# Pairwise Weakly Regular-Lindelöf Spaces

Authors: Adem Kılıçman; Zabidin Salleh
Journal: Abstract and Applied Analysis
 (2008)

---

## 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.


---

## Body

## 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 Section3,
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 Section4,
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 τi and τj,
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 τi,
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 τi for each i=1,2,
respectively.IfS⊆A⊆X,
then i-intA(S) and i-clA(S) will be used to denote the interior and
closure of S with respect to topology τi 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,τi) is a Lindelöf space, where i∈{1,2}.Definition 2.1 (See [11, 12]).
A subsetS 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 τi 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 τi 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
spaceX 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
spaceX 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 spaceX is said to be ij-nearly Lindelöf [15] (resp., ij-almost Lindelöf [16], ij-weakly Lindelöf [8]) if for every i-open cover {Uα:α∈Δ} of X there exists a countable subset {αn:n∈ℕ} of Δ such that (2.1)X=⋃n∈ℕi-int(j-cl(Uαn))(resp.,X=⋃n∈ℕj-cl(Uαn),X=j-cl(⋃n∈ℕUαn)),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
subsetS of a bitopological space X is said to be ij-weakly Lindelöf relative to X if for every cover {Uα:α∈Δ} of S by i-open subsets of X such that S⊆⋃α∈ΔUα there exists a countable subset {αn:n∈ℕ} of Δ such that S⊆j-cl(⋃n∈ℕUαn). 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 spaceX 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]).
Ani-open cover {Uα: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 Cα of X such that Cα⊆Uα and X=⋃α∈Δi-int(Cα). {Uα:α∈Δ} is said pairwise regular cover if it is both ij-regular cover and ji-regular cover.Definition 3.2.
A bitopological spaceX is said to be ij-almost regular-Lindelöf [17] (resp., ij-nearly regular-Lindelöf [18]) if for every ij-regular cover {Uα:α∈Δ} of X there exists a countable subset {αn:n∈ℕ} of Δ
such
that (3.1)X=⋃n∈ℕj-cl(Uαn)(resp.,X=⋃n∈ℕi-int(j-cl(Uαn))),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 spaceX is said to be ij-weakly regular-Lindelöf if for every ij-regular cover {Uα:α∈Δ} of X,
there exists a countable subset {αn:n∈ℕ} of Δ such that X=j-cl(⋃n∈ℕUαn). X
is said pairwise weakly regular-Lindelöf if it
is both ij-weakly regular-Lindelöf and ji-weakly regular-Lindelöf.Obviously,
everyij-weakly Lindelöf space is ij-weakly regular-Lindelöf, and every ij-almost regular-Lindelöf space is ij-weakly regular-Lindelöf.Question.
Isij-weakly regular-Lindelöf spaces implies ij-weakly Lindelöf?Question.
Isij-weakly regular-Lindelöf spaces implies ij-almost regular-Lindelöf?The authors
expected that the answer of these questions is no. We can answer Question1.
by some restrictions on the space with the following proposition. First of all,
we need the following lemmas.Lemma 3.4 (See [17]).
LetX 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 spaceX is ij-regular if and only if it is ij-almost regular and ij-semiregular.Proposition 3.6.
Anij-weakly regular-Lindelöf and ij-regular space X is ij-weakly Lindelöf.Proof.
Let{Uα:α∈Δ} be an ij-regular open cover of X.
For each x∈X,
there exists αx∈Δ such that x∈Uαx.
Since X is ij-almost regular, there exist two ij-regular open subsets Vαx and Wαx of X such that x∈Vαx⊆j-cl(Vαx)⊆Wαx⊆j-cl(Wαx)⊆Uαx by
Lemma 3.4. Since for each α∈Δ,
there exists a ji-regular closed set j-cl(Vαx) in X such that j-cl(Vαx)⊆Wαx and X=⋃α∈ΔVαx=⋃α∈Δi-int(j-cl(Vαx)),
the family {Wαx:x∈X} is an ij-regular cover of X.
Since X is ij-weakly regular-Lindelöf, there exists a countable
set of points {xn:n∈ℕ} of X such that X=j-cl(⋃n∈ℕWαxn)⊆j-cl(⋃n∈ℕUαxn). So, X=j-cl(⋃n∈ℕUαxn) and since X is ij-semiregular, therefore X is ij-weakly Lindelöf.Corollary 3.7.
A pairwise weakly regular-Lindelöf and pairwise regular spaceX is pairwise weakly Lindelöf.Proposition3.6
implies the following corollaries.Corollary 3.8.
LetX be an ij-regular space. Then, X is ij-weakly regular-Lindelöf if and only if it
is ij-weakly Lindelöf.Corollary 3.9.
LetX 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 spaceX is called ij-weak P-space if for each countable family {Un:n∈ℕ} of i-open sets in X,
we have j-cl(⋃n∈ℕUαn)=⋃n∈ℕj-cl(Uαn) 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 inij-weak P-spaces, ij-almost regular-Lindelöf property equivalent
to ij-weakly regular-Lindelöf property.Proposition 3.11.
LetX 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 inij-weak P-spaces, ⋃n∈ℕj-cl(Uαn)=j-cl(⋃n∈ℕUαn) for any countable family {Un:n∈ℕ} of i-open sets in X.Corollary 3.12.
LetX be a pairwise weak P-spaces. Then, X
is pairwise almost regular-Lindelöf if and
only if X is pairwise weakly regular-Lindelöf.IfX 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.
Inij-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 Proposition3.11 and the previous fact.Corollary 3.14.
In pairwise almost regular and pairwise weakP-spaces, pairwise weakly regular-Lindelöf
property is equivalent to pairwise nearly Lindelöf property.Lemma 3.15 (See [17]).
Anij-regular and ij-almost regular-Lindelöf space X is i-Lindelöf.Corollary 3.16.
Inij-regular and ij-weak P-spaces, ij-weakly regular-Lindelöf property is
equivalent to i-Lindelöf property.Proof.
This is
a direct consequence of Proposition3.11 and Lemma 3.15.Corollary 3.17.
In pairwise regular and pairwise weakP-spaces, pairwise weakly regular-Lindelöf
property is equivalent to Lindelöf property.Definition 3.18 (See [8]).
A
subsetE 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 spaceX 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 spaceX is j-separable, then it is ij-weakly Lindelöf.Lemma 3.21 (See [18]).
Anij-regular and ij-nearly regular-Lindelöf space X is i-Lindelöf.It is clear
that everyij-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.
Letℬ
be the collection of closed-open intervals in
the real line ℝ: (3.2)ℬ={[a,b):a,b∈ℝ,a<b}. 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 everyij-almost Lindelöf is ij-weakly Lindelöf, but the converse is not true
as in the following example show.Lemma 3.23 (See [16]).
Anij-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.
Example3.24 solves the open problem in [8, Question 1].Lemma 3.26 (See [8]).
Anij-weakly Lindelöf, ij-regular, and ij-nearly paracompact bitopological space X is i-Lindelöf.Proposition 3.27.
LetX 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.
LetX 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.
LetX
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
ofij-weakly regular-Lindelöf spaces.Theorem 3.29.
A bitopological spacesX is ij-weakly regular-Lindelöf if and only if for
every family {Cα:α∈Δ} of i-closed subsets of X such that for each α∈Δ, there exists a j-open subset Aα of X with Aα⊇Cα and ⋂α∈Δi-cl(Aα)=∅,
there exists a countable subfamily {Cαn:n∈ℕ} such that j-int(⋂n∈ℕCαn)=∅.Proof.
Let{Cα:α∈Δ} be a family of i-closed subsets of X such that for each α∈Δ there exists a j-open subset Aα of X with Aα⊇Cα and ⋂α∈Δi-cl(Aα)=∅.
It follows that X=X∖(⋂α∈Δi-cl(Aα))=⋃α∈Δ(X∖i-cl(Aα))=⋃α∈Δi-int(X∖Aα).
Since Cα⊆Aα⊆j-int(i-cl(Aα))⊆i-cl(Aα),
then X∖i-cl(Aα)⊆X∖j-int(i-cl(Aα))⊆X∖Cα, that is, i-int(X∖Aα)⊆j-cl(i-int(X∖Aα))⊆X∖Cα.
Therefore,(3.3)X=⋃α∈Δi-int(X∖Aα)⊆⋃α∈Δ(X∖Cα).So X=⋃α∈Δ(X∖Cα) and the family {X∖Cα:α∈Δ} is an ij-regular cover of X.
Since X is ij-weakly regular-Lindelöf, there exists a
countable subfamily {X∖Cαn:n∈ℕ} such that(3.4)X=j-cl(⋃n∈ℕX∖Cαn)=j-cl(X∖(⋂n∈ℕCαn))=X∖(j-int(⋂n∈ℕCαn)).Therefore, j-int(⋂n∈ℕCαn)=∅.
Conversely, let{Uα:α∈Δ} be an ij-regular cover of X.
By Definition 3.1, for each α∈Δ, Uα is i-open set in X and there exists a ji-regular closed subset Cα of X such that Cα⊆Uα and X=⋃α∈Δi-int(Cα).
The family {X∖Uα:α∈Δ} of i-closed subsets of X is satisfying the condition, for each α∈Δ, there exists a j-open subset X∖Cα of X such that X∖Cα⊇X∖Uα and
(3.5)⋂α∈Δi-cl(X∖Cα)=⋂α∈Δ(X∖i-int(Cα))=X∖(⋃α∈Δi-int(Cα))=X∖X=∅.By hypothesis, there exists a
countable subset {αn:n∈ℕ} of Δ such that	j-int(⋂n∈ℕ(X∖Uαn))=∅, that is, j-int(X∖⋃n∈ℕUαn)=∅.
So X∖j-cl(⋃n∈ℕUαn)=∅ and, therefore, X=j-cl(⋃n∈ℕUαn). This completes the proof.Corollary 3.30.
A bitopological spacesX
is pairwise weakly regular-Lindelöf if and
only if for every family {Cα:α∈Δ} of closed subsets of X such that for each α∈Δ, there exists an open subset Aα of X with Aα⊇Cα and ⋂α∈Δcl(Aα)=∅, there exists a countable subfamily {Cαn:n∈ℕ} such that int(⋂n∈ℕCαn)=∅.The following
diagram illustrates the relationship among the generalizations of pairwise
Lindelöf spaces and the generalizations of pairwise regular-Lindelöf spaces in
terms ofij-:
(3.6)


## 4. Pairwise Weakly Regular-Lindelöf Subspaces and Subsets

A subsetS 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]).
LetS be a subset of a bitopological space X.
A cover {Uα:α∈Δ} of S by i-open subsets of X such that S⊆⋃α∈ΔUα 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 Cα of X such that Cα⊆Uα and S⊆⋃α∈Δi-int(Cα). {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.Definition 4.2 (See [17]).
A
subsetS of a bitopological space X is said to be ij-almost regular-Lindelöf relative to X if for every ij-regular cover {Uα:α∈Δ} of S by i-open subsets of X there exists a countable subset {αn:n∈ℕ} of Δ such that S⊆⋃n∈ℕj-cl(Uαn). 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 subsetS of a bitopological space X is said to be ij-weakly regular-Lindelöf relative to X if for every ij-regular cover {Uα:α∈Δ} of S by i-open subsets of X there exists a countable subset {αn:n∈ℕ} of Δ such that S⊆j-cl(⋃n∈ℕUαn). 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,
everyij-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.
Isij-weakly regular-Lindelöf relative to the space
implies ij-weakly Lindelöf relative to the space?Question.
Isij-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 subsetS of a bitopological spaces X is ij-weakly regular-Lindelöf relative to X if and only if for every family {Cα:α∈Δ} of i-closed subsets of X such that for each α∈Δ there exists a j-open subset Aα of X with Aα⊇Cα and (⋂α∈Δi-cl(Aα))∩S=∅ there exists a countable subfamily {Cαn:n∈ℕ} such that (j-int(⋂n∈ℕCαn))∩S=∅.Proof.
Let{Cα:α∈Δ} be a family of i-closed subsets of X such that for each α∈Δ there exists a j-open subset Aα of X with Aα⊇Cα and (⋂α∈Δi-cl(Aα))∩S=∅. It follows that S⊆X∖(⋂α∈Δi-cl(Aα))=⋃α∈Δ(X∖i-cl(Aα))=⋃α∈Δi-int(X∖Aα).
Since Cα⊆Aα⊆j-int(i-cl(Aα))⊆i-cl(Aα),
then X∖i-cl(Aα)⊆X∖j-int(i-cl(Aα))⊆X∖Cα,
that is, i-int(X∖Aα)⊆j-cl(i-int(X∖Aα))⊆X∖Cα.
Therefore, S⊆⋃α∈Δi-int(X∖Aα)⊆⋃α∈Δ(X∖Cα).
So j-cl(i-int(X∖Aα)) is a ji-regular closed subset of X satisfying the condition of Definition 4.1.
Thus, the family {X∖Cα: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∖Cαn:n∈ℕ} such that(4.1)S⊆j-cl(⋃n∈ℕ(X∖Cαn))=j-cl(X∖⋂n∈ℕCαn)=X∖j-int(⋂n∈ℕCαn).Therefore, (j-int(⋂n∈ℕCαn))∩S=∅.
Conversely, let{Uα: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 Cα of X such that Cα⊆Uα and S⊆⋃α∈Δi-int(Cα).
The family {X∖Uα:α∈Δ} of i-closed subsets of X is satisfying the condition, for each α∈Δ, there exists a j-open set X∖Cα⊇X∖Uα with(4.2)S⊆⋃α∈Δi-int(Cα)=X∖(⋂α∈ΔX∖i-int(Cα))=X∖(⋂α∈Δi-cl(X∖Cα)),then it follows that, (⋂α∈Δi-cl(X∖Cα))∩S=∅. By hypothesis, there exists a countable subset {αn:n∈ℕ} of Δ such that(4.3)(j-int(⋂n∈ℕ(X∖Uαn)))∩S=∅,thatis,(j-int(X∖⋃n∈ℕUαn))∩S=∅.Thus we have, (X∖j-cl(⋃n∈ℕUαn))∩S=∅ and, therefore, S⊆j-cl(⋃n∈ℕUαn). This completes the proof.Corollary 4.5.
A subsetS of a bitopological spaces X
is pairwise weakly regular-Lindelöf relative
to X if and only if for every family {Cα:α∈Δ} of closed subsets of X such that for each α∈Δ there exists an open subset Aα of X with Aα⊇Cα and (⋂α∈Δcl(Aα))∩S=∅, there exists a countable subfamily {Cαn:n∈ℕ} such that (int(⋂n∈ℕCαn))∩S=∅.Proposition 4.6.
A subsetS of a space X is ij-weakly regular-Lindelöf relative to X if and only if for every family {Uα:α∈Δ} of ij-regular open subsets of X satisfying the conditions S⊆⋃α∈ΔUα and for each α∈Δ there exists a nonempty ji-regular closed subset Cα of X such that Cα⊆Uα and S⊆⋃α∈Δi-int(Cα),
then there exists a countable subset {αn:n∈ℕ} of Δ such that S⊆j-cl(⋃n∈ℕUαn).Proof.
The necessity is obvious by the
Definitions4.1 and 4.2 since every ij-regular open set in X is i-open. For the sufficiency, let {Uα:α∈Δ} be a family of i-open sets in X satisfying the conditions of Definition 4.1
above. Then {i-int(j-cl(Uα)):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 Cα⊆Uα⊆i-int(j-cl(Uα)).
By hypothesis, there exists a countable subset {αn:n∈ℕ} of Δ such that(4.4)S⊆j-cl(⋃n∈ℕ(i-int(j-cl(Uαn))))⊆j-cl(⋃n∈ℕj-cl(Uαn))⊆j-cl(j-cl(⋃n∈ℕUαn))=j-cl(⋃n∈ℕUαn).This implies that S is ij-weakly regular-Lindelöf relative to X and completes the proof.Corollary 4.7.
A subsetS of a space X
is pairwise weakly regular-Lindelöf relative
to X if and only if for every family {Uα:α∈Δ} of pairwise regular open subsets of X satisfying the conditions S⊆⋃α∈ΔUα and for each α∈Δ there exists a nonempty pairwise regular
closed subset Cα of X such that Cα⊆Uα and S⊆⋃α∈Δint(Cα),
then there exists a countable subset {αn:n∈ℕ} of Δ such that S⊆cl(⋃n∈ℕUαn).Proposition 4.8.
If{Ak:k∈ℕ} is a countable family of subsets of a space X such that each Ak is ij-weakly regular-Lindelöf relative to X,
then ⋃{Ak:k∈ℕ} is ij-weakly regular-Lindelöf relative to X.Proof.
Let{Uα:α∈Δ} be an ij-regular cover of ⋃{Ak:k∈ℕ} by i-open subsets of X.
Then for each α∈Δ,
there exists a nonempty ji-regular closed subset Cα of X such that Cα⊆Uα and ⋃k∈ℕAk⊆⋃α∈Δi-int(Cα).
Let Δk={α∈Δ:Uα∩Ak≠∅},
then for each αk∈Δk⊆Δ there exists a nonempty ji-regular closed subset Cαk of X such that Cαk⊆Uαk and Ak⊆⋃αk∈Δki-int(Cαk).
So {Uαk:αk∈Δk} is an ij-regular cover of Ak by i-open subsets of X.
Since Ak is ij-weakly regular-Lindelöf relative to X,
there exists a countable subfamily {Uαkn:n∈ℕ} such that Ak⊆j-cl(⋃n∈ℕUαkn). But a countable union of countable sets is countable,
so(4.5)⋃k∈ℕAk⊆⋃k∈ℕ(j-cl(⋃n∈ℕUαkn))⊆j-cl(⋃k∈ℕ(⋃n∈ℕUαkn))=j-cl(⋃n∈ℕUαkn).This implies that ⋃{Ak:k∈ℕ} is ij-weakly regular-Lindelöf relative to X and completes the proof.Corollary 4.9.
If{Ak:k∈ℕ} is a countable family of subsets of a space X such that each Ak is pairwise weakly regular-Lindelöf relative
to X,
then ⋃{Ak:k∈ℕ}
is pairwise weakly regular-Lindelöf relative
to X.Proposition 4.10.
IfS is an ij-weakly regular-Lindelöf subspace of a
bitopological space X,
then S is ij-weakly regular-Lindelöf relative to X.Proof.
Let{Uα: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 Cα of X such that Cα⊆Uα and S⊆⋃α∈Δi-intX(Cα).
For each α∈Δ,
we have i-intX(Cα)∩S and Uα∩S are i-open sets in S,
and Cα∩S is j-closed set in S.
Since for each α∈Δ,
there exists a ji-regular closed set j-clS(i-intX(Cα)∩S) in S such that j-clS(i-intX(Cα)∩S)⊆Cα∩S⊆Uα∩S and(4.6)S=(⋃α∈Δi-intX(Cα))∩S=⋃α∈Δ(i-intX(Cα)∩S)⊆⋃α∈Δi-intS(j-clS(i-intX(Cα)∩S)),that is, S=⋃α∈Δi-intS(j-clS(i-intX(Cα)∩S)),
then the family {Uα∩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 {αn:n∈ℕ} of Δ such that(4.7)S=j-clS(⋃n∈ℕ(Uαn∩S))=(j-clX(⋃n∈ℕ(Uαn∩S)))∩S⊆j-clX(⋃n∈ℕUαn).This shows that S is ij-weakly regular-Lindelöf relative to X.Corollary 4.11.
IfS
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 Proposition4.10 above true?The authors
expected that the answer is no.Theorem 4.12.
If everyij-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{Uα:α∈Δ} be an ij-regular cover of X.
For each α∈Δ,
there exists a nonempty ji-regular closed subset Cα of X such that Cα⊆Uα and X=⋃α∈Δi-int(Cα).
Fix an arbitrary α0∈Δ and let Δ∗=Δ∖{α0}.
Put K=X∖(i-int(Cα0)),
then K is an ij-regular closed subset of X and K⊆⋃α∈Δ∗i-int(Cα).
Therefore, {Uα: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 {αn:n∈ℕ∗} of Δ∗ such that K⊆j-cl(⋃n∈ℕ∗Uαn). So, we have(4.8)X=K∪(i-int(Cα0))⊆K∪(j-cl(Uα0))⊆(j-cl(⋃n∈ℕ∗Uαn))∪(j-cl(Uα0))=⋃n∈ℕj-cl(Uαn).So X=j-cl(⋃n∈ℕUαn) and this shows that X is ij-weakly regular-Lindelöf.Corollary 4.13.
If every pairwise regular closed proper subset of a bitopological
spaceX
is pairwise weakly regular-Lindelöf relative
to X,
then X is pairwise weakly regular-Lindelöf.It is very
clear that Theorem4.12 implies the following corollaries.Corollary 4.14.
If everyij-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 spaceX
is pairwise weakly regular-Lindelöf relative
to X,
then X is pairwise weakly regular-Lindelöf.Note that, the
spaceX 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.
LetX 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{Uα:α∈Δ} be an ij-regular cover of A by i-open subsets of X.
Hence the family {Uα: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,Uα1,Uα2,…} such that(4.9)X=j-cl(⋃n∈ℕUαn)∪j-cl(X∖A)=(j-cl(⋃n∈ℕUαn))∪(X∖A).But A and X∖A are disjoint; therefore, we have A⊆j-cl(⋃n∈ℕUαn). This completes the proof.Corollary 4.17.
LetX
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 Theorem4.16 implies the following
corollary.Corollary 4.18.
LetX 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.
LetX
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.
Isi-closed subspace of an ij-weakly regular-Lindelöf space Xij-weakly regular-Lindelöf?Question 7.
Isij-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
Theorem4.16 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.
IfA is a countable subset of ordinals Ω not containing ω1, where ω1 being the first uncountable ordinal, then supA<ω1.Example 4.21.
LetΩ
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,αn):n∈ℕ} such that Ω0⊆2-cl(⋃n∈ℕ[1,αn))=⋃n∈ℕ[1,αn). 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 anij-weakly regular-Lindelöf property is not
hereditary property and, therefore, pairwise weakly regular-Lindelöf property
is not so.



---

*Source: 184243-2008-05-05.xml*