Pairwise Weakly Regular-Lindelöf Spaces

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 regularLindelöf property is not a hereditary property. Some counterexamples will be considered in order to establish some of their relations.


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 öf spaces, and these are studied separately for different reasons and purposes.In 1959, Frolík 2 introduced the notion of weakly Lindel öf spaces and in 1996, Cammaroto and Santoro 3 studied and gave further new results about these spaces followed by Kılıc ¸man and Fawakhreh 4 .In the same paper, Cammaroto and Santoro introduced the notion of weakly regular-Lindel öf 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 öf spaces and obtained some results.Then, Kılıc ¸man and Fawakhreh 6 studied subspaces of this spaces and obtained some results.
Recently, the authors studied pairwise Lindel öfness in 7 and introduced and studied the notion of pairwise weakly Lindel öf 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 öf bitopological spaces.The purpose of this paper is to define the notion of weakly regular-Lindel öf 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 öf subspaces and subsets and also investigate some of their characterizations.
In Section 4, we will define the concept of pairwise weakly regular-Lindel öf subspaces and subsets.We will define the concept of pairwise weakly regular-Lindel öf relative to a bitopological space by investigating the ij-weakly regular-Lindel öf property and obtain some results.The main result obtained is pairwise, and weakly regular-Lindel öf property is not a hereditary property by a counterexample given.

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 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-Lindel öf space which mean that X, τ i is a Lindel öf space, where i ∈ {1, 2}.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.
Definition 3.3.A bitopological space X is said to be ij-weakly regular-Lindel öf if for every ijregular cover {U α : α ∈ Δ} of X, there exists a countable subset {α n : n ∈ N} of Δ such that X j-cl n∈N U α n .X is said pairwise weakly regular-Lindel öf if it is both ij-weakly regular-Lindel öf and ji-weakly regular-Lindel öf.
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 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 Lemma 3.4.Since for each α ∈ Δ, there exists a ji-regular closed set j-cl V α x in X such that j-cl V α x ⊆ W α x and X α∈Δ V α x α∈Δ i-int j-cl V α x , the family {W α x : x ∈ X} is an ij-regular cover of X.Since X is ij-weakly regular-Lindel öf, there exists a countable set of points 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∈N U α n n∈N j-cl U α n 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-Lindel öf property equivalent to ij-weakly regular-Lindel öf 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∈N j-cl U α n jcl 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 ij-almost regular space, then X is ij-almost regular-Lindel öf if and only if it is ij-nearly Lindel öf see 17 .Thus, we have the following corollary.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-Lindel öf is ij-weakly regular-Lindel öf and every ij-almost regular-Lindel öf space is ij-weakly regular-Lindel öf, but the converses are not true in general as the following example show.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 12-regular and 2-separable.Thus, R×R is 12-weakly Lindel öf by Lemma 3.20, and so R × R is 12-weakly regular-Lindel öf.It is known that R × R is not 1-Lindel öf since the 1-closed subspace L { x, y : y −x} is not 1-Lindel öf for it is a discrete subspace see 19 .Since R × R is 12-regular, but not 1-Lindel öf, then it is neither 12-almost regular-Lindel öf nor 12-nearly regular-Lindel öf by Lemmas 3.15 and 3.21.
It is clear that every ij-almost Lindel öf is ij-weakly Lindel öf, but the converse is not true as in the following example show.
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-Lindel öf spaces.

Theorem 3.29. A bitopological spaces X is ij-weakly regular-Lindelöf if and only if for every family
Proof.Let {C α : α ∈ Δ} be a family of i-closed subsets of X such that for each α ∈ Δ there exists 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-Lindel öf, there exists a countable subfamily {X \ C α n : n ∈ N} such that Therefore, j-int n∈N C α n ∅.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 By hypothesis, there exists a countable subset ∅ and, therefore, X j-cl 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
The following diagram illustrates the relationship among the generalizations of pairwise Lindel öf spaces and the generalizations of pairwise regular-Lindel öf spaces in terms of ij-: ij-nearly Lindel öf ij-almost Lindel öf ij-weakly Lindel öf ij-nearly ij-almost ij-weakly regular-Lindel öf regular-Lindel öf regular-Lindel öf 3.6

Pairwise weakly regular-Lindel öf subspaces and subsets
A subset S of a bitopological space X is said to be ij-weakly regular-Lindel öf resp., pairwise weakly regular-Lindel öf if S is ij-weakly regular-Lindel öf resp., pairwise weakly regular-Lindel öf as a subspace of X, that is, S is ij-weakly regular-Lindel öf resp., pairwise weakly regular-Lindel öf with respect to the inducted bitopology from the bitopology of X.
of X by Definition 4.1.By hypothesis, K is ij-weakly regular-Lindel öf relative to X, hence there exists a countable subset {α n : n ∈ N * } of Δ * such that K ⊆ j-cl n∈N * U α n .So, we have

4.8
So X j-cl n∈N U α n and this shows that X is ij-weakly regular-Lindel öf.
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-Lindel öf, 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-Lindel öf, there exists a countable subfamily {X \ A, U α 1 , U α 2 , . ..} such that But A and X \ A are disjoint; therefore, we have A ⊆ j-cl 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 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.

Corollary 3 . 13 .Corollary 3 . 16 .
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.In ij-regular and ij-weak P -spaces, ij-weakly regular-Lindelöf property is equivalent to i-Lindelöf property.

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 Lindel öf 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 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-with > 0 such that x ∈ a, b-⊆ 2-cl a, b-a, b-⊆ a, b .
Definition 2.4 see 1, 11 .A bitopological space X, τ 1 , τ 2 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 2.7.A bitopological space X is said to be ij-nearly Lindel öf 15 resp., ij-almost Lindel öf 16 , ij-weakly Lindel öf 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 and X is said pairwise nearly Lindel öf resp., pairwise almost Lindel öf, pairwise weakly Lindel öf if it is both ij-nearly Lindel öf resp., ij-almost Lindel öf, ij-weakly Lindel öf and jinearly Lindel öf resp., ji-almost Lindel öf, ji-weakly Lindel öf .Definition 2.8 see 8 .A subset S of a bitopological space X is said to be ij-weakly Lindel öf 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-cl n∈N U α n .S is said pairwise weakly Lindel öf relative to X if it is both ij-weakly Lindel öf relative to X and jiweakly Lindel öf relative to 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 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∈Δ i-int C α .{Uα: 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.then it follows that, α∈Δ i-cl X \ C α ∩ S ∅.By hypothesis, there exists a countable subset {α n : n ∈ N} of Δ such that Thus we have, X \ j-cl n∈N U α n ∩ S ∅ and, therefore, S ⊆ j-cl n∈N U α n .This completes the proof.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 cl A α ∩ S ∅, there exists a countable subfamily {C α n : n ∈ N} such that int n∈N C α n ∩ S ∅.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∈Δ U 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∈Δ iint C α , then there exists a countable subset {α n : n ∈ N} of Δ such that S ⊆ j-cl n∈N U α n .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 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 int C α , then there exists a countable subset {α n 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 ⊆ iint j-cl U α .By hypothesis, there exists a countable subset {α n : n ∈ N} of Δ such that This implies that S is ij-weakly regular-Lindel öf 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