Academic Editor: Agacik Zafer

Copyright © 2008 Adem Kılıçman and Zabidin Salleh. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 K 𝚤 l 𝚤 çman 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 K 𝚤 l 𝚤 çman [5] studied this new generalization of Lindelöf spaces and obtained some results. Then, K 𝚤 l 𝚤 çman 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], K 𝚤 l 𝚤 çman 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 𝑖 𝑗 -weakly regular-Lindelöf property and some results obtained. Furthermore, we study the relation between 𝑖 𝑗 -nearly Lindelöf, 𝑖 𝑗 -almost Lindelöf, 𝑖 𝑗 -weakly Lindelöf, 𝑖 𝑗 -almost regular-Lindelöf, 𝑖 𝑗 -nearly regular-Lindelöf, and 𝑖 𝑗 -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 𝑖 𝑗 -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 , 𝜏 1 , 𝜏 2 ) (or simply 𝑋 ) are always mean topological spaces and bitopological spaces, respectively, unless explicitly stated. We always use 𝑖 𝑗 - to denote the certain properties with respect to topology 𝜏 𝑖 and 𝜏 𝑗 , where 𝑖 , 𝑗 ∈ { 1 , 2 } and 𝑖 ≠ 𝑗 . By 𝑖 - i n t ( 𝐴 ) and 𝑖 - c l ( 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 i n t ( 𝐴 ) and c l ( 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 - i n t 𝐴 ( 𝑆 ) and 𝑖 - c l 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 𝑖 𝑗 -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 , 𝜏 1 , 𝜏 2 ) is said to be 𝑖 𝑗 -regular open (resp., 𝑖 𝑗 -regular closed) if 𝑖 - i n t ( 𝑗 - c l ( 𝑆 ) ) = 𝑆 (resp., 𝑖 - c l ( 𝑗 - i n t ( 𝑆 ) ) = 𝑆 ) , and 𝑆 is said pairwise regular open (resp., pairwise regular closed) if it is both 𝑖 𝑗 -regular open and 𝑗 𝑖 -regular open (resp., 𝑖 𝑗 -regular closed and 𝑗 𝑖 -regular closed).

Definition 2.2. Let ( 𝑋 , 𝜏 1 , 𝜏 2 ) 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 ∈ 𝜏 1 ∩ 𝜏 2 ;(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 ∈ 𝜏 1 ∩ 𝜏 2 ;(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) 𝑖 𝑗 -clopen if 𝐹 is 𝑖 -closed and 𝑗 -open set in 𝑋 , 𝐹 is said clopen if it is both 𝑖 𝑗 -clopen and 𝑗 𝑖 -clopen in 𝑋 .

Definition 2.3 (See [13]). A bitopological space ( 𝑋 , 𝜏 1 , 𝜏 2 ) is said to be Lindelöf if the topological space ( 𝑋 , 𝜏 1 ) and ( 𝑋 , 𝜏 2 ) are both Lindelöf. Equivalently, ( 𝑋 , 𝜏 1 , 𝜏 2 ) 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 , 𝜏 1 , 𝜏 2 ) is said to be 𝑖 𝑗 -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 - c l ( 𝑈 ) ⊆ 𝑉 , and 𝑋 is said to be pairwise regular if it is both 𝑖 𝑗 -regular and 𝑗 𝑖 -regular.

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

Definition 2.6 (See [11, 12]). A bitopological space 𝑋 is said to be 𝑖 𝑗 -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 - i n t ( 𝑗 - c l ( 𝑈 ) ) ⊆ 𝑉 , and 𝑋 is said pairwise semiregular if it is both 𝑖 𝑗 -semiregular and 𝑗 𝑖 -semiregular.

Definition 2.7. A bitopological space 𝑋 is said to be 𝑖 𝑗 -nearly Lindelöf [15] (resp., 𝑖 𝑗 -almost Lindelöf [16], 𝑖 𝑗 -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 ∈ ℕ 𝑖 - i n t ( 𝑗 - c l ( 𝑈 𝛼 𝑛 ) ) ( r e s p . , 𝑋 = 𝑛 ∈ ℕ 𝑗 - c l ( 𝑈 𝛼 𝑛 ) , 𝑋 = 𝑗 - c l ( 𝑛 ∈ ℕ 𝑈 𝛼 𝑛 ) ) , ( 2 . 1 ) and 𝑋 is said pairwise nearly Lindelöf (resp., pairwise almost Lindelöf, pairwise weakly Lindelöf) if it is both 𝑖 𝑗 -nearly Lindelöf (resp., 𝑖 𝑗 -almost Lindelöf, 𝑖 𝑗 -weakly Lindelöf) and 𝑗 𝑖 -nearly Lindelöf (resp., 𝑗 𝑖 -almost Lindelöf, 𝑗 𝑖 -weakly Lindelöf).

Definition 2.8 (See [8]). A subset 𝑆 of a bitopological space 𝑋 is said to be 𝑖 𝑗 -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 ∈ Δ 𝑈 𝛼 there exists a countable subset { 𝛼 𝑛 ∶ 𝑛 ∈ ℕ } of Δ such that 𝑆 ⊆ 𝑗 - ⋃ c l ( 𝑛 ∈ ℕ 𝑈 𝛼 𝑛 ) . 𝑆 is said pairwise weakly Lindelöf relative to 𝑋 if it is both 𝑖 𝑗 -weakly Lindelöf relative to 𝑋 and 𝑗 𝑖 -weakly Lindelöf relative to 𝑋 .

Definition 2.9 (See [8]). A bitopological space 𝑋 is said to be 𝑖 𝑗 -nearly paracompact if every cover of 𝑋 by 𝑖 𝑗 -regular open sets admits a locally finite refinement. 𝑋 is said pairwise nearly paracompact if it is both 𝑖 𝑗 -nearly paracompact and 𝑗 𝑖 -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 𝑖 𝑗 -regular cover if for every 𝛼 ∈ Δ there exists a nonempty 𝑗 𝑖 -regular closed subset 𝐶 𝛼 of 𝑋 such that 𝐶 𝛼 ⊆ 𝑈 𝛼 and ⋃ 𝑋 = 𝛼 ∈ Δ 𝑖 - i n t ( 𝐶 𝛼 ) . { 𝑈 𝛼 ∶ 𝛼 ∈ Δ } is said pairwise regular cover if it is both 𝑖 𝑗 -regular cover and 𝑗 𝑖 -regular cover.

Definition 3.2. A bitopological space 𝑋 is said to be 𝑖 𝑗 -almost regular-Lindelöf [17] (resp., 𝑖 𝑗 -nearly regular-Lindelöf [18]) if for every 𝑖 𝑗 -regular cover { 𝑈 𝛼 ∶ 𝛼 ∈ Δ } of 𝑋 there exists a countable subset { 𝛼 𝑛 ∶ 𝑛 ∈ ℕ } of Δ such that 𝑋 = 𝑛 ∈ ℕ 𝑗 - c l ( 𝑈 𝛼 𝑛 ) ( r e s p . , 𝑋 = 𝑛 ∈ ℕ 𝑖 - i n t ( 𝑗 - c l ( 𝑈 𝛼 𝑛 ) ) ) , ( 3 . 1 ) then 𝑋 is said pairwise almost regular-Lindelöf (resp., pairwise nearly regular-Lindelöf) if it is both 𝑖 𝑗 -almost regular-Lindelöf (resp., 𝑖 𝑗 -nearly regular-Lindelöf) and 𝑗 𝑖 -almost regular-Lindelöf (resp., 𝑗 𝑖 -nearly regular-Lindelöf).

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

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

Question. Is 𝑖 𝑗 -weakly regular-Lindelöf spaces implies 𝑖 𝑗 -weakly Lindelöf?

Question. Is 𝑖 𝑗 -weakly regular-Lindelöf spaces implies 𝑖 𝑗 -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 𝑖 𝑗 -almost regular space. Then, for each 𝑥 ∈ 𝑋 and for each 𝑖 𝑗 -regular open subset 𝑊 of 𝑋 containing 𝑥 there exist two 𝑖 𝑗 -regular open subsets 𝑈 and 𝑉 of 𝑋 such that 𝑥 ∈ 𝑈 ⊆ 𝑗 - c l ( 𝑈 ) ⊆ 𝑉 ⊆ 𝑗 - c l ( 𝑉 ) ⊆ 𝑊 .

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

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

Proof. Let { 𝑈 𝛼 ∶ 𝛼 ∈ Δ } be an 𝑖 𝑗 -regular open cover of 𝑋 . For each 𝑥 ∈ 𝑋 , there exists 𝛼 𝑥 ∈ Δ such that 𝑥 ∈ 𝑈 𝛼 𝑥 . Since 𝑋 is 𝑖 𝑗 -almost regular, there exist two 𝑖 𝑗 -regular open subsets 𝑉 𝛼 𝑥 and 𝑊 𝛼 𝑥 of 𝑋 such that 𝑥 ∈ 𝑉 𝛼 𝑥 ⊆ 𝑗 - c l ( 𝑉 𝛼 𝑥 ) ⊆ 𝑊 𝛼 𝑥 ⊆ 𝑗 - c l ( 𝑊 𝛼 𝑥 ) ⊆ 𝑈 𝛼 𝑥 by Lemma 3.4. Since for each 𝛼 ∈ Δ , there exists a 𝑗 𝑖 -regular closed set 𝑗 - c l ( 𝑉 𝛼 𝑥 ) in 𝑋 such that 𝑗 - c l ( 𝑉 𝛼 𝑥 ) ⊆ 𝑊 𝛼 𝑥 and ⋃ 𝑋 = 𝛼 ∈ Δ 𝑉 𝛼 𝑥 = ⋃ 𝛼 ∈ Δ 𝑖 - i n t ( 𝑗 - c l ( 𝑉 𝛼 𝑥 ) ) , the family { 𝑊 𝛼 𝑥 ∶ 𝑥 ∈ 𝑋 } is an 𝑖 𝑗 -regular cover of 𝑋 . Since 𝑋 is 𝑖 𝑗 -weakly regular-Lindelöf, there exists a countable set of points { 𝑥 𝑛 ∶ 𝑛 ∈ ℕ } of 𝑋 such that 𝑋 = 𝑗 - ⋃ c l ( 𝑛 ∈ ℕ 𝑊 𝛼 𝑥 𝑛 ) ⊆ 𝑗 - ⋃ c l ( 𝑛 ∈ ℕ 𝑈 𝛼 𝑥 𝑛 ) . So, 𝑋 = 𝑗 - ⋃ c l ( 𝑛 ∈ ℕ 𝑈 𝛼 𝑥 𝑛 ) and since 𝑋 is 𝑖 𝑗 -semiregular, therefore 𝑋 is 𝑖 𝑗 -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 𝑖 𝑗 -regular space. Then, 𝑋 is 𝑖 𝑗 -weakly regular-Lindelöf if and only if it is 𝑖 𝑗 -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 𝑖 𝑗 -weak 𝑃 -space if for each countable family { 𝑈 𝑛 ∶ 𝑛 ∈ ℕ } of 𝑖 -open sets in 𝑋 , we have 𝑗 - ⋃ c l ( 𝑛 ∈ ℕ 𝑈 𝛼 𝑛 ⋃ ) = 𝑛 ∈ ℕ 𝑗 - c l ( 𝑈 𝛼 𝑛 ) then 𝑋 is called pairwise weak 𝑃 -space if it is both 𝑖 𝑗 -weak 𝑃 -space and 𝑗 𝑖 -weak 𝑃 -space.

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

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

Proof. The proof follows immediately from the fact that in 𝑖 𝑗 -weak 𝑃 -spaces, ⋃ 𝑛 ∈ ℕ 𝑗 - c l ( 𝑈 𝛼 𝑛 ) = 𝑗 - ⋃ c l ( 𝑛 ∈ ℕ 𝑈 𝛼 𝑛 ) 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 𝑖 𝑗 -almost regular space, then 𝑋 is 𝑖 𝑗 -almost regular-Lindelöf if and only if it is 𝑖 𝑗 -nearly Lindelöf (see [17]). Thus, we have the following corollary.

Corollary 3.13. In 𝑖 𝑗 -almost regular and 𝑖 𝑗 -weak 𝑃 -spaces, 𝑖 𝑗 -weakly regular-Lindelöf property is equivalent to 𝑖 𝑗 -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 𝑖 𝑗 -regular and 𝑖 𝑗 -almost regular-Lindelöf space 𝑋 is 𝑖 -Lindelöf.

Corollary 3.16. In 𝑖 𝑗 -regular and 𝑖 𝑗 -weak 𝑃 -spaces, 𝑖 𝑗 -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 - c l ( 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 𝑖 𝑗 -weakly Lindelöf.

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

It is clear that every 𝑖 𝑗 -nearly regular-Lindelöf is 𝑖 𝑗 -weakly regular-Lindelöf and every 𝑖 𝑗 -almost regular-Lindelöf space is 𝑖 𝑗 -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 𝜏 1 on ℝ . Choose usual topology as topology 𝜏 2 on ℝ . Thus, ( ℝ , 𝜏 1 , 𝜏 2 ) 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 , 𝜏 1 , 𝜏 2 ) is 1 2 -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 - c l [ 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 , 𝜏 1 × 𝜏 1 , 𝜏 2 × 𝜏 2 ) is 1 2 -regular and 2 -separable. Thus, ℝ × ℝ is 1 2 -weakly Lindelöf by Lemma 3.20, and so ℝ × ℝ is 1 2 -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 1 2 -regular, but not 1 -Lindelöf, then it is neither 1 2 -almost regular-Lindelöf nor 1 2 -nearly regular-Lindelöf by Lemmas 3.15 and 3.21.

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

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

Example 3.24. Let ( ℝ , 𝜏 1 , 𝜏 2 ) be a bitopological space defined as in Example 3.22 above. Example 3.22 shows that ℝ × ℝ is 1 2 -weakly Lindelöf, but not 1 -Lindelöf. Since ℝ × ℝ is 1 2 -regular, but not 1 -Lindelöf, then it is nor 1 2 -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 𝑖 𝑗 -weakly Lindelöf, 𝑖 𝑗 -regular, and 𝑖 𝑗 -nearly paracompact bitopological space 𝑋 is 𝑖 -Lindelöf.

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

Proof. Let 𝑋 be an 𝑖 𝑗 -regular, 𝑖 𝑗 -nearly paracompact, and 𝑖 𝑗 -weakly regular-Lindelöf space. Then, 𝑋 is 𝑖 𝑗 -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 𝑖 𝑗 -weakly regular-Lindelöf spaces.

Theorem 3.29. A bitopological spaces 𝑋 is 𝑖 𝑗 -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 ∈ Δ 𝑖 - c l ( 𝐴 𝛼 ) = ∅ , there exists a countable subfamily { 𝐶 𝛼 𝑛 ∶ 𝑛 ∈ ℕ } such that 𝑗 - ⋂ i n t ( 𝑛 ∈ ℕ 𝐶 𝛼 𝑛 ) = ∅ .

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 ∈ Δ 𝑖 - c l ( 𝐴 𝛼 ) = ∅ . It follows that ⋂ 𝑋 = 𝑋 ⧵ ( 𝛼 ∈ Δ 𝑖 - c l ( 𝐴 𝛼 ⋃ ) ) = 𝛼 ∈ Δ ( 𝑋 ⧵ 𝑖 - c l ( 𝐴 𝛼 ⋃ ) ) = 𝛼 ∈ Δ 𝑖 - i n t ( 𝑋 ⧵ 𝐴 𝛼 ) . Since 𝐶 𝛼 ⊆ 𝐴 𝛼 ⊆ 𝑗 - i n t ( 𝑖 - c l ( 𝐴 𝛼 ) ) ⊆ 𝑖 - c l ( 𝐴 𝛼 ) , then 𝑋 ⧵ 𝑖 - c l ( 𝐴 𝛼 ) ⊆ 𝑋 ⧵ 𝑗 - i n t ( 𝑖 - c l ( 𝐴 𝛼 ) ) ⊆ 𝑋 ⧵ 𝐶 𝛼 , that is, 𝑖 - i n t ( 𝑋 ⧵ 𝐴 𝛼 ) ⊆ 𝑗 - c l ( 𝑖 - i n t ( 𝑋 ⧵ 𝐴 𝛼 ) ) ⊆ 𝑋 ⧵ 𝐶 𝛼 . Therefore, 𝑋 = 𝛼 ∈ Δ 𝑖 - i n t ( 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 𝑖 𝑗 -regular cover of 𝑋 . Since 𝑋 is 𝑖 𝑗 -weakly regular-Lindelöf, there exists a countable subfamily { 𝑋 ⧵ 𝐶 𝛼 𝑛 ∶ 𝑛 ∈ ℕ } such that 𝑋 = 𝑗 - c l ( 𝑛 ∈ ℕ 𝑋 ⧵ 𝐶 𝛼 𝑛 ) = 𝑗 - c l ( 𝑋 ⧵ ( 𝑛 ∈ ℕ 𝐶 𝛼 𝑛 ) ) = 𝑋 ⧵ ( 𝑗 - i n t ( 𝑛 ∈ ℕ 𝐶 𝛼 𝑛 ) ) . ( 3 . 4 ) Therefore, 𝑗 - ⋂ i n t ( 𝑛 ∈ ℕ 𝐶 𝛼 𝑛 ) = ∅ .
Conversely, let { 𝑈 𝛼 ∶ 𝛼 ∈ Δ } be an 𝑖 𝑗 -regular cover of 𝑋 . By Definition 3.1, for each 𝛼 ∈ Δ , 𝑈 𝛼 is 𝑖 -open set in 𝑋 and there exists a 𝑗 𝑖 -regular closed subset 𝐶 𝛼 of 𝑋 such that 𝐶 𝛼 ⊆ 𝑈 𝛼 and ⋃ 𝑋 = 𝛼 ∈ Δ 𝑖 - i n t ( 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 ∈ Δ 𝑖 - c l ( 𝑋 ⧵ 𝐶 𝛼 ) = 𝛼 ∈ Δ ( 𝑋 ⧵ 𝑖 - i n t ( 𝐶 𝛼 ) ) = 𝑋 ⧵ ( 𝛼 ∈ Δ 𝑖 - i n t ( 𝐶 𝛼 ) ) = 𝑋 ⧵ 𝑋 = ∅ . ( 3 . 5 ) By hypothesis, there exists a countable subset { 𝛼 𝑛 ∶ 𝑛 ∈ ℕ } of Δ such that 𝑗 - ⋂ i n t ( 𝑛 ∈ ℕ ( 𝑋 ⧵ 𝑈 𝛼 𝑛 ) ) = ∅ , that is, 𝑗 - ⋃ i n t ( 𝑋 ⧵ 𝑛 ∈ ℕ 𝑈 𝛼 𝑛 ) = ∅ . So 𝑋 ⧵ 𝑗 - ⋃ c l ( 𝑛 ∈ ℕ 𝑈 𝛼 𝑛 ) = ∅ and, therefore, 𝑋 = 𝑗 - ⋃ c l ( 𝑛 ∈ ℕ 𝑈 𝛼 𝑛 ) . 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 ∈ Δ c l ( 𝐴 𝛼 ) = ∅ , there exists a countable subfamily { 𝐶 𝛼 𝑛 ∶ 𝑛 ∈ ℕ } such that ⋂ i n t ( 𝑛 ∈ ℕ 𝐶 𝛼 𝑛 ) = ∅ .

The following diagram illustrates the relationship among the generalizations of pairwise Lindelöf spaces and the generalizations of pairwise regular-Lindelöf spaces in terms of 𝑖 𝑗 -: 83846.fig.002(3.6)

4. Pairwise Weakly Regular-Lindelöf Subspaces and Subsets

A subset 𝑆 of a bitopological space 𝑋 is said to be 𝑖 𝑗 -weakly regular-Lindelöf (resp., pairwise weakly regular-Lindelöf) if 𝑆 is 𝑖 𝑗 -weakly regular-Lindelöf (resp., pairwise weakly regular-Lindelöf) as a subspace of 𝑋 , that is, 𝑆 is 𝑖 𝑗 -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 ∈ Δ 𝑈 𝛼 is said to be 𝑖 𝑗 -regular cover of 𝑆 by 𝑖 -open subsets of 𝑋 if for each 𝛼 ∈ Δ , there exists a nonempty 𝑗 𝑖 -regular closed subset 𝐶 𝛼 of 𝑋 such that 𝐶 𝛼 ⊆ 𝑈 𝛼 and ⋃ 𝑆 ⊆ 𝛼 ∈ Δ 𝑖 - i n t ( 𝐶 𝛼 ) . { 𝑈 𝛼 ∶ 𝛼 ∈ Δ } is said pairwise regular cover by open subsets of 𝑋 if it is both 𝑖 𝑗 -regular cover of 𝑆 by 𝑖 -open subsets of 𝑋 and 𝑗 𝑖 -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 𝑖 𝑗 -almost regular-Lindelöf relative to 𝑋 if for every 𝑖 𝑗 -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 ∈ ℕ 𝑗 - c l ( 𝑈 𝛼 𝑛 ) . 𝑆 is said pairwise almost regular-Lindelöf relative to 𝑋 if it is both 𝑖 𝑗 -almost regular-Lindelöf relative to 𝑋 and 𝑗 𝑖 -almost regular-Lindelöf relative to 𝑋 .

Definition 4.3. A subset 𝑆 of a bitopological space 𝑋 is said to be 𝑖 𝑗 -weakly regular-Lindelöf relative to 𝑋 if for every 𝑖 𝑗 -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 - ⋃ c l ( 𝑛 ∈ ℕ 𝑈 𝛼 𝑛 ) . 𝑆 is said pairwise weakly regular-Lindelöf relative to 𝑋 if it is both 𝑖 𝑗 -weakly regular-Lindelöf relative to 𝑋 and 𝑗 𝑖 -weakly regular-Lindelöf relative to 𝑋 .

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

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

Question. Is 𝑖 𝑗 -weakly regular-Lindelöf relative to the space implies 𝑖 𝑗 -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 𝑖 𝑗 -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 ∈ Δ 𝑖 - c l ( 𝐴 𝛼 ) ) ∩ 𝑆 = ∅ there exists a countable subfamily { 𝐶 𝛼 𝑛 ∶ 𝑛 ∈ ℕ } such that ⋂ ( 𝑗 - i n t ( 𝑛 ∈ ℕ 𝐶 𝛼 𝑛 ) ) ∩ 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 ∈ Δ 𝑖 - c l ( 𝐴 𝛼 ) ) ∩ 𝑆 = ∅ . It follows that ⋂ 𝑆 ⊆ 𝑋 ⧵ ( 𝛼 ∈ Δ 𝑖 - c l ( 𝐴 𝛼 ⋃ ) ) = 𝛼 ∈ Δ ( 𝑋 ⧵ 𝑖 - c l ( 𝐴 𝛼 ⋃ ) ) = 𝛼 ∈ Δ 𝑖 - i n t ( 𝑋 ⧵ 𝐴 𝛼 ) . Since 𝐶 𝛼 ⊆ 𝐴 𝛼 ⊆ 𝑗 - i n t ( 𝑖 - c l ( 𝐴 𝛼 ) ) ⊆ 𝑖 - c l ( 𝐴 𝛼 ) , then 𝑋 ⧵ 𝑖 - c l ( 𝐴 𝛼 ) ⊆ 𝑋 ⧵ 𝑗 - i n t ( 𝑖 - c l ( 𝐴 𝛼 ) ) ⊆ 𝑋 ⧵ 𝐶 𝛼 , that is, 𝑖 - i n t ( 𝑋 ⧵ 𝐴 𝛼 ) ⊆ 𝑗 - c l ( 𝑖 - i n t ( 𝑋 ⧵ 𝐴 𝛼 ) ) ⊆ 𝑋 ⧵ 𝐶 𝛼 . Therefore, ⋃ 𝑆 ⊆ 𝛼 ∈ Δ 𝑖 - i n t ( 𝑋 ⧵ 𝐴 𝛼 ⋃ ) ⊆ 𝛼 ∈ Δ ( 𝑋 ⧵ 𝐶 𝛼 ) . So 𝑗 - c l ( 𝑖 - i n t ( 𝑋 ⧵ 𝐴 𝛼 ) ) is a 𝑗 𝑖 -regular closed subset of 𝑋 satisfying the condition of Definition 4.1. Thus, the family { 𝑋 ⧵ 𝐶 𝛼 ∶ 𝛼 ∈ Δ } is an 𝑖 𝑗 -regular cover of 𝑆 by 𝑖 -open subsets of 𝑋 . Since 𝑋 is 𝑖 𝑗 -weakly regular-Lindelöf relative to 𝑋 , there exists a countable subfamily { 𝑋 ⧵ 𝐶 𝛼 𝑛 ∶ 𝑛 ∈ ℕ } such that 𝑆 ⊆ 𝑗 - c l ( 𝑛 ∈ ℕ ( 𝑋 ⧵ 𝐶 𝛼 𝑛 ) ) = 𝑗 - c l ( 𝑋 ⧵ 𝑛 ∈ ℕ 𝐶 𝛼 𝑛 ) = 𝑋 ⧵ 𝑗 - i n t ( 𝑛 ∈ ℕ 𝐶 𝛼 𝑛 ) . ( 4 . 1 ) Therefore, ⋂ ( 𝑗 - i n t ( 𝑛 ∈ ℕ 𝐶 𝛼 𝑛 ) ) ∩ 𝑆 = ∅ .
Conversely, let { 𝑈 𝛼 ∶ 𝛼 ∈ Δ } be an 𝑖 𝑗 -regular cover of 𝑆 by 𝑖 -open subsets of 𝑋 . By Definition 4.1, for each 𝛼 ∈ Δ , there exists a 𝑗 𝑖 -regular closed subset 𝐶 𝛼 of 𝑋 such that 𝐶 𝛼 ⊆ 𝑈 𝛼 and ⋃ 𝑆 ⊆ 𝛼 ∈ Δ 𝑖 - i n t ( 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 with 𝑆 ⊆ 𝛼 ∈ Δ 𝑖 - i n t ( 𝐶 𝛼 ) = 𝑋 ⧵ ( 𝛼 ∈ Δ 𝑋 ⧵ 𝑖 - i n t ( 𝐶 𝛼 ) ) = 𝑋 ⧵ ( 𝛼 ∈ Δ 𝑖 - c l ( 𝑋 ⧵ 𝐶 𝛼 ) ) , ( 4 . 2 ) then it follows that, ( ⋂ 𝛼 ∈ Δ 𝑖 - c l ( 𝑋 ⧵ 𝐶 𝛼 ) ) ∩ 𝑆 = ∅ . By hypothesis, there exists a countable subset { 𝛼 𝑛 ∶ 𝑛 ∈ ℕ } of Δ such that ( 𝑗 - i n t ( 𝑛 ∈ ℕ ( 𝑋 ⧵ 𝑈 𝛼 𝑛 ) ) ) ∩ 𝑆 = ∅ , t h a t i s , ( 𝑗 - i n t ( 𝑋 ⧵ 𝑛 ∈ ℕ 𝑈 𝛼 𝑛 ) ) ∩ 𝑆 = ∅ . ( 4 . 3 ) Thus we have, ⋃ ( 𝑋 ⧵ 𝑗 - c l ( 𝑛 ∈ ℕ 𝑈 𝛼 𝑛 ) ) ∩ 𝑆 = ∅ and, therefore, 𝑆 ⊆ 𝑗 - ⋃ c l ( 𝑛 ∈ ℕ 𝑈 𝛼 𝑛 ) . 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 ∈ Δ c l ( 𝐴 𝛼 ) ) ∩ 𝑆 = ∅ , there exists a countable subfamily { 𝐶 𝛼 𝑛 ∶ 𝑛 ∈ ℕ } such that ⋂ ( i n t ( 𝑛 ∈ ℕ 𝐶 𝛼 𝑛 ) ) ∩ 𝑆 = ∅ .

Proposition 4.6. A subset 𝑆 of a space 𝑋 is 𝑖 𝑗 -weakly regular-Lindelöf relative to 𝑋 if and only if for every family { 𝑈 𝛼 ∶ 𝛼 ∈ Δ } of 𝑖 𝑗 -regular open subsets of 𝑋 satisfying the conditions ⋃ 𝑆 ⊆ 𝛼 ∈ Δ 𝑈 𝛼 and for each 𝛼 ∈ Δ there exists a nonempty 𝑗 𝑖 -regular closed subset 𝐶 𝛼 of 𝑋 such that 𝐶 𝛼 ⊆ 𝑈 𝛼 and ⋃ 𝑆 ⊆ 𝛼 ∈ Δ 𝑖 - i n t ( 𝐶 𝛼 ) , then there exists a countable subset { 𝛼 𝑛 ∶ 𝑛 ∈ ℕ } of Δ such that 𝑆 ⊆ 𝑗 - ⋃ c l ( 𝑛 ∈ ℕ 𝑈 𝛼 𝑛 ) .

Proof. The necessity is obvious by the Definitions 4.1 and 4.2 since every 𝑖 𝑗 -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 - i n t ( 𝑗 - c l ( 𝑈 𝛼 ) ) ∶ 𝛼 ∈ Δ } is a family of 𝑖 𝑗 -regular open sets in 𝑋 satisfying the conditions of the theorem, since for each 𝛼 ∈ Δ , we have 𝐶 𝛼 ⊆ 𝑈 𝛼 ⊆ 𝑖 - i n t ( 𝑗 - c l ( 𝑈 𝛼 ) ) . By hypothesis, there exists a countable subset { 𝛼 𝑛 ∶ 𝑛 ∈ ℕ } of Δ such that 𝑆 ⊆ 𝑗 - c l ( 𝑛 ∈ ℕ ( 𝑖 - i n t ( 𝑗 - c l ( 𝑈 𝛼 𝑛 ) ) ) ) ⊆ 𝑗 - c l ( 𝑛 ∈ ℕ 𝑗 - c l ( 𝑈 𝛼 𝑛 ) ) ⊆ 𝑗 - c l ( 𝑗 - c l ( 𝑛 ∈ ℕ 𝑈 𝛼 𝑛 ) ) = 𝑗 - c l ( 𝑛 ∈ ℕ 𝑈 𝛼 𝑛 ) . ( 4 . 4 ) This implies that 𝑆 is 𝑖 𝑗 -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 ∈ Δ 𝑈 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 ∈ Δ i n t ( 𝐶 𝛼 ) , then there exists a countable subset { 𝛼 𝑛 ∶ 𝑛 ∈ ℕ } of Δ such that ⋃ 𝑆 ⊆ c l ( 𝑛 ∈ ℕ 𝑈 𝛼 𝑛 ) .

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

Proof. Let { 𝑈 𝛼 ∶ 𝛼 ∈ Δ } be an 𝑖 𝑗 -regular cover of ⋃ { 𝐴 𝑘 ∶ 𝑘 ∈ ℕ } by 𝑖 -open subsets of 𝑋 . Then for each 𝛼 ∈ Δ , there exists a nonempty 𝑗 𝑖 -regular closed subset 𝐶 𝛼 of 𝑋 such that 𝐶 𝛼 ⊆ 𝑈 𝛼 and ⋃ 𝑘 ∈ ℕ 𝐴 𝑘 ⊆ ⋃ 𝛼 ∈ Δ 𝑖 - i n t ( 𝐶 𝛼 ) . Let Δ 𝑘 = { 𝛼 ∈ Δ ∶ 𝑈 𝛼 ∩ 𝐴 𝑘 ≠ ∅ } , then for each 𝛼 𝑘 ∈ Δ 𝑘 ⊆ Δ there exists a nonempty 𝑗 𝑖 -regular closed subset 𝐶 𝛼 𝑘 of 𝑋 such that 𝐶 𝛼 𝑘 ⊆ 𝑈 𝛼 𝑘 and 𝐴 𝑘 ⊆ ⋃ 𝛼 𝑘 ∈ Δ 𝑘 𝑖 - i n t ( 𝐶 𝛼 𝑘 ) . So { 𝑈 𝛼 𝑘 ∶ 𝛼 𝑘 ∈ Δ 𝑘 } is an 𝑖 𝑗 -regular cover of 𝐴 𝑘 by 𝑖 -open subsets of 𝑋 . Since 𝐴 𝑘 is 𝑖 𝑗 -weakly regular-Lindelöf relative to 𝑋 , there exists a countable subfamily { 𝑈 𝛼 𝑘 𝑛 ∶ 𝑛 ∈ ℕ } such that 𝐴 𝑘 ⊆ 𝑗 - ⋃ c l ( 𝑛 ∈ ℕ 𝑈 𝛼 𝑘 𝑛 ) . But a countable union of countable sets is countable, so 𝑘 ∈ ℕ 𝐴 𝑘 ⊆ 𝑘 ∈ ℕ ( 𝑗 - c l ( 𝑛 ∈ ℕ 𝑈 𝛼 𝑘 𝑛 ) ) ⊆ 𝑗 - c l ( 𝑘 ∈ ℕ ( 𝑛 ∈ ℕ 𝑈 𝛼 𝑘 𝑛 ) ) = 𝑗 - c l ( 𝑛 ∈ ℕ 𝑈 𝛼 𝑘 𝑛 ) . ( 4 . 5 ) This implies that ⋃ { 𝐴 𝑘 ∶ 𝑘 ∈ ℕ } is 𝑖 𝑗 -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 𝑖 𝑗 -weakly regular-Lindelöf subspace of a bitopological space 𝑋 , then 𝑆 is 𝑖 𝑗 -weakly regular-Lindelöf relative to 𝑋 .

Proof. Let { 𝑈 𝛼 ∶ 𝛼 ∈ Δ } be an 𝑖 𝑗 -regular cover of 𝑆 by 𝑖 -open subsets of 𝑋 . Then, for each 𝛼 ∈ Δ there exists a nonempty 𝑗 𝑖 -regular closed subset 𝐶 𝛼 of 𝑋 such that 𝐶 𝛼 ⊆ 𝑈 𝛼 and ⋃ 𝑆 ⊆ 𝛼 ∈ Δ 𝑖 - i n t 𝑋 ( 𝐶 𝛼 ) . For each 𝛼 ∈ Δ , we have 𝑖 - i n t 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 𝑗 𝑖 -regular closed set 𝑗 - c l 𝑆 ( 𝑖 - i n t 𝑋 ( 𝐶 𝛼 ) ∩ 𝑆 ) in 𝑆 such that 𝑗 - c l 𝑆 ( 𝑖 - i n t 𝑋 ( 𝐶 𝛼 ) ∩ 𝑆 ) ⊆ 𝐶 𝛼 ∩ 𝑆 ⊆ 𝑈 𝛼 ∩ 𝑆 and 𝑆 = ( 𝛼 ∈ Δ 𝑖 - i n t 𝑋 ( 𝐶 𝛼 ) ) ∩ 𝑆 = 𝛼 ∈ Δ ( 𝑖 - i n t 𝑋 ( 𝐶 𝛼 ) ∩ 𝑆 ) ⊆ 𝛼 ∈ Δ 𝑖 - i n t 𝑆 ( 𝑗 - c l 𝑆 ( 𝑖 - i n t 𝑋 ( 𝐶 𝛼 ) ∩ 𝑆 ) ) , ( 4 . 6 ) that is, ⋃ 𝑆 = 𝛼 ∈ Δ 𝑖 - i n t 𝑆 ( 𝑗 - c l 𝑆 ( 𝑖 - i n t 𝑋 ( 𝐶 𝛼 ) ∩ 𝑆 ) ) , then the family { 𝑈 𝛼 ∩ 𝑆 ∶ 𝛼 ∈ Δ } is an 𝑖 𝑗 -regular cover of 𝑆 . Since 𝑆 is an 𝑖 𝑗 -weakly regular-Lindelöf subspace of 𝑋 , there exists a countable subset { 𝛼 𝑛 ∶ 𝑛 ∈ ℕ } of Δ such that 𝑆 = 𝑗 - c l 𝑆 ( 𝑛 ∈ ℕ ( 𝑈 𝛼 𝑛 ∩ 𝑆 ) ) = ( 𝑗 - c l 𝑋 ( 𝑛 ∈ ℕ ( 𝑈 𝛼 𝑛 ∩ 𝑆 ) ) ) ∩ 𝑆 ⊆ 𝑗 - c l 𝑋 ( 𝑛 ∈ ℕ 𝑈 𝛼 𝑛 ) . ( 4 . 7 ) This shows that 𝑆 is 𝑖 𝑗 -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 𝑖 𝑗 -regular closed proper subset of a bitopological space 𝑋 is 𝑖 𝑗 -weakly regular-Lindelöf relative to 𝑋 , then 𝑋 is 𝑖 𝑗 -weakly regular-Lindelöf.

Proof. Let { 𝑈 𝛼 ∶ 𝛼 ∈ Δ } be an 𝑖 𝑗 -regular cover of 𝑋 . For each 𝛼 ∈ Δ , there exists a nonempty 𝑗 𝑖 -regular closed subset 𝐶 𝛼 of 𝑋 such that 𝐶 𝛼 ⊆ 𝑈 𝛼 and ⋃ 𝑋 = 𝛼 ∈ Δ 𝑖 - i n t ( 𝐶 𝛼 ) . Fix an arbitrary 𝛼 0 ∈ Δ and let Δ ∗ = Δ ⧵ { 𝛼 0 } . Put 𝐾 = 𝑋 ⧵ ( 𝑖 - i n t ( 𝐶 𝛼 0 ) ) , then 𝐾 is an 𝑖 𝑗 -regular closed subset of 𝑋 and ⋃ 𝐾 ⊆ 𝛼 ∈ Δ ∗ 𝑖 - i n t ( 𝐶 𝛼 ) . Therefore, { 𝑈 𝛼 ∶ 𝛼 ∈ Δ ∗ } is an 𝑖 𝑗 -regular cover of 𝐾 by 𝑖 -open subsets of 𝑋 by Definition 4.1. By hypothesis, 𝐾 is 𝑖 𝑗 -weakly regular-Lindelöf relative to 𝑋 , hence there exists a countable subset { 𝛼 𝑛 ∶ 𝑛 ∈ ℕ ∗ } of Δ ∗ such that 𝐾 ⊆ 𝑗 - ⋃ c l ( 𝑛 ∈ ℕ ∗ 𝑈 𝛼 𝑛 ) . So, we have 𝑋 = 𝐾 ∪ ( 𝑖 - i n t ( 𝐶 𝛼 0 ) ) ⊆ 𝐾 ∪ ( 𝑗 - c l ( 𝑈 𝛼 0 ) ) ⊆ ( 𝑗 - c l ( 𝑛 ∈ ℕ ∗ 𝑈 𝛼 𝑛 ) ) ∪ ( 𝑗 - c l ( 𝑈 𝛼 0 = ) ) 𝑛 ∈ ℕ 𝑗 - c l ( 𝑈 𝛼 𝑛 ) . ( 4 . 8 ) So 𝑋 = 𝑗 - ⋃ c l ( 𝑛 ∈ ℕ 𝑈 𝛼 𝑛 ) and this shows that 𝑋 is 𝑖 𝑗 -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 𝑖 𝑗 -regular closed subset of a bitopological space 𝑋 is 𝑖 𝑗 -weakly regular-Lindelöf relative to 𝑋 , then 𝑋 is 𝑖 𝑗 -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 𝑖 𝑗 -weakly regular-Lindelöf, we have the following results.

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

Proof. Let { 𝑈 𝛼 ∶ 𝛼 ∈ Δ } be an 𝑖 𝑗 -regular cover of 𝐴 by 𝑖 -open subsets of 𝑋 . Hence the family { 𝑈 𝛼 ∶ 𝛼 ∈ Δ } ∪ { 𝑋 ⧵ 𝐴 } is an 𝑖 𝑗 -regular cover of 𝑋 since 𝑋 ⧵ 𝐴 is a proper 𝑗 𝑖 -clopen subset of 𝑋 is also a 𝑗 𝑖 -regular closed subset of 𝑋 . Since 𝑋 is 𝑖 𝑗 -weakly regular-Lindelöf, there exists a countable subfamily { 𝑋 ⧵ 𝐴 , 𝑈 𝛼 1 , 𝑈 𝛼 2 , … } such that 𝑋 = 𝑗 - c l ( 𝑛 ∈ ℕ 𝑈 𝛼 𝑛 ) ∪ 𝑗 - c l ( 𝑋 ⧵ 𝐴 ) = ( 𝑗 - c l ( 𝑛 ∈ ℕ 𝑈 𝛼 𝑛 ) ) ∪ ( 𝑋 ⧵ 𝐴 ) . ( 4 . 9 ) But 𝐴 and 𝑋 ⧵ 𝐴 are disjoint; therefore, we have 𝐴 ⊆ 𝑗 - ⋃ c l ( 𝑛 ∈ ℕ 𝑈 𝛼 𝑛 ) . 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 𝑖 𝑗 -weakly regular-Lindelöf space. If 𝐴 is an 𝑖 𝑗 -clopen subset of 𝑋 , then 𝐴 is 𝑖 𝑗 -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 𝑖 𝑗 -weakly regular-Lindelöf space 𝑋 𝑖 𝑗 -weakly regular-Lindelöf?

Question 7.Is 𝑖 𝑗 -regular closed subspace of an 𝑖 𝑗 -weakly regular-Lindelöf space 𝑋 𝑖 𝑗 -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 𝑖 𝑗 -clopen is necessary and it is not sufficient to be only 𝑖 -open or 𝑖 𝑗 -regular open as example below shows. Arbitrary subspaces of 𝑖 𝑗 -weakly regular-Lindelöf spaces need not be 𝑖 𝑗 -weakly regular-Lindelöf nor 𝑖 𝑗 -weakly regular-Lindelöf relative to the spaces. An 𝑖 -open or 𝑖 𝑗 -regular open subset of an 𝑖 𝑗 -weakly regular-Lindelöf space is neither 𝑖 𝑗 -weakly regular-Lindelöf nor 𝑖 𝑗 -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 𝜔 1 , where 𝜔 1 being the first uncountable ordinal, then s u p 𝐴 < 𝜔 1 .

Example 4.21. Let Ω denote the set of ordinals which are less than or equal to the first uncountable ordinal number 𝜔 1 , that is, Ω = [ 1 , 𝜔 1 ] . This Ω is an uncountable well-ordered set with a largest element 𝜔 1 , having the property that if 𝛼 ∈ Ω with 𝛼 < 𝜔 1 , 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 𝜏 1 . Choose discrete topology as another topology for Ω denoted by 𝜏 2 . So ( Ω , 𝜏 1 , 𝜏 2 ) form a bitopological space. Now it is known that Ω is a 1 -Lindelöf space [20], so it is 1 2 -weakly Lindelöf and thus 1 2 -weakly regular-Lindelöf. The subspace Ω 0 = Ω ⧵ { 𝜔 1 } = [ 1 , 𝜔 1 ) , however, is not 1 -Lindelöf (see [20]). We notice that Ω 0 is 1 -open subspace of Ω and also 1 2 -regular open subset of Ω . Observe that Ω 0 is not 1 2 -weakly regular-Lindelöf by Corollary 3.16 since it is 1 2 -regular and 1 2 -weak 𝑃 -space. Moreover, Ω 0 is not 1 2 -weakly regular-Lindelöf relative to Ω . In fact, the family { [ 1 , 𝛼 ) ∶ 𝛼 ∈ Ω 0 } of 1 -open sets in Ω is 1 2 -regular cover of Ω 0 by 1 -open subsets of Ω because Ω 0 ⊆ ⋃ 𝛼 ∈ Ω 0 [ 1 , 𝛼 ) and for each 𝛼 ∈ Ω 0 , there exists a nonempty 2 1 -regular closed subset [ 1 , 𝛼 ) of Ω such that [ 1 , 𝛼 ) ⊆ [ 1 , 𝛼 ) and Ω 0 ⊆ ⋃ 𝛼 ∈ Ω 0 ⋃ [ 1 , 𝛼 ) = 𝛼 ∈ Ω 0 1 - i n t ( [ 1 , 𝛼 ) ) . But the family { [ 1 , 𝛼 ) ∶ 𝛼 ∈ Ω 0 } has no countable subfamily { [ 1 , 𝛼 𝑛 ) ∶ 𝑛 ∈ ℕ } such that Ω 0 ⊆ 2 - ⋃ c l ( 𝑛 ∈ ℕ [ 1 , 𝛼 𝑛 ⋃ ) ) = 𝑛 ∈ ℕ [ 1 , 𝛼 𝑛 ) . For if { [ 1 , 𝛼 1 ) , [ 1 , 𝛼 2 ) , … } satisfy the condition: 2 -closures of unions of it elements cover Ω 0 , then s u p { 𝛼 1 , 𝛼 2 , … } = 𝜔 1 which is impossible by Lemma 4.20.

So we can conclude that an 𝑖 𝑗 -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.

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