Pairwise Weakly Regular-Lindel ¨of Spaces

doi:10.1155/2008/184243

Research Article

Pairwise Weakly Regular-Lindel ¨of Spaces

Adem Kılıc¸man¹and 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

Correspondence should be addressed to Adem Kılıc¸man,[email protected] Received 19 December 2007; Accepted 4 April 2008

Recommended by Agacik Zafer

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

Copyrightq2008 A. Kılıc¸man and Z. 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.

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 Lindel ¨of spaces, and these are studied separately for different reasons and purposes. In 1959, Frol´ık2introduced the notion of weakly Lindel ¨of spaces and in 1996, Cammaroto and Santoro3studied and gave further new results about these spaces followed by Kılıc¸man and Fawakhreh4. In the same paper, Cammaroto and Santoro introduced the notion of weakly regular-Lindel ¨of spaces by using regular covers and leave open the study of this new concept. In 2001, Fawakhreh and Kılıc¸man 5 studied this new generalization of Lindel ¨of spaces and obtained some results.

Then, Kılıc¸man and Fawakhreh6studied subspaces of this spaces and obtained some results.

Recently, the authors studied pairwise Lindel ¨ofness in7and introduced and studied the notion of pairwise weakly Lindel ¨of spaces in bitopological spaces, see 8, where the authors extended some results that were due to Cammaroto and Santoro 3, Kılıc¸man and Fawakhreh 4, and Fawakhreh 9. In 10, the authors also studied the mappings and pairwise continuity on pairwise Lindel ¨of bitopological spaces. The purpose of this paper is to define the notion of weakly regular-Lindel ¨of 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-Lindel ¨of subspaces and subsets and also investigate some of their characterizations.

InSection 3, we will introduce the concept of pairwise weakly regular-Lindel ¨of bitopo- logical spaces by using pairwise regular cover. This study begin by investigating theij-weakly regular-Lindel ¨of property and some results obtained. Furthermore, we study the relation betweenij-nearly Lindel ¨of,ij-almost Lindel ¨of,ij-weakly Lindel ¨of,ij-almost regular-Lindel ¨of, ij-nearly regular-Lindel ¨of, andij-weakly regular-Lindel ¨of spaces, wherei, j1 or 2,i /j.

InSection 4, we will define the concept of pairwise weakly regular-Lindel ¨of subspaces and subsets. We will define the concept of pairwise weakly regular-Lindel ¨of relative to a bitopological space by investigating theij-weakly regular-Lindel ¨of property and obtain some results. The main result obtained is pairwise, and weakly regular-Lindel ¨of 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 useij- to denote the certain properties with respect to topologyτi andτj, wherei, j ∈ {1,2}

andi /j. Byi-intAandi-clA, we will mean the interior and the closure of a subsetAofX with respect to topologyτ_i, respectively. We denote by intAand clAfor the interior and the closure of a subsetAofXwith respect to topologyτifor eachi1,2, respectively.

IfS⊆A⊆X, theni-intASandi-clASwill be used to denote the interior and closure ofSwith respect to topologyτ_iin the subspaceA, respectively. Byi-open cover ofX, we mean that the cover ofXbyi-open sets inX; similar for theij-regular open cover ofXand so forth.

We will use the notation,X isi-Lindel ¨of space which mean that X, τiis a Lindel ¨of space, wherei∈ {1,2}.

Definition 2.1see11,12. A subsetSof a bitopological spaceX, τ1, τ2is said to beij-regular open resp., ij-regular closedif i-intj-clS S resp., i-clj-intS S, and S is said pairwise regular openresp., pairwise regular closedif it is bothij-regular open andji-regular openresp.,ij-regular closed andji-regular closed.

Definition 2.2. LetX, τ1, τ2be a bitopological space. A subsetFofXis said to be

ii-open ifFis open with respect toτiinX,Fis said open inXif it is both 1-open and 2-open inX, or equivalently,F∈τ1∩τ2;

iii-closed ifF is closed with respectτ_i inX,F is said closed inX if it is both 1-closed and 2-closed inX, or equivalently,X\F∈τ1∩τ2;

iiii-clopen ifF is bothi-closed andi-open set inX,F is said clopen inX if it is both 1-clopen and 2-clopen inX;

ivij-clopen ifFisi-closed andj-open set inX,Fis said clopen if it is bothij-clopen and ji-clopen inX.

Definition 2.3see13. A bitopological spaceX, τ1, τ₂is said to be Lindel ¨of if the topological spaceX, τ1andX, τ2are both Lindel ¨of. Equivalently,X, τ1, τ₂is Lindel ¨of if everyi-open cover ofXhas a countable subcover for eachi1,2.

Definition 2.4see1,11. A bitopological spaceX, τ1, τ2is said to beij-regular if for each pointx∈Xand for eachi-open setV ofXcontainingxthere exists ani-open setUsuch that x∈U⊆j-clU⊆V,andXis said to be pairwise regular if it is bothij-regular andji-regular.

Definition 2.5see11,14. A bitopological spaceX is said to beij-almost regular if for each x ∈Xand for eachij-regular open setV ofX containingxthere is anij-regular open setU such thatx∈U⊆j-clU⊆V,thenXis said to be pairwise almost regular if it is bothij-almost regular andji-almost regular.

Definition 2.6see 11,12. A bitopological space X is said to be ij-semiregular if for each x ∈ X and for each i-open set V ofX containingx there is ani-open set U such that x ∈ U ⊆ i-intj-clU ⊆ 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 Lindel ¨of15 resp., ij-almost Lindel ¨of16,ij-weakly Lindel ¨of8if for everyi-open cover{Uα:α∈Δ}ofXthere exists a countable subset{αn:n∈N}ofΔsuch that

X

n∈N

i-int j-cl

U_αn

resp.,X

n∈N

j-cl U_αn

,Xj-cl

n∈N

U_αn

, 2.1

and X is said pairwise nearly Lindel ¨of resp., pairwise almost Lindel ¨of, pairwise weakly Lindel ¨ofif it is bothij-nearly Lindel ¨ofresp.,ij-almost Lindel ¨of,ij-weakly Lindel ¨ofandji- nearly Lindel ¨ofresp.,ji-almost Lindel ¨of,ji-weakly Lindel ¨of.

Definition 2.8see8. A subsetSof a bitopological spaceXis said to beij-weakly Lindel ¨of relative toXif for every cover{Uα:α∈Δ}ofSbyi-open subsets ofXsuch thatS⊆

α∈ΔU_α there exists a countable subset {αn : n ∈ N} of Δ such thatS ⊆ j-cl

n∈NUαn. S is said pairwise weakly Lindel ¨of relative toX if it is bothij-weakly Lindel ¨of relative to X andji- weakly Lindel ¨of relative toX.

Definition 2.9see8. A bitopological spaceX is said to beij-nearly paracompact if every cover ofXbyij-regular open sets admits a locally finite refinement.Xis said pairwise nearly paracompact if it is bothij-nearly paracompact andji-nearly paracompact.

3. Pairwise weakly regular-Lindel ¨of spaces

Definition 3.1see17. Ani-open cover{Uα : α ∈ Δ}of a bitopological space X is said to beij-regular cover if for everyα∈Δthere exists a nonemptyji-regular closed subsetCαofX such thatC_α ⊆ U_α andX

α∈Δi-intCα.{Uα : α∈ Δ}is said pairwise regular cover if it is bothij-regular cover andji-regular cover.

Definition 3.2. A bitopological spaceXis said to beij-almost regular-Lindel ¨of17 resp.,ij- nearly regular-Lindel ¨of 18 if for everyij-regular cover{Uα : α ∈ Δ} ofX there exists a countable subset{αn:n∈N}ofΔsuch that

X

n∈N

j-cl Uαn

resp.,X

n∈N

i-int j-cl

Uαn

, 3.1

then X is said pairwise almost regular-Lindel ¨ofresp., pairwise nearly regular-Lindel ¨of if it is bothij-almost regular-Lindel ¨ofresp.,ij-nearly regular-Lindel ¨ofandji-almost regular- Lindel ¨ofresp.,ji-nearly regular-Lindel ¨of.

Definition 3.3. A bitopological spaceXis said to beij-weakly regular-Lindel ¨of if for everyij- regular cover{Uα :α ∈Δ}ofX, there exists a countable subset{αn : n∈ N}ofΔ such that X j-cl

n∈NUαn.X is said pairwise weakly regular-Lindel ¨of if it is bothij-weakly regular- Lindel ¨of andji-weakly regular-Lindel ¨of.

Obviously, everyij-weakly Lindel ¨of space isij-weakly regular-Lindel ¨of, and everyij- almost regular-Lindel ¨of space isij-weakly regular-Lindel ¨of.

Question 1. Isij-weakly regular-Lindel ¨of spaces impliesij-weakly Lindel ¨of?

Question 2. Isij-weakly regular-Lindel ¨of spaces impliesij-almost regular-Lindel ¨of?

The authors expected that the answer of these questions is no. We can answerQuestion 1.

by some restrictions on the space with the following proposition. First of all, we need the following lemmas.

Lemma 3.4see17. LetX be anij-almost regular space. Then, for eachx ∈ X and for eachij- regular open subsetW ofX containingxthere exist twoij-regular open subsetsUandV ofX such thatx∈U⊆j-clU⊆V ⊆j-clV⊆W.

Lemma 3.5see17. A spaceXisij-regular if and only if it isij-almost regular andij-semiregular.

Proposition 3.6. Anij-weakly regular-Lindel¨of andij-regular spaceXisij-weakly Lindel¨of.

Proof. Let{Uα :α∈Δ}be anij-regular open cover ofX. For eachx∈X, there existsα_x ∈Δ such thatx∈U_αx. SinceXisij-almost regular, there exist twoij-regular open subsetsV_αx and Wαx ofX such thatx ∈ Vαx ⊆ j-clVαx ⊆ Wαx ⊆ j-clWαx ⊆ Uαx by Lemma 3.4. Since for eachα ∈ Δ, there exists aji-regular closed setj-clVαxinX such thatj-clVαx ⊆ Wαx and

X

α∈ΔV_αx

α∈Δi-intj-clVαx, the family{Wαx :x∈X}is anij-regular cover ofX. Since Xisij-weakly regular-Lindel ¨of, there exists a countable set of points{xn :n∈ N}ofX such thatXj-cl

n∈NWα_xn⊆j-cl

n∈NUα_xn. So,Xj-cl

n∈NUα_xnand sinceXisij-semiregular, thereforeXisij-weakly Lindel ¨of.

Corollary 3.7. A pairwise weakly regular-Lindel¨of and pairwise regular spaceX is pairwise weakly Lindel¨of.

Proposition 3.6implies the following corollaries.

Corollary 3.8. LetX be anij-regular space. Then,X isij-weakly regular-Lindel¨of if and only if it is ij-weakly Lindel¨of.

Corollary 3.9. LetXbe a pairwise regular space. Then,Xis pairwise weakly regular-Lindel¨of if and only if it is pairwise weakly Lindel¨of.

Definition 3.10see8. A bitopological spaceXis calledij-weakP-space if for each countable family{Un:n∈N}ofi-open sets inX, we havej-cl

n∈NU_αn

n∈Nj-clUαnthenXis called pairwise weakP-space if it is bothij-weakP-space andji-weakP-space.

The following proposition shows that in ij-weak P-spaces,ij-almost regular-Lindel ¨of property equivalent toij-weakly regular-Lindel ¨of property.

Proposition 3.11. LetXbe anij-weakP-spaces. Then,Xisij-almost regular-Lindel¨of if and only ifX isij-weakly regular-Lindel¨of.

Proof. The proof follows immediately from the fact that inij-weakP-spaces,

n∈Nj-clUαn j- cl

n∈NU_αnfor any countable family{Un:n∈N}ofi-open sets inX.

Corollary 3.12. LetXbe a pairwise weakP-spaces. Then,Xis pairwise almost regular-Lindel¨of if and only ifXis pairwise weakly regular-Lindel¨of.

IfXis anij-almost regular space, thenXisij-almost regular-Lindel ¨of if and only if it is ij-nearly Lindel ¨ofsee17. Thus, we have the following corollary.

Corollary 3.13. In ij-almost regular and ij-weak P-spaces, ij-weakly regular-Lindel¨of property is equivalent toij-nearly Lindel¨of property.

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

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

Lemma 3.15see17. Anij-regular andij-almost regular-Lindel¨of spaceXisi-Lindel¨of.

Corollary 3.16. Inij-regular andij-weakP-spaces,ij-weakly regular-Lindel¨of property is equivalent toi-Lindel¨of property.

Proof. This is a direct consequence ofProposition 3.11andLemma 3.15.

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

Definition 3.18see8. A subsetEof a bitopological spaceXis said to bei-dense inXor is an i-dense subset ofXifi-clE X.Eis said dense inXor is a dense subset ofXif it isi-dense inXor is ani-dense subset ofXfor eachi1,2.

Definition 3.19 see 8. A bitopological space X is said to be i-separable if there exists a countablei-dense subset ofX.Xis said separable if it isi-separable for eachi1,2.

Lemma 3.20see8. If the bitopological spaceXisj-separable, then it isij-weakly Lindel¨of.

Lemma 3.21see18. Anij-regular andij-nearly regular-Lindel¨of spaceXisi-Lindel¨of.

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

Example 3.22. LetBbe the collection of closed-open intervals in the real lineR:

B

a, b:a, b∈R, a < b . 3.2 Hence,Bis a base for the lower limit topologyτ₁onR. Choose usual topology as topologyτ₂ onR. Thus,R, τ1, τ2is a Lindel ¨of bitopological spacesee19. Note that, sets of the form

-∞, a,a, bora,∞are both 1-open and 1-closed inR, and sets of the forma, banda,∞ are 1-open inRsee19. It is easy to check thatR, τ1, τ2is 12-regular since for eachx∈Rand for each 1-open set of the forma, binRcontainingx, there exists a 1-open seta, b-with >0 such thatx∈a, b-⊆2-cla, b- a, b- ⊆a, b. We left to the reader to check for other forms of 1-open sets inR. It is clear thatRis 2-separable since the rational numbers are a countable 2-dense subset ofR. SoR×R, τ1×τ1, τ2×τ2is 12-regular and 2-separable. Thus,R×R is 12-weakly Lindel ¨of byLemma 3.20, and soR×Ris 12-weakly regular-Lindel ¨of. It is known thatR×Ris not 1-Lindel ¨of since the 1-closed subspaceL{x, y:y−x}is not 1-Lindel ¨of for it is a discrete subspacesee19. SinceR×Ris 12-regular, but not 1-Lindel ¨of, then it is neither 12-almost regular-Lindel ¨of nor 12-nearly regular-Lindel ¨of by Lemmas3.15and3.21.

It is clear that everyij-almost Lindel ¨of isij-weakly Lindel ¨of, but the converse is not true as in the following example show.

Lemma 3.23see16. Anij-regular space isij-almost Lindel¨of if and only if it isi-Lindel¨of.

Example 3.24. Let R, τ1, τ₂ be a bitopological space defined as in Example 3.22 above.

Example 3.22shows thatR×Ris 12-weakly Lindel ¨of, but not 1-Lindel ¨of. SinceR×Ris 12- regular, but not 1-Lindel ¨of, then it is nor 12-almost Lindel ¨of byLemma 3.23.

Remark 3.25. Example 3.24solves the open problem in8, Question 1.

Lemma 3.26 see 8. Anij-weakly Lindel¨of, ij-regular, and ij-nearly paracompact bitopological spaceXisi-Lindel¨of.

Proposition 3.27. LetX be anij-regular andij-nearly paracompact spaces. Then,X isi-Lindel¨of if and only ifXisij-weakly regular-Lindel¨of.

Proof. Let X be an ij-regular, ij-nearly paracompact, and ij-weakly regular-Lindel ¨of space.

Then, X is ij-weakly Lindel ¨of by Proposition 3.6. So X is i-Lindel ¨of 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 Lindel¨of if and only ifXis pairwise weakly regular-Lindel¨of.

Now, we give a characterization ofij-weakly regular-Lindel ¨of spaces.

Theorem 3.29. A bitopological spacesX isij-weakly regular-Lindel¨of if and only if for every family {Cα :α∈Δ}ofi-closed subsets ofX such that for eachα ∈Δ,there exists aj-open subsetAαofX withAα ⊇ Cαand

α∈Δi-clAα ∅, there exists a countable subfamily{Cαn : n ∈ N}such that j-int

n∈NC_α

n ∅.

Proof. Let {Cα : α ∈ Δ}be a family ofi-closed subsets of X such that for eachα ∈ Δ there exists aj-open subset A_α ofXwithA_α ⊇ C_α and

α∈Δi-clAα ∅. It follows thatX X\

α∈Δi-clAα

α∈ΔX\i-clAα

α∈Δi-intX\Aα. SinceCα ⊆Aα ⊆j-inti-clAα⊆i- clAα, thenX\i-clAα⊆X\j-inti-clAα⊆X\Cα, that is,i-intX\Aα⊆j-cli-intX\Aα⊆ X\C_α. Therefore,

X

α∈Δ

i-int X\Aα

⊆

α∈Δ

X\Cα

. 3.3

SoX

α∈ΔX\Cαand the family{X\Cα :α ∈Δ}is anij-regular cover ofX. SinceXis ij-weakly regular-Lindel ¨of, there exists a countable subfamily{X\Cαn:n∈N}such that

Xj-cl

n∈N

X\Cαn

j-cl

X\

n∈N

Cαn

X\

j-int

n∈N

Cαn

. 3.4

Therefore,j-int

n∈NCαn ∅.

Conversely, let{Uα:α∈Δ}be anij-regular cover ofX. ByDefinition 3.1, for eachα∈Δ, U_αisi-open set inXand there exists aji-regular closed subsetC_αofXsuch thatC_α⊆U_αand

X

α∈Δi-intCα. The family {X \Uα : α ∈ Δ}of i-closed subsets ofX is satisfying the condition, for eachα∈Δ,there exists aj-open subsetX\CαofXsuch thatX\Cα ⊇X\Uα

and

α∈Δ

i-cl X\Cα

α∈Δ

X\i-int Cα

X\

α∈Δ

i-int Cα

X\X∅. 3.5 By hypothesis, there exists a countable subset{αn:n∈N}ofΔsuch thatj-int

n∈NX\Uαn

∅, that is,j-intX\

n∈NUαn ∅. SoX\j-cl

n∈NU_α

n ∅and, therefore,Xj-cl

n∈NUαn. This completes the proof.

Corollary 3.30. A bitopological spacesXis pairwise weakly regular-Lindel¨of if and only if for every family{Cα :α ∈Δ}of closed subsets ofXsuch that for eachα∈ Δ,there exists an open subsetAα

ofXwithA_α⊇C_αand

α∈ΔclAα ∅, there exists a countable subfamily{Cαn :n∈N}such that int

n∈NC_α

n ∅.

The following diagram illustrates the relationship among the generalizations of pairwise Lindel ¨of spaces and the generalizations of pairwise regular-Lindel ¨of spaces in terms ofij-:

ij-nearly Lindel ¨of ij-almost Lindel ¨of ij-weakly Lindel ¨of

ij-nearly ij-almost ij-weakly

regular-Lindel ¨of regular-Lindel ¨of regular-Lindel ¨of

3.6

4. Pairwise weakly regular-Lindel ¨of subspaces and subsets

A subsetSof a bitopological spaceX is said to beij-weakly regular-Lindel ¨ofresp., pairwise weakly regular-Lindel ¨ofif Sisij-weakly regular-Lindel ¨ofresp., pairwise weakly regular- Lindel ¨ofas a subspace ofX, that is,Sisij-weakly regular-Lindel ¨ofresp., pairwise weakly regular-Lindel ¨ofwith respect to the inducted bitopology from the bitopology ofX.

Definition 4.1see17. LetSbe a subset of a bitopological spaceX. A cover{Uα :α∈Δ}of Sbyi-open subsets ofX such thatS ⊆

α∈ΔU_α is said to beij-regular cover of Sbyi-open subsets ofX if for eachα∈Δ, there exists a nonemptyji-regular closed subsetCαofX such thatC_α ⊆ U_α andS ⊆

α∈Δi-intCα.{Uα : α ∈ Δ}is said pairwise regular cover by open subsets ofXif it is bothij-regular cover ofSbyi-open subsets ofXandji-regular cover ofS byj-open subsets ofX.

Definition 4.2see17. A subsetSof a bitopological spaceXis said to beij-almost regular- Lindel ¨of relative toXif for everyij-regular cover{Uα : α ∈Δ}ofSbyi-open subsets ofX there exists a countable subset{αn:n∈N}ofΔsuch thatS⊆

n∈Nj-clUαn.Sis said pairwise almost regular-Lindel ¨of relative toXif it is bothij-almost regular-Lindel ¨of relative toX and ji-almost regular-Lindel ¨of relative toX.

Definition 4.3. A subsetSof a bitopological spaceX is said to beij-weakly regular-Lindel ¨of relative toXif for everyij-regular cover{Uα :α∈Δ}ofSbyi-open subsets ofXthere exists a countable subset{αn : n∈ N}ofΔsuch thatS ⊆ j-cl

n∈NU_αn.Sis said pairwise weakly regular-Lindel ¨of relative toXif it is bothij-weakly regular-Lindel ¨of relative toXandji-weakly regular-Lindel ¨of relative toX.

Obviously, everyij-weakly Lindel ¨of relative to the space isij-weakly regular-Lindel ¨of relative to the space and every ij-almost regular-Lindel ¨of relative to the space isij-weakly regular-Lindel ¨of relative to the space.

Question 3. Is ij-weakly regular-Lindel ¨of relative to the space implies ij-weakly Lindel ¨of relative to the space?

Question 4. Is ij-weakly regular-Lindel ¨of relative to the space implies ij-almost regular- Lindel ¨of relative to the space?

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

Theorem 4.4. A subsetSof a bitopological spacesXisij-weakly regular-Lindel¨of relative toXif and only if for every family{Cα : α∈Δ}ofi-closed subsets ofXsuch that for eachα∈ Δthere exists a j-open subsetAαofXwithAα⊇Cαand

α∈Δi-clAα∩S∅there exists a countable subfamily {Cαn :n∈N}such thatj-int

n∈NC_α

n∩S∅.

Proof. Let {Cα : α ∈ Δ}be a family ofi-closed subsets of X such that for eachα ∈ Δ there exists aj-open subsetA_α ofX withA_α ⊇ C_α and

α∈Δi-clAα∩S ∅. It follows thatS ⊆ X\

α∈Δi-clAα

α∈ΔX\i-clAα

α∈Δi-intX\Aα. SinceCα⊆Aα⊆j-inti-clAα⊆i- clAα, thenX\i-clAα⊆X\j-inti-clAα⊆X\Cα, that is,i-intX\Aα⊆j-cli-intX\Aα⊆ X\C_α. Therefore,S⊆

α∈Δi-intX\A_α⊆

α∈ΔX\C_α. Soj-cli-intX\A_αis aji-regular closed subset ofXsatisfying the condition ofDefinition 4.1. Thus, the family{X\C_α:α∈Δ}

is anij-regular cover ofSbyi-open subsets ofX. SinceXisij-weakly regular-Lindel ¨of relative toX, there exists a countable subfamily{X\Cαn :n∈N}such that

S⊆j-cl

n∈N

X\C_αn j-cl

X\

n∈N

C_αn

X\j-int

n∈N

C_αn

. 4.1

Therefore,j-int

n∈NC_αn∩S∅.

Conversely, let{Uα : α ∈ Δ} be anij-regular cover of Sby i-open subsets ofX. By Definition 4.1, for eachα∈Δ,there exists aji-regular closed subsetCαofXsuch thatCα⊆Uα

andS ⊆

α∈Δi-intCα. The family{X\Uα :α∈Δ}ofi-closed subsets ofXis satisfying the condition, for eachα∈Δ,there exists aj-open setX\C_α⊇X\U_αwith

S⊆

α∈Δ

i-int Cα

X\

α∈Δ

X\i-int Cα

X\

α∈Δ

i-cl X\Cα

, 4.2

then it follows that,

α∈Δi-clX\Cα∩S∅. By hypothesis, there exists a countable subset {αn:n∈N}ofΔsuch that

j-int

n∈N

X\U_αn

∩S∅, that is,

j-int

X\

n∈N

U_αn

∩S∅. 4.3 Thus we have,X\j-cl

n∈NU_αn∩S ∅and, therefore,S⊆j-cl

n∈NU_α

n. This completes the proof.

Corollary 4.5. A subsetSof a bitopological spacesXis pairwise weakly regular-Lindel¨of relative toX if and only if for every family{Cα:α∈Δ}of closed subsets ofXsuch that for eachα∈Δthere exists an open subsetA_αofXwithA_α⊇C_αand

α∈ΔclAα∩S∅, there exists a countable subfamily {Cαn :n∈N}such thatint

n∈NC_α

n∩S∅.

Proposition 4.6. A subsetSof a spaceXisij-weakly regular-Lindel¨of relative toXif and only if for every family{Uα:α∈Δ}ofij-regular open subsets ofXsatisfying the conditionsS⊆

α∈ΔU_αand for eachα∈Δthere exists a nonemptyji-regular closed subsetCαofXsuch thatCα⊆UαandS⊆

α∈Δi- intCα, then there exists a countable subset{αn:n∈N}ofΔsuch thatS⊆j-cl

n∈NU_αn.

Proof. The necessity is obvious by the Definitions4.1and4.2since everyij-regular open set in Xisi-open. For the sufficiency, let{Uα :α∈Δ}be a family ofi-open sets inXsatisfying the conditions ofDefinition 4.1above. Then{i-intj-clUα:α∈Δ}is a family ofij-regular open sets inXsatisfying the conditions of the theorem, since for eachα∈Δ,we haveCα ⊆Uα ⊆i- intj-clUα. By hypothesis, there exists a countable subset{αn:n∈N}ofΔsuch that

S⊆j-cl

n∈N

i-int j-cl

Uαn

⊆j-cl

n∈N

j-cl Uαn

⊆j-cl

j-cl

n∈N

U_αn j-cl

n∈N

U_αn

.

4.4

This implies thatSisij-weakly regular-Lindel ¨of relative toXand completes the proof.

Corollary 4.7. A subsetSof a spaceX is pairwise weakly regular-Lindel¨of relative toX if and only if for every family{Uα : α∈ Δ}of pairwise regular open subsets ofX satisfying the conditionsS ⊆

α∈ΔUαand for eachα∈Δthere exists a nonempty pairwise regular closed subsetCαofXsuch that C_α ⊆ U_αandS ⊆

α∈ΔintCα, then there exists a countable subset{αn : n ∈ N}ofΔ such that S⊆cl

n∈NU_αn.

Proposition 4.8. If{Ak :k ∈N}is a countable family of subsets of a spaceX such that eachA_kis ij-weakly regular-Lindel¨of relative toX, then

{Ak:k∈N}isij-weakly regular-Lindel¨of relative toX.

Proof. Let{Uα :α∈Δ}be anij-regular cover of

{Ak :k ∈N}byi-open subsets ofX. Then for eachα∈Δ, there exists a nonemptyji-regular closed subsetCαofXsuch thatCα⊆Uαand k∈NA_k⊆

α∈Δi-intCα. LetΔk{α∈Δ:U_α∩A_k/∅}, then for eachα_k∈Δk⊆Δthere exists a nonemptyji-regular closed subsetC_αk ofXsuch thatC_αk ⊆U_αkandA_k ⊆

αk∈Δki-intCαk. So{Uαk :αk ∈Δk}is anij-regular cover ofAkbyi-open subsets ofX. SinceAkisij-weakly

regular-Lindel ¨of relative to X, there exists a countable subfamily{Uα_kn : n ∈ N} such that Ak⊆j-cl

n∈NU_αkn. But a countable union of countable sets is countable, so

k∈N

A_k⊆

k∈N

j-cl

n∈N

U_αkn

⊆j-cl

k∈N

n∈N

U_αkn

j-cl

n∈N

U_αkn

. 4.5

This implies that{Ak :k ∈N}isij-weakly regular-Lindel ¨of relative toXand completes the proof.

Corollary 4.9. If{Ak : k ∈ N}is a countable family of subsets of a spaceX such that eachA_k is pairwise weakly regular-Lindel¨of relative toX, then

{Ak:k∈N}is pairwise weakly regular-Lindel¨of relative toX.

Proposition 4.10. IfSis anij-weakly regular-Lindel¨of subspace of a bitopological spaceX, thenSis ij-weakly regular-Lindel¨of relative toX.

Proof. Let{Uα:α∈Δ}be anij-regular cover ofSbyi-open subsets ofX. Then, for eachα∈Δ there exists a nonemptyji-regular closed subsetC_α ofXsuch thatC_α ⊆ U_αandS ⊆

α∈Δi- int_XCα. For eachα∈Δ, we havei-int_XCα∩SandU_α∩Sarei-open sets inS, andC_α∩Sis j-closed set inS. Since for eachα∈Δ, there exists aji-regular closed setj-clSi-intXCα∩Sin Ssuch thatj-clSi-intXCα∩S⊆Cα∩S⊆Uα∩Sand

S

α∈Δ

i-int

X

Cα

∩S

α∈Δ

i-int

X

Cα

∩S

⊆

α∈Δ

i-int

S

j-clS

i-int

X

Cα

∩S

, 4.6

that is,S

α∈Δi-intSj-clSi-intXCα∩S, then the family{Uα∩S:α∈Δ}is anij-regular cover ofS. SinceS is anij-weakly regular-Lindel ¨of subspace ofX, there exists a countable subset{αn:n∈N}ofΔsuch that

Sj-cl

S

n∈N

Uαn∩S

j-clX

n∈N

Uαn∩S

∩S⊆j-cl

X

n∈N

Uαn

. 4.7

This shows thatSisij-weakly regular-Lindel ¨of relative toX.

Corollary 4.11. IfSis a pairwise weakly regular-Lindel¨of subspace of a bitopological spaceX, thenS is pairwise weakly regular-Lindel¨of relative toX.

Question 5. Is the converse ofProposition 4.10above true?

The authors expected that the answer is no.

Theorem 4.12. If everyij-regular closed proper subset of a bitopological spaceXisij-weakly regular- Lindel¨of relative toX, thenXisij-weakly regular-Lindel¨of.

Proof. Let{Uα :α∈ Δ}be anij-regular cover ofX. For eachα ∈Δ, there exists a nonempty ji-regular closed subset C_α ofX such thatC_α ⊆ U_α andX

α∈Δi-intCα. Fix an arbitrary α₀∈Δand letΔ^∗ Δ\ {α0}. PutKX\i-intCα0, thenKis anij-regular closed subset ofX andK⊆

α∈Δ^∗i-intCα. Therefore,{Uα:α∈Δ^∗}is anij-regular cover ofKbyi-open subsets

ofXbyDefinition 4.1. By hypothesis,Kisij-weakly regular-Lindel ¨of relative toX, hence there exists a countable subset{αn:n∈N^∗}ofΔ^∗such thatK⊆j-cl

n∈N^∗Uαn. So, we have

XK∪ i-int

C_α0

⊆K∪ j-cl

U_α0

⊆

j-cl

n∈N^∗

U_αn

∪ j-cl

U_α0

n∈N

j-cl U_αn

.

4.8

SoXj-cl

n∈NU_αnand this shows thatXisij-weakly regular-Lindel ¨of.

Corollary 4.13. If every pairwise regular closed proper subset of a bitopological spaceX is pairwise weakly regular-Lindel¨of relative toX, thenXis pairwise weakly regular-Lindel¨of.

It is very clear thatTheorem 4.12implies the following corollaries.

Corollary 4.14. If everyij-regular closed subset of a bitopological spaceXisij-weakly regular-Lindel¨of relative toX, thenXisij-weakly regular-Lindel¨of.

Corollary 4.15. If every pairwise regular closed subset of a bitopological spaceXis pairwise weakly regular-Lindel¨of relative toX, thenXis pairwise weakly regular-Lindel¨of.

Note that, the spaceXin above propositions is any bitopological space. If we considerX itself is anij-weakly regular-Lindel ¨of, we have the following results.

Theorem 4.16. LetX be anij-weakly regular-Lindel¨of space. IfAis a properij-clopen subset ofX, thenAisij-weakly regular-Lindel¨of relative toX.

Proof. Let{Uα :α∈ Δ}be anij-regular cover ofAbyi-open subsets ofX. Hence the family {Uα :α∈Δ} ∪ {X\A}is anij-regular cover ofXsinceX\Ais a properji-clopen subset of Xis also aji-regular closed subset ofX. SinceXisij-weakly regular-Lindel ¨of, there exists a countable subfamily{X\A, Uα1, Uα2, . . .}such that

Xj-cl

n∈N

U_αn

∪j-clX\A

j-cl

n∈N

U_αn

∪X\A. 4.9

ButAandX\Aare disjoint; therefore, we haveA⊆j-cl

n∈NUαn. This completes the proof.

Corollary 4.17. LetXbe a pairwise weakly regular-Lindel¨of space. IfAis a proper clopen subset ofX, thenAis pairwise weakly regular-Lindel¨of relative toX.

It is very clear thatTheorem 4.16implies the following corollary.

Corollary 4.18. LetXbe anij-weakly regular-Lindel¨of space. IfAis anij-clopen subset ofX, thenA isij-weakly regular-Lindel¨of relative toX.

Corollary 4.19. LetXbe a pairwise weakly regular-Lindel¨of space. IfAis a clopen subset ofX, thenA is pairwise weakly regular-Lindel¨of relative toX.

Question 6. Isi-closed subspace of anij-weakly regular-Lindel ¨of spaceX ij-weakly regular- Lindel ¨of?

Question 7. Isij-regular closed subspace of an ij-weakly regular-Lindel ¨of spaceX ij-weakly regular-Lindel ¨of?

The authors expected that the answer of both questions is no. Observe that the condition inTheorem 4.16that a subset should beij-clopen is necessary and it is not sufficient to be only i-open orij-regular open as example below shows. Arbitrary subspaces ofij-weakly regular- Lindel ¨of spaces need not beij-weakly regular-Lindel ¨of norij-weakly regular-Lindel ¨of relative to the spaces. Ani-open orij-regular open subset of anij-weakly regular-Lindel ¨of space is neitherij-weakly regular-Lindel ¨of norij-weakly regular-Lindel ¨of relative to the spaces as in the following example also show. We need the following lemmasee20, page 11.

Lemma 4.20. IfAis a countable subset of ordinals Ωnot containing ω1, whereω1 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 numberω1, that is,Ω 1, ω1. ThisΩis an uncountable well-ordered set with a largest elementω1, having the property that ifα∈Ωwithα < ω1, then{β∈Ω:β≤α}

is countable. SinceΩ 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 byτ2. So Ω, τ1, τ2form a bitopological space. Now it is known that Ω is a 1- Lindel ¨of space 20, so it is 12-weakly Lindel ¨of and thus 12-weakly regular-Lindel ¨of. The subspaceΩ0 Ω\ {ω1} 1, ω1, however, is not 1-Lindel ¨ofsee20. We notice thatΩ0is 1-open subspace ofΩand also 12-regular open subset ofΩ. Observe thatΩ0is not 12-weakly regular-Lindel ¨of by Corollary 3.16since it is 12-regular and 12-weakP-space. Moreover,Ω0

is not 12-weakly regular-Lindel ¨of relative to Ω. In fact, the family {1, α : α ∈ Ω0} of 1- open sets in Ω is 12-regular cover ofΩ0 by 1-open subsets of Ωbecause Ω0 ⊆

α∈Ω01, α and for each α ∈ Ω0, there exists a nonempty 21-regular closed subset1, αofΩsuch that 1, α ⊆1, αandΩ0 ⊆

α∈Ω01, α

α∈Ω01-int1, α. But the family{1, α :α ∈Ω0}has no countable subfamily{1, αn : n∈ N}such thatΩ0 ⊆ 2-cl

n∈N1, αn

n∈N1, αn. For if{1, α1,1, α2, . . .}satisfy the condition: 2-closures of unions of it elements coverΩ0, then sup{α1, α2, . . .}ω1which is impossible byLemma 4.20.

So we can conclude that an ij-weakly regular-Lindel ¨of property is not hereditary property and, therefore, pairwise weakly regular-Lindel ¨of 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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