Note on p 1 -Lindel¨of Spaces Which are not Contra Second Countable Spaces in Bitopology

: In this article we show that a contra second countable bitopological space is a p1-Lindel¨of space, but the converse part is not necessarily true in general. We provide suitable example with the help of concepts of nest and interlocking from other areas related to bitopology. The relation between pairwise regular spaces and p 1 -normal spaces has been investigated. Finally, we propose some open problems which may enrich various concepts related to Lindel¨ofness in a bitopological space and other areas of mathematical ideas.

It can be found that countability, Lindelöfness are old concepts in general topology and suitable relations along with applications can be found almost in every area of science and social science, but they are different for the case of bitopological space since bitopological space has more than two types of Lindelöf spaces viz.pairwise Lindelöf space, p-Lindelöf space, p 1 -Lindelöf space etc.
Literature survey on bitoppological space suggests possible scopes of establishing re-lations between various types of Lindelöfness viz.p 1 -Lindelöf space, p-Lindelöf space and pairwise Lindelöf space under suitable conditions.
In this paper, we give answer to the following question which was raised from literature reviews related to bitopological space."What type of a countable space in a bitopological space is a p 1 -Lindelöf space?" If in addition U contains at least one non-empty member of τ 1 and at least one nonempty member of τ 2 , then it is called pairwise open cover [8].Pairwise compactness was defined by Fletcher et al. [8].Notions of p-compact space, p 1 -compact space, p-Lindelöf space and p 1 -Lindelöf space can be found in the paper of Kilićkman, Salleh [13] and Birsan [2].According to Reilly [20], (X, τ 1 , τ 2 ) is pairwise Lindelöf space if each pairwise open cover has a countable subcover.Cooke and Reilly [5] investigated the relation between semi-compactness and pairwise compactness in bitopological spaces.Kilićkman and Salleh ( [4], [13], [14], [15], [16], [17], [22] ) also studied various properties of pairwise Lindelöfness.Cocompactness, cotopology, (i, j)-Baire space etc. were all studied by Dvalishvili [6].
From the prospective of applications, one may find uses of various concepts of bitopological spaces in science and social science.In 2010, Salama [21] worked on lower and upper approximations of rough sets by using a class of generalized closed sets of bitopological space to detect rheumatic fever.Recently, Acharjee and Tripathy [1] used the concept of bitopological space to reveal poverty patterns and equilibria between strategies of consumers and governments.For some other applications, one may refer to the extensive works of Bosi [3].
In this article, we consider p 1 -Lindelöf space due to Birsan [2] and we consider notions of Good and Papadopoulos [10] to give counter example 3.1.One may find recent application of results of LOTS in Minkowski Space of relativity [33].A systematic study on various types of Lindelöf spaces w.r.t.generalized continuous functions of a bitopological space can be found in [22].
One may refer [25], [26] for some results in the direction of this article.
Definition 2.3.([10], Definition 1) Let X be a set.We say that a collection of subsets S of X: 1. T 0 -separates X, if and only if for all x, y ∈ X, such that x = y, there exists S ∈ S such that x ∈ S,y / ∈ S or y ∈ S ,x / ∈ S.
2. T 1 -separates X, if and only if for all x, y ∈ X, such that x = y, there exist S, T ∈ S, such that x ∈ S, y / ∈ S and y ∈ T ,x / ∈ T .
One can easily see that a space is T 0 (resp.T 1 ) if and only if its topology is generated by a T 0 -(resp.T 1 -) separating subbase, but the statement of Definition 2.3 is not valid for the T 2 separation axiom, if one defines a T 2 -separating subbase in an analogous way.Definition 2.4.([10],Definition 2) Let X be a set and let L ⊆ P(X).The order L on X is defined by declaring x L y, if and only if there exists some L ∈ L, such that x ∈ L and y / ∈ L.
Theorem 2.2.( [10], Theorem 8) Let X be a set and let L be a T 0 -separating nest on X.The following are equivalent: (iii) for all L ∈ L, either L has no L -maximal element or X − L has a L -minimal element.(ii) X is a GO space if and only if there are two nests, L and R, of open sets whose union is T 1 -separating and forms a subbase for τ .
(iii) X is a LOTS if and only if there are two interlocking nests L and R, of open sets whose union is T 1 -separating and forms a subbase for τ .

Main results.
In this section we shall show that a contra second countable space is a p 1 -Lindelöf space but the converse is not true.Definition 3.1 Let (X, τ 1 , τ 2 ) be a bitopological space, then: (i) (X, τ 1 , τ 2 ) is said to be an (i, j)-second countable bitopological space, if (X, τ i ) is second countable with respect to τ j where i, j ∈ {1, 2}.
Let U = {U α : α ∈ ∆} be any τ 1 -open cover of X.Then for every x ∈ X, there exists U x ∈ U such that x ∈ U x .Since (X, τ 1 , τ 2 ) is a contra second countable bitopological space, it is both (1, 2)-second countable and (2, 1)-second countable.So, for each x ∈ U x and U x ∈ U , there exists Similarly, one can prove the other part.
Proof.The proof follows from Theorem 3.1.and Theorem 8 of [13].The following example shows that a p 1 -Lindelöf space is not a contra second countable space.In this example, we use some order set theoretical notions, which are due to Good and Pa-padopoulos [10].
Example 3.1.We consider a bitopological space (X, τ 1 , τ 2 ), such that τ 1 makes X a LOTS.Thus, there exists a subbase for τ 1 which consists of the union of two nests L and R, such both L and R are interlocking and L ∪ R obviously T 1 -separates X.Having stated this, we consider all elements of L that have a maximal element.Then, according to the property of interlocking, the complement L c of L will have a minimal element.We isolate all these L c s that have a minimal element to form a nest.We do the same for the corresponding R c s for the nest R. The union of the two nests consisting of L c s and R c s form a subbase for the topology τ 2 , which violates the definition of contra-second countability of (X, τ 1 , τ 2 ).It shows that a p 1 -Lindelöf space is not a contra second countable space.

Open questions.
Thus, at the end of this article, we want to propose some open questions.One may find from literature survey of bitopological space, that following questions are still unanswered.The questions may be simple in nature but they may help to unify concepts of Pairwise Lindelöfness, p-Lindelöfness and p 1 -Lindelöfness.They are as follows: Q.1.On which conditions p 1 -Lindelöf space =⇒ p-Lindelöf space and vice-versa?Q.2.On which condition, p 1 -Lindelöf space ⇐⇒ p-Lindelöf space?
In Example 2 of [13], it can be found that a p-normal space is a p 1 -normal space but the converse is not true in general.Thus, we have the following questions on the concept of normality in a bitopological space.Q.3.On which condition, p 1 -normal space =⇒p-normal space?Q.4.What is the unified relation between p 1 -Lindelöf space, p-Lindelöf space and pairwise Lindelöf space?

Conclusion.
In this paper, we provide answer and some results related to p 1 -Lindelöf space and countability with the help of an example of LOTS specially nest and interlocking.This paper may find some impacts where bitopological space plays crucial roles.Since, rough set version of bitopological space is developed in [31], thus it is natural to find scopes of this paper in computer science in near future.

Theorem 2 . 3 .
([10], Theorem 10) Let (X, τ ) be a topological space.Then: (i) If L and R are two nests of open sets whose union is T 1 -separating, then every L -order open set is open in X.