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Some properties on generalizations of Lindelof spaces in bitopological space.

Abstract

We introduce the concept of pairwise regular Lindelof spaces and investigate the several properties of such spaces. Standard results in Lindelof spaces in unital topology are extended to bitopological setting and prove some properties of the pairwise regular Lindelof spaces and also weakness properties of Lindelof.

AMS Subject Classification: 54E55.

Keywords: pairwise Lindelof, pairwise regular cover, pairwise regular Lindelof, pairwise almost regular Lindelof, pairwise semi-regularization.

1. Introduction

Among the various covering properties of bitopological spaces a lot of attention has been given to those covers which involve open and regularly open (closed) sets. Kelly [9] introduced and studied the notion of bitopological spaces. A set equipped with two topologies is called a bitopological space. Since then several authors continued investigating such spaces. Lindelofness for bitopological spaces has been defined by Reilly [11]. In 1996 Cammaroto and Santoro [5] introduced the concept of nearly (almost & weakly) regular Lindelof spaces by using regular covers which were introduced by Cammaroto and Lo Faro [4] for unital topological spaces which are considered as one of the main generalizations of Lindelof spaces. Moreover, Fawakhreha and Kilicman [7] continued the study of [5], and they gave further results on these generalizations of regular Lindelof spaces. Every regularly open cover of topological spaces admits a countable subcover. The purpose of this paper is to introduce the concept of pairwise nearly (almost & weakly) regular Lindelof spaces on using pairwise regular covers for bitopological spaces and prove some properties of the pairwise regular Lindelof spaces. Implications among them are also established.

2. Preliminaries

Throughout the paper, by space X, we shall mean bitopological space (X, [[tau].sub.1], [[tau].sub.2]), where [[tau].sub.1], [[tau].sub.2] are arbitrary topologies on X. For [subset] (X, [[tau].sub.1], [[tau].sub.2]), [[tau].sub.1] -int A and [[tau].sub.i], -clA, will respectively stand for the interior and closure of A in (X, [[tau].sub.1] where i = 1,2 A subset A of (X, [[tau].sub.1], [[tau].sub.2]) is said to be the ([[tau].sub.i], [[tau].sub.j])-regular open [12] if A = [[tau].sub.i] -int ([[tau].sub.j] -clA) and the complement of ([[tau].sub.i], [[tau].sub.j])-regular open is called ([[tau].sub.j], [[tau].sub.1])-regular closed [12] for i, j = 1,2. & i [not equal to] j. Let [B.sub.1] be the family of all (1,2)-regular open subsets of X and let [B.sub.2] be the family of all (2,1)-regular open subsets of X. Since the intersection of two ([[tau].sub.i], [[tau].sub.j])-regular open subsets of X is again ([[tau].sub.i], [[tau].sub.j]) -regular open set. Therefore, [B.sub.1] and [B.sub.2] both generate topologies for X say [[tau].sup.*.sub.1] and [[tau].sup.*.sub.2] respectively. Thus with every bitopological space (X, [[tau].sub.1], [[tau].sub.2]) there is associated another bitopological space (X, [[tau].sub.1], [[tau].sub.2]) there is associated another bitopological space (X, [[tau].sup.*.sub.1], [[tau].sup.*.sub.2], called the pairwise semiregularization of [[tau].sub.1] and [[tau].sub.2]. The space (X, [[tau].sub.1], [[tau].sub.2]) is said to be pairwise semiregular if [[tau].sub.1 = [[tau].sup.*.sub.1] and [[tau].sub.2] [[tau].sup.*.sub.2] [12]. A space X is pairwise almost regular [12] if for every ([[tau].sub.1], [[tau].sub.2])-regular closed set F and a point x [not member of] F, there exists a [[tau].sub.1]-open set V and a disjoint [[tau].sub.2]-open set U such that x [member of] U and V [subset] V. Note that a space X is pairwise regular iff it is pairwise semi-regular and pairwise almost regular. A cover U of the bitopological space (X, [[tau].sub.1], [[tau].sub.2]) is called pairwise open if u = [[tau].sub.1 [union] [[tau].sub.2] and U contains a non-empty member of [[tau].sub.1] and non-empty member of [[tau].sub.2]. A space (X, [tau]) is quasi regular open [4], if every non-empty open set U [member of] [[tau].sub.i]\[phi] of X there exists a non-empty open set V [member of] [[tau].sub.i]\[phi] in X such that [[tau].sub.i]-clV [subset]U. A bitopological space (X, [[tau].sub.1], [[tau].sub.2]) is said to be a pairwise Lindelof [11] of every pairwise open cover of X admits a countable subcover. In [6, 10], the theorem state that "(X, [[tau].sub.1], [[tau].sub.2]) is pairwise extremally disconnected if for every pair of disjoint sets A and B, where a [member of] [[tau].sub.1] and B [member of] [[tau].sub.2], [[tau].sub.2] -cl A [intersection][[tau].sub.1] -cl B = [phi] holds". It is easy to show that a space (X, [[tau].sub.1], [[tau].sub.2]) is pairwise extremally disconnected iff, given any regular open subsets U [member of] [[tau].sub.1] and V [member of] [[tau].sub.2] with U [intersection] V = [phi], [[tau].sub.2] -clU[intersection][[tau].sub.1] clV = [phi]

Definition 2.1[2] Let U is a pairwise open cover of (X, [[tau].sub.1], [[tau].sub.2]), then the pairwise open cover V of (X, [[tau].sub.1], [[tau].sub.2]) is a parallel refinement of U if every [[tau].sub.1] open set in V is contained in some [[tau].sub.1] open set of U and every [[tau].sub.2] open set in V is contained in some [[tau].sub.2] open set of U

Definition 2.2 [10] A bitopological space (X, [[tau].sub.1], [[tau].sub.2]) is said to be [[tau].sub.i] -extermally disconnected w. r. t [[tau].sub.j] iff for every [[tau].sub.i] open set A of X, {[[tau].sub.j] - clA}- [[tau]sub.i] open, then X is called pairwise extermally disconnected iff it is [[tau].sub.1] extermally disconnected w. r. t. [[tau].sub.2] and [[tau].sub.2] extermally disconnected w. r. t [[tau].sub.1].

Lemma 2.3 [8] Let (X, [tau] be a topological space and (X, [[tau].sup.*] its semiregularization. Then

(a) regularly open sets of (X, [tau]) are the same as regularly open sets of (X, [[tau].sup.*]).

(b) regularly closed sets of (X, [tau]) are the same as regularly closed sets of (X, [tau].sup.*]).

(c) [tau] - int C = [[tau].sup.*] int C for any regularly closed set C.

(d) [tau] - cl A = [[tau].sup.*] -cl A for every A [member of] [tau].

Definition 2.4 [3] Let (X, [tau]) be a topological space and (X, [[tau].sup.*]) its semiregularization. A topological property P is called semiregular provided that [tau] has the property P iff [[tau].sup.*] has the property P.

If(X, [[tau].sub.1], [[tau].sub.2]) is a bitopological space and P is some topological property, then (i, j)-P denotes an analog of this property for [[tau].sub.i] with respect to [[tau].sub.j], and p-P denotes the conjunction (1, 2)-P [conjunction] (2, 1)-P, i.e., p-P denotes an "absolute" bitopological analog of P, where p is the abbreviation for "pairwise". Sometimes (1, 2)-P [??] (2, 1)-P (and thus [??] p-P), so that it is suffices to consider one of these three bitopological analogs. Also note that (X, [[tau].sub.i]) has a property P [??] (X, [[tau].sub.1], [[tau].sub.2]) has a property I - P, and d-P [??] 1- P [conjunction] 2- P, where d is abbreviation for "double". By analogy, this observed in the case of semiregular: A bitopological property P is called semiregular provided that [[tau].sub.1] and [[tau].sub.2] has the property P iff [[tau].sup.*.sub.2] [[tau].sup.*.sub.2] has the property P.

Definition 2.5 [1] A bitopological space (X, [[tau].sub.1], [[tau].sub.2]) which has a property R is minimal R (maximal R) if for each [[tau].sub.3] [subset] [[tau].sub.1], [[tau].sub.4] [subset] [[tau].sub.2]] R ([[tau].sub.3] [contains][[tau].sub.1], [[tau]sub.4] [contains] [[tau].sub.2] such that (X, [[tau].sub.3], [[tau].sub.4]) has R we have [[tau].sub.3] = [[tau].sub.1], [[tau]sub.4], = [[tau].sub.2].

Theorem 2.6 [1] Let R, R' be any two properties for a bitopological space (x, [[tau].sub.1], [[tau].sub.2]) and R implies R'. Then (X, [[tau].sub.1], [[tau].sub.2]) is maximal (minimal) R' implies (X, [[tau].sub.1], [[tau].sub.2]) is maximal (minimal) R.

3. Pairwise Regular Lindelof Spaces

We have the following definitions,

Definition 3.1 A bitopological space (X, [[tau].sub.1], [[tau].sub.2]) is said to be ([[tau].sub.i],[[tau].sub.j])- nearly (almost & weakly) Lindelof spaces iff for every pairwise open cover {[U.sub.[alpha]: [alpha] [member of] [DELTA]} of X there exists a countable subfamily {[[alpha].sub.n] : n [member of] N} [subset or equal to] [DELTA] such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Then X is called pairwise nearly (almost & weakly) Lindelof iff it is (1, 2) - and (2, 1) -nearly (almost & weakly) Lindelof.

Definition 3.2 A pairwise open cover {[U.sub.[alpha]][member of][DELTA]} of a bitopological space (X, [[tau].sub.1], [[tau].sub.2]) is called [[tau].sub.i]- regular cover w. r. t[[tau].sub.j] if, for every [alpha] [member of] [DELTA], there exists a non-empty ([[tau].sub.i], [[tau].sub.j])-regularly closed subset [C.sub.[alpha]] of X such that [C.sub.[alpha]] [subset or equal to] [U.sub.[[alpha] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is a [[tau].sub.i]-quasi regular open. (i [not equal to] j & i, j = 1,2). Then X is called pairwise regular cover iff it is (1, 2) - and (2, 1)-regular cover.

Example 3.3 Let X = (0, 1], we define a topology [[tau].sub.1]- discrete topology and [[tau].sub.2] = {[phi], X, (a,1], [alpha] [member of] X}. If A = (a, 1] be a pairwise open cover of X, then [[tau].sub.2]-int(A)=(a,1)and [[tau].sub.1] - cl([[tau].sub.2] - int A) = (a, 1]= A be a ([[tau].sub.1], [[tau].sub.2])-regular closed set. Now A [subset] (a, 1]and [union] {[[tau].sub.2] - int A}= [union](a,1] = X. Hence A is a pairwise regular open cover of X.

Definition 3.4 A space (X, [[tau].sub.1], [[tau].sub.2]) is called pairwise regular Lindelof if every cover of {[U.sub.[alpha]]: [alpha] [member of] [DELTA]} of the space (X, [[tau].sub.1], [[tau].sub.2]) by pairwise regular covers contains a countable subfamily [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

It is clear that every pairwise regular Lindelof space is pairwise Lindelof, but the converse is not true as the following example shows.

Example 3.5 Let X = [0, 1], we define a topology [[tau].sub.1] = {[phi], {0}, X} and [[tau].sub.2] = {[phi], {1}, X} on X, then (X, [[tau].sub.1], [[tau].sub.2]) is pairwise Lindelof, but it is not pairwise regular Lindelof because the pairwise regular cover [union] = {{0, x}, {1}: x [member of] X} has no countable subcover.

Example 3.6 Let X = [0, 1], we define a topology [[tau].sub.1] be a discrete topology and [[tau].sub.2] = {A [subset or equal to] X|A = [phi] or X\A is countable}. Then X is automatically pairwise Lindelof in this topology, since every open set only omits countably many points of X. But it is not a pairwise regular Lindelof.

In particular, the above converse statement is true by using the maximality (minimality) of bitopological properties: we have the implication property, Pairwise Lindelof [??] pairwise regular Lindelof and so maximal pairwise regular Lindelof [??] maximal pairwise Lindelof

Theorem 3.7 A space (X, [[tau].sub.1], [[tau].sub.2]) is regular Lindelof iff every family U of [[tau].sub.i], [[tau].sub.j]-regular open subsets of (X, [[tau].sub.1], [[tau].sub.2]) with [intersection] {[U.sub.[alpha]]: [alpha] [member of] v} = [phi] contains a countable subfamily U' such that [intersection]{[[tau].sub.1] -clU; U [member of] V'} for i [not equal to] j and i, j = 1, 2.)

Proof: Necessity: Let V : {[U.sub.[alpha]]: [alpha][member of][DELTA] be a family of [[tau].sub.i], [[tau].sub.j]-regular open subsets w. r. t [[tau].sub.j] of (X, [[tau].sub.1], [[tau].sub.2]) such that [intersection]{[U.sub.[alpha]] : [alpha][member of] V}=[phi]. Since {X\[U.sub.[alpha]] : [alpha] [member of] [DELTA] is [[tau].sub.j], [[tau].sub.i]-regularly closed w. r. t [[tau].sub.j] and hence the family {X\[U.sub.[alpha]]: [alpha] [member of] [DELTA]} is a regular cover of X by [[tau].sub.j]- regularly closed w. r. t [[tau].sub.i]. Since X is pairwise regular Lindelof, there exists a countable number of indices {[[alpha].sub.n] : n [member of] N} such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Sufficient: Let V = {[V.sub.[alpha]: [alpha] [member of] [DELTA]} be a pairwise regular cover of the space X by regularly closed subsets [[tau].sub.i] w. r. t. [[tau].sub.j]. Then {X][V.sub.[alpha]]: [alpha][member of] [DELTA]} is a family of [[tau].sub.j] regular open subsets w. r. t [[tau].sub.i] with [intersection]{X\[V.sub.[alpha]]: [alpha] [member of][DELTA]}=[phi]. By hypothesis, there exists a countable subset [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

So [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and i, j = 1,2. This proves that (X, [[tau].sub.1], [[tau].sub.2]) pairwise regular Lindelof

Theorem 3.8 Every pairwise regular Lindelof space (X, [[tau].sub.1], [[tau].sub.2]) is an extremally disconnected.

Proof: Assume that (X, [[tau].sub.1], [[tau].sub.2]) is not [[tau].sub.i]-extermally disconnected w. r. t [[tau].sub.j]. Let A and B respectively, be ([[tau].sub.i], [[tau].sub.j])-regular open sets such that A[intersection]B-[phi] then [[tau].sub.j] -cl A[member of] [[tau].sub.i], but [[tau].sub.j]) -cl A [intersection][[tau].sub.i] -cl B [not equal to] [phi], say t [member of] [[tau].sub.j] - cl A [intersection][[tau].sub.i] -cl B.Hence X\A and X\B be a ([[tau].sub.j], [[tau].sub.i])-regularly closed and ([[tau].sub.i], [[tau].sub.j])-regularly closed sets. Now, the family {X\A, X\B} forms a pairwise regular cover of the space (X, [[tau].sub.1], [[tau].sub.2]) by pairwise regular closed subsets. Since X is a pairwise regular Lindelof, thus X = [[tau].sub.i]-int(X\A)[union][[tau].sub.j]-int(X\B). Consider t [member of] [[tau].sub.i]-int(X\A), but t [member of] [[tau].sub.j] -cl A and therefore [phi][not equal to] [[tau].sub.i]- int(X\A}[intersection][[tau].sub.j] -cl A [subset or equal to] (X\A}[intersection]A, a contradiction. Similarly, (X, [[tau].sub.1], [[tau].sub.2]) is [[tau].sub.j] extremally disconnected w. r. t [[tau].sub.i]. The following theorem states that regular Lindelofness is a property shared by both (X, [[tau].sub.1], [[tau].sub.2]) and (X, [[tau].sup.*.sub.1], [[tau].sup.*.sub.2]. We have the analogue result from [8].

Theorem 3.9 Let (X, [[tau].sub.1], [[tau].sub.2]) be a space. Then (X, [[tau].sub.1], [[tau].sub.2]) is regular regular Lindelof iff (X, [[tau].sup.*.sub.1], [[tau].sup.*.sub.2] is regular Lindelof.

Proof: Suppose that (X, [[tau].sub.1], [[tau].sub.2]) is regular Lindelof and let {[U.sub.[alpha]]:[alpha][member of] [DELTA]} be any pairwise regular cover of X by regularly closed sets in [[tau].sup.*.sub.1] and [[tau].sup.*.sub.2]. To show that (X, [[tau].sup.*.sub.1], [[tau].sup.*.sub.2]) is regular Lindelof. Since [[tau].sup.*.sub.1] [subset or equal to] [[tau].sub.1] and [[tau].sup.*.sub.2] [subset or equal to] [[tau].sub.2] and Lemma 2.3 (a, band d), we have, {[U.sub.[alpha]: [alpha] [member of][DELTA]} is a pairwise regular cover of the pairwise regular Lindelof space (X, [[tau].sub.1], [[tau].sub.2]). So, there exists a countable subfamily [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (by Lemma 2.3(c)). This implies that (X, [[tau].sup.*.sub.1], [[tau].sup.*.sub.2]) is regular Lindelof. Conversely, suppose that [[tau].sup.*.sub.1], [[tau].sup.*.sub.2]) is regular Lindelof and let{[U.sub.[alpha]]:[alpha][member of][DELTA]} be a pairwise regular cover of (X, [[tau].sub.1], [[tau].sub.2]). Then since [U.sub.[alpha]][subset or equal to][[tau].sub.j]-int ([[tau].sub.i]- cl[U.sub.[alpha]], by Lemma 2.3 (c and d), we have{[[tau].sub.j]-int([[tau].sub.i]-cl[U.sub.[alpha]]): [alpha][member of] [DELTA]} is a pairwise regular cover of the pairwise regular Lindelof (X, [[tau].sup.*.sub.1], [[tau].sup.*.sub.2]). So, there exists a countable subfamily [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (by Lemma 2.3 (a and c)). This implies that (X, [[tau].sub.1], [[tau].sub.2]) is pairwise regular Lindelof.

Theorem 3.10 A bitopological space (X, [[tau].sub.1], [[tau].sub.2]) is pairwise regular Lindelof iff every proper ([[tau].sub.1], [[tau].sub.2])-regularly open subset of X is a [[tau].sub.j], [[tau].sub.1])-regular Lindelof

Proof: Let X be pairwise regular Lindelof and let A be any proper ([[tau].sub.i], [[tau].sub.j])-regularly open subset of X. Then (X\A) is a nonempty [[tau].sub.j]-regularly closed set w. r. t[[tau].sub.i]. Let {[U.sub.[alpha]]: [alpha][member of][DELTA]} be a regular cover of A by sets that are regularly closed [[tau].sub.j] then (X\A)[union]{[U.sub.[alpha]: [alpha][member of][DELTA]} is a pairwise regular cover of A which is pairwise regular Lindelof, so there exists a countable number of indices {[[alpha].sub.n] : n [member of] N} such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Therefore [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], this shows that A is a ([[tau].sub.j],[[tau].sub.i])-regular Lindelof. Conversely, let V = {[U.sub.[alpha]: [alpha] [member of] [DELTA]} be a pairwise regular cover of X. Let the ([[tau].sub.i],[[tau].sub.j])-regularly closed sets in U be {[U.sub.[beta]]: [beta] [member of] [[DELTA].sub.1]} and ([[tau].sub.j],[[tau].sub.i])- regulary closed sets in U be{[V.sub.[alpha]]: [alpha][member of][[DELTA].sub.2]}, then two cases arise: arise: (i)[union]{[V.sub.[alpha]]: [alpha][member of] [[DELTA].sub.2]} = X. Choose a[[beta].sub.0] [member of] [[DELTA].sub.1], such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Then {[V.sub.[alpha]]: [alpha][member of][[DELTA].sub.2]} is a ([[tau].sub.j], [[tau].sub.i])- regularly closed cover of the ([[tau].sub.j], [[tau].sub.i])- regularly open proper subset [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. By hypothesis, X is ([[tau].sub.j], [[tau].sub.i])-regular Lindelof. Then there exists a countable set of indices {[[alpha].sub.n]: n [member of] N} such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Hence [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and X is pairwise regular Lindelof. (ii) [union]{[V.sub.[alpha]]: [alpha][member of][[DELTA].sub.2]} [not equal to] X. Choose a[[beta].sub.0][member of] [[DELTA].sub.1], such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Then A=X- [union]{[V.sub.[alpha]:[alpha][member of][[DELTA]sub.2]} is a proper ([[tau].sub.j], [[tau].sub.i])-regularly open subset of X and A [subset][union]{[U.sub.[beta]:[beta][member of][[DELTA].sub.1]), then there exists a countable subset {[beta].sub.n]: n [member of] N} such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Similarly we have [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], for some countable subsets{[alpha].sub.n]: n [member of] N}, consequently, we have [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. This shows that (X, ([[tau].sub.j], [[tau].sub.i]) is pairwise regular Lindelof.

Pairwise Almost Regular Lindelof Spaces

Definition 4.1 A bitopological space (X, ([[tau].sub.j], [[tau].sub.i]) is said to be ([[tau].sub.j], [[tau].sub.i])- nearly (almost & weakly) regular Lindelof, if for every pairwise regular cover {[U.sub.[alpha]: [alpha] [member of] [DELTA]} of X admits accountable subfamily [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Then X is called pairwise nearly (almost & weakly) regular Lindelof if it is (1, 2)--and (2, 1)--pairwise nearly (almost & weakly) regular Lindelof.

Obviously we have the following implications:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Remark 4.2 (a) Every pairwise regular cover {[U.sub.[alpha]: [alpha][member of] [DELTA]} of x admits a refinement of ([[tau].sub.j], [[tau].sub.i])-regularly open sets in X which covers X. Since if {[U.sub.[alpha]]: [alpha][member of] [DELTA]} is a pairwise regular cover of X, then for each [alpha] [member of][DELTA] there exists a non-empty ([[tau].sub.j], [[tau].sub.i])-regularly closed subset [C.sub.[alpha]] with [C.sub.[alpha]] [subset not equal to] [U.sub.[alpha] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is a pairwise regularly open cover of X refines {[U.sub.[alpha]]: [alpha][member of] [DELTA]}.

(b) If (X, ([[tau].sub.j], [[tau].sub.i]) is a pairwise almost regular space then every pairwise regularly open cover {[U.sub.[alpha]: [alpha][member of] [DELTA]} of X admits a pairwise regular cover {[W.sub.[gamma]: [gamma] [member of] [GAMMA]} refines {[U.sub.[alpha]: [alpha][member of][DELTA]}. Since if {[U.sub.[alpha]]:[alpha][member of][DELTA]} is a pairwise regularly open cover of X and [U.sub.[alpha], containing x, there exist two ([[tau].sub.j], [[tau].sub.i])-regularly open subsets [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. So family [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is a pairwise regular cover of X that refines {[U.sub.[alpha]: [alpha][member of] [DELTA]}.

Theorem 4.3 If the space (X, ([[tau].sub.j], [[tau].sub.i]) is pairwise almost Lindelof, then it is pairwise nearly regular Lindelof.

Proof: Suppose X is pairwise almost Lindelof. We have to prove that X is pairwise nearly regular Lindelof. Let {[U.sub.[alpha]: [alpha][member of][DELTA]} be a pairwise regular cover of X. By remark 4.2(a), there exists a family ([[tau].sub.j], [[tau].sub.i])-regularly closed sets [alpha]= {[C.sub.[alpha]: [alpha] [member of] [DELTA]} with [C.sub.[alpha]] [subset not equal to] [U.sub.[alpha]] such that {[[tau].sublj]-int [C.sub.[alpha]]: [alpha] [member of] [DELTA]}. But X is pairwise almost Lindelof, there exists countable subfamily {[a.sub.n]: n [member of] N} such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. But [alpha] is a family of ([[tau].sub.j], [[tau].sub.i])-regular closed set. Hence [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] By [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] from (4.3.2), (4.3.3) and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] this implies that X is a pairwise nearly regular Lindelof.

Theorem 4.4 Let (X, ([[tau].sub.j], [[tau].sub.i]) be an extremally disconnected and pairwise almost regular Lindelof space, then it is pairwise nearly Lindelof.

Proof: Let {[U.sub.[alpha]]: [alpha] [member of] [DELTA]} be a pairwise regular cover of X. Since X is extremally disconnected, {[[tau].sub.j]-cl[U.sub.[alpha]]: [alpha][member of][DELTA]} is a pairwise regular covers of X. Since X is pairwise almost regular- Lindelof space, so there exists a countable subset [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. This implies that X is pairwise nearly Lindelof.

Theorem 4.5 If the space (X, ([[tau].sub.j], [[tau].sub.i]) is pairwise almost regular Lindelof iff every pairwise (quasi) regularly open subset of X having the countable intersection property has a non-empty intersection.

Proof: Let X be a pairwise almost regular Lindelof and {[U.sub.[alpha]]: [alpha][member of][DELTA]} be a pairwise regularly open family of subsets of X having the countable intersection property. Suppose that [intersection]{[U.sub.[alpha]]: [alpha] [member of] [DELTA]}=[phi]. Then the family (X\[U.sub.[alpha]]} [alpha] [member of][DELTA]} forms a pairwise regular cover of X with [intersection] [U.sub.[alpha]]: [alpha] [member of] [DELTA]}=[phi]. Since X is pairwise almost regular Lindelof, then there exists a ([[tau].sub.i][[tau].sub.j]}-regularly closed set X\[C.sub.[alpha]] of X and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. For some countable subset{[[alpha].sub.n]: n [member of] N}, we have [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], and now [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] hence [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] this contradiction implies that [intersection]{[U.sub.[alpha]:[alpha] [member of] [DELTA]}[not equal to][phi]. Conversely, assume that (X,[[tau].sub.1], [[tau].sub.2]) be not almost regular Lindelof. Then there exists a pairwise regularly cover {[U.sub.[alpha]:[alpha] [member of] [DELTA]} of X which has no countable subcover. Thus, {X\[U.sub.[alpha]:[alpha] [member of] [DELTA]} is a pairwise regular open family of subsets which has the countable intersection property. Since {[U.sub.[alpha]:[alpha] [member of] [DELTA]} is a cover of X, we have [intersection]{X\[U.sub.[alpha]:[alpha] [member of] [DELTA]}=[phi] which completes the proof.

Theorem 4.6 Let Y be a pairwise quasi regular open subspace of a bitopological space (X,[[tau].sub.1],[[tau].sub.2]) and [subset] Y. Then A is a pairwise almost regular Lindelof subset of X iff A is a pairwise almost regular Lindelof subset of Y.

Proof: Let A be a pairwise almost regular Lindelof subset of X and {[U.sub.[alpha]:[alpha] [member of] [DELTA]} be a pairwise regular cover (in Y) of A. Then {[U.sub.[alpha]:[alpha] [member of] [DELTA]} is a pairwise regular cover (in X) of A, there exists a ([[tau].sub.i], [[tau].sub.j])-regularly closed [C.sub.[alpha]] of X such that [[tau].sub.j]- int [C.sub.[alpha]] [subset] [C.sub.[alpha]] [subset][U.sub.[alpha]] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. By hypothesis, there exists a countable subset of indices {[[alpha].sub.n]: n [member of] N} such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. This shows that A is a pairwise almost regular Lindelof subset of Y. Conversely, let A be a pairwise almost regular Lindelof subset of Y and {[U.sub.[alpha]:[alpha] [member of] [DELTA]} be a pairwise regular cover of A. Then A [subset] {Y [intersection] {[U.sub.[alpha]:[alpha] [member of] [DELTA]} is a pairwise regular cover (in Y) of A. Then there exists a countable subset of {[[alpha].sub.n]: n [member of] N} such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Hence, A is pairwise almost regular Lindelof

Theorem 4.7 If the space (X, [[tau].sub.1][[tau].sub.2]) is pairwise almost Lindelof, then it is pairwise nearly regular Lindelof.

Proof: Suppose X is pairwise almost Lindelof. We have to prove that X is pairwise nearly regular Lindelof. Let {[U.sub.[alpha]:[alpha] [member of] [DELTA]} be a pairwise regular cover of X. By remark 4.2 (a), there exists a family of ([[tau].sub.i][[tau].sub.j])-regularly closed sets {[C.sub.[alpha]]: [alpha] [member of] [DELTA]} with {[C.sub.[alpha] [alpha] [member of] [DELTA]} [subset] {[U.sub.[alpha]:[alpha] [member of] [DELTA]} ... (4.7.1) such that {[[tau].sub.j]-int [C.sub.[alpha]]: [alpha] [member of] [DELTA]} is a pairwise regular open cover of X that refines {[U.sub.[alpha]:[alpha] [member of] [DELTA]}. But X is pairwise almost Lindelof, there exists countable subfamily {[[alpha].sub.n]: n [member of] N} such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. But [C.sub.[alpha]] is a family of ([[tau].sub.1],[[tau].sub.j])-regular closed set, hence [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] now by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] this implies that X is a pairwise nearly regular Lindelof and completes the proof.

Lemma 4.8 Every pairwise nearly regular Lindelof is pairwise almost regular Lindelof

Proof: The proof is obvious.

In the lemma 4.8, the converse is not in general as the following example shows.

Example 4.9 The bitopological space (X, [[tau].sub.1],[[tau].sub.2]), where X = R, [[tau].sub.1] the discrete topology and [[tau].sub.2] the topology of countable complements on R, is clearly pairwise almost regular Lindelof, but it is not a pairwise nearly regular Lindelof.

Theorem 4.10 If the space is pairwise almost regular and pairwise nearly regular Lindelof, then it is pairwise nearly Lindelof.

Proof: By lemma 4.8 and theorem 4.7, clearly the space is pairwise nearly Lindelof.

Lemma 4.11 If the space (X, [[tau].sub.1],[[tau].sub.2]) is pairwise semi-regular and any pairwise cover of X by ([[tau].sub.1], [[tau].sub.j])-regularly open sets of X admits a countable subfamily with dense union of X, then it is pairwise weakly Lindelof.

Proof: The proof is obvious.

Theorem 4.12 A pairwise regular and pairwise weakly regular Lindelof space is pairwise weakly Lindelof.

Proof: Suppose that (X, [[tau].sub.1],[[tau].sub.2]) is pairwise regular and pairwise weakly regular Lindelof. We have to prove that X is pairwise weakly Lindelof. Let {[U.sub.[alpha]:[alpha] [member of] [DELTA]} be a pairwise regular open cover of X. For each x [member of] X, there exists [[alpha].sub.x] [member of] [DELTA] such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] but X is pairwise almost regular, then by remark 4.2 (b), there exists a family [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]-regularly open sets in X that forms a pairwise regular cover of X, since X is pairwise weakly regular Lindelof, there exists a countable set of points {[s.xub.1],[x.sub.2], ..., [x.sub.n], ...} of X such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Hence by (4.12.1), [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Since the space X is semi-regular and by lemma 4.11, the space X is pairwise weakly Lindelof.

We give now a characterization of pairwise almost (nearly & weakly) regular Lindelof spaces.

Theorem 4.13 The following conditions are equivalent:

(i) A bitopological space [(X, [[tau].sub.1], [[tau].sub.2]) is pairwise almost regular Lindelof.

(ii) For every family {[C.sub.[alpha]]: [alpha] [member of] [DELTA]} [[tau].sub.i]-closed subsets of X, such that for each [alpha] [member of] [DELTA], there exists a [[tau].sub.j]-open set [A.sub.[alpha]] [contains] [C.sub.[alpha]] with [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] there exists a countable subfamily {[[alpha].sub.n]: n [member of] N} such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for i [not equal to] and i, j = 1,2.

Proof: (i) [??] (ii) Suppose that (i)holds, (X,[[tau].sub.1], [[tau].sub.2]) is pairwise almost regular Lindelof. Let {[C.sub.[alpha]:[alpha] [member of] [DELTA]} be a family of [[tau].sub.i]-closed subsets of X such that for each [alpha] [member of] [DELTA]}, there exists an [[tau].sub.j]-open set [A.sub.[alpha]] [contains] [C.sub.[alpha]] with [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. It follows that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Since [A.sub.[alpha]] [contains] [C.sub.[alpha]] for some [[tau].sub.j]-open set [A.sub.[alpha]]. But [A.sub.[alpha]] [subset] [[tau].sub.i]-cl[A.sub.[alpha]], hence [C.sub.[alpha]] [subset] [A.sub.[alpha]] [subset] [[tau].sub.i] - cl[A.sub.[alpha]] this implies [??][[tau].sub.j]-int[A.sub.[[alpha]] = [A.sub.[alpha]]) [C.sub.[alpha]] [subset] [[tau].sub.j]-int[A.sub.[[alpha]] [subset][[tau].sub.j]-int([[tau].sub.i]- cl[A.sub.[alpha]]) and now [C.sub.[alpha]] [subset] [A.sub.[[alpha]] [subset][[tau].sub.j]-int([[tau].sub.i]- cl[A.sub.[alpha]] [subset][[tau].sub.i] - cl[A.sub.[alpha]]. Taking complements for the inclusion, [[tau].sublj- int([[tau].sub.i]-cl[A.sub.[alpha]] [subset][[tau].sub.i]-cl[A.sub.[alpha]] we get, X\[[tau].sub.j]-int([[tau].sub.i]- cl[A.sub.[alpha]]) [contains] X\[[tau].sub.i]-cl[A.sub.[alpha]] ... (4.13.1) again taking complements for the inclusion [C.sub.[alpha]] [subset [[tau].sub.j]-int([[tau].sub.i]-cl[A.sub.[alpha]]) wet get X\[C.sub.[alpha]][contains] x\[[tau].sub.j]-int([[tau].sub.i]-cl[A.sub.[alpha]]) ... (4.13.2) but [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (by 4.13.2). The family {X\[C.sub.[alpha]]: [alpha][member of] [DELTA]} is a pairwise regular cover of X. Since X is pairwise almost regular Lindelof, there exists a countable subfamily such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and therefore [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(ii)[??](i) Suppose that (ii) is true. We have to prove that (X,[[tau].sub.1][[tau].sub.2]) is pairwise almost regular Lindelof. Let {[U.sub.[alpha]:[alpha] [member of] [DELTA]}be a pairwise regular cover of X. For each [alpha] [member of] [DELTA]}, there exists a ([[tau].sub.i],[[tau].sub.j])-regularly closed set [C.sub.[alpha] of X such that [[tau].sub.j]-int[C.sub.[alpha]][subset][C.sub.[alpha]][subset][U.sub.[alpha]] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. The family {X\[U.sub.[alpha]:[alpha] [member of] [DELTA]}of [[tau].sub.i]-closed sets is such that for each [alpha] [member of] [DELTA], there exists the [[tau].sub.j]-open set X\[C.sub.[alpha]] [contains] X\[U.sub.[alpha]] such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and therefore [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] this is true (ii). Hence the conditions are equivalent.

The following theorems give a characterization of pairwise nearly & weakly regular Lindelof spaces. The proofs of these theorems are similar to the proof of theorem 4.13.

Theorem 4.14 The following two conditions are equivalent:

(i) A bitopological space (x,[[tau].sub.1], [[tau].sub.2]) is pairwise nearly regular Lindelof.

(ii) For every family {[C.sub.[alpha]]: [alpha] [member of] [DELTA]} of ([[tau].sub.i],[[tau].sub.j])-regularly closed subsets of X, such that for each [alpha] [member of] [DELTA], there exists a - [[tau].sub.j]-open set [A.sub.[alpha]] [contains][C.sub.[alpha]] with [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] there exists a countable subfamily {[alpha].sub.n]: n [member of] N} such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and i, j = 1,2.

Theorem 4.15 A bitopological space (X, [[tau].sub.1],[[tau].sub.2]) is pairwise weakly regular Lindelof iff every family {[C.sub.[alpha]:[alpha] [member of] [DELTA]}of([[tau].sub.i],[[tau].sub.j])-regularly closed subsets of X, such that for each [alpha] [member of] [DELTA], there exists a [[tau].sublj]-open set [A.sub.[alpha]] [contains] [C.sub.[alpha]] with [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] there exists a countable subfamily {[[alpha].sub.n]: n [member of] N} such that [[tau].sub.j]-int [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Theorem 4.16 Let (X,[[tau].sub.1],[[tau].sub.2]) be a space. Then(X, [[tau].sub.1],[[tau].sub.2]) is almost regular Lindelof iff (X,[[tau].sup.*.sub.], [[tau].sup.*.sub.2]) is almost regular Lindelof.

Proof: Suppose that (X,[[tau].sub.1],[[tau].sub.2]) is almost regular Lindelof and let {[U.sub.[alpha]:[alpha] [member of] [DELTA]} be any pairwise regular cover of X by regularly closed sets in [[tau].sup.*.sub.1] and [[tau].sup.*.sub.2]. To show that (X,[[tau].sup.*.sub.1],[[tau].sup.*.sub.2]) is regular Lindelof. Since [[tau].sup.*.sub.1] [subset or equal to] [[tau].sub.1] and [[tau].sup.*.sub.2] [subset or equal to] [[tau].sub.2] and Lemma 2.3 (a, band d), we have, {[U.sub.[alpha]:[alpha] [member of] [DELTA]} is a pairwise regular cover of the pairwise almost regular Lindelof space ([X,[[tau].sub.1],[[tau].sub.2]). So, there exists a countable subfamily [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (by Lemma 2.3 (d)). This implies that (X,[[tau].sup*.sub.1],[[tau].sup.*.sub.2]) is regular Lindelof. Conversely, suppose that (X,[[tau].sup.*.sub.1],[[tau].sup.*.sub.2]) is almost regular Lindelof and let {[U.sub.[alpha]:[alpha] [member of] [DELTA]} be a pairwise regular cover of (X,[[tau].sub.1][[tau].sub.2]). Then since [U.sub.[alpha]] [subset o equal to] [[tau].sub.j]-int([[tau].sub.i]-cl[U.sub.[alpha]]), by Lemma 2.3 (c and d), we have {[[tau].sublj]- int([[tau].sub.i]-cl[U.sub.[alpha]]):[alpha][member of][DELTA]} is a pairwise regular cover of the pairwise almost regular Lindelof([X,[[tau].sup.*.sub.1],[[tau].sup.*.sub.2]). So, there exists a countable subfamily [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (by Lemma 2.3) a and c). This implies that (X,[[tau].sub.1],[[tau].sub.2]) is pairwise regular Lindelof.

References

[1] M.E. Abd El-Monsef and A.M. Kozae, 1987, On maximal and minimal bitopological properties, Bull. Fac. Sci. Alexandria Univ., 27, no A (1), 2234.

[2] M.E. Abd El-Monsef and A.S. Ramadan, 1991, On pairwise paracompact spaces, Proc. Math. Phys. Soc. Egypt 66, 1-11.

[3] D.E. Cameron, 1978, Properties of S-closed spaces, Proc. Amer. Math. Soc. 72, 581-586.

[4] F. Cammaroto and G. Lo Faro, 1981, Spazi weakly compact, Riv. Mat. Univ. Parma 14 no. 7, 383-395.

[5] F. Cammaroto and G. Santoro, 1996, Some counterexamples and properties on generalizations of Lindelof spaces, Int. J. Math & Math. Sci. 19 no. 4, 737-746.

[6] R. Engelking, 1976, General Topology, PWN, Warszawa Fawakhreha, A. J. and Kilicman, A., 2001, On generalizations of regular Lindelof spaces, Int. J. Math & Math. Sci. 27, 535-539.

[7] A.J. Fawakhreha and A. Kilicman, 2004, Semiregular properties and generalized Lindelof spaces, Math. Vesnik. 56, 77-80.

[8] J.C. Kelly, 1963, Bitopological spaces, Proc. London Mathematical Society, 13, 71-81.

[9] M.N. Mukherjee, 1985, On pairwise S - Closed bitopological spaces, Int. J. Math & Math. Sci 8, no. 4, 729 - 745

[10] I.V. Reilly, 1973, Pairwise Lindelof bitopological spaces, Kyungpook Math. J., 19,1-4.

[11] A.R. Singal and S.P. Arya, 1971, On pairwise almost regular spaces, Glasnik Math., 6, no 26, 335-338.

J. Kavikumar, Abd Wahid Md Raji and Azme Khamis

Science Studies Centre, Universiti Tun Hussein Onn Malaysia,

86400 Parit Raja, Batu Pahat, Johor, Malaysia

E-mail: kaviphd@gmail.com, abdwahid@uthm.edu.my, azme@uthm.edu.my
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