ON PAIRWISE L-CLOSED SPACES IN BITOPOLOGICAL SPACES

The purpose of this paper is to introduce some further properties of the concept of pairwise L-closed spaces as a continuation of a previous study of this notion.


Introduction
Kelly [1] introduced and studied the notion of bitopological spaces in 1963.He defined pairwise Hausdorff, pairwise regular and pairwise normal spaces.
Several mathematicians studied various concepts in bitopological spaces which are turned to be an important field in general topology.In this paper, we study the notion of pairwise L-closed spaces in bitopological spaces and their relations with other bitopological concepts.
We use R and N to denote the set of all real and natural numbers respectively, p-to denote pairwise, cl to denote the closure of a set, and τ s , τ l to denote Sorgenfrey and left ray topologies on R or N.

Pairwise L-Closed spaces
Definition 2.1: A bitopological space (X,τ 1 ,τ 2 ) is said to be pairwise L-closed space if each Definition 2.2: [5] A bitopological space (X,τ 1 ,τ 2 ) is called pairwise T 1 space if for each pair of distinct points x,y in X, there exists a τ 1 -neighbourhood U of x and a τ 2 -neighbourhood is a countable subset of X and hence closed by the assumption,see [6] .
Each point of A is an isolated point.Thus A is discrete.
is first (second) countable and (X,τ 2 ) is first (second) countable.Definition 2.8: A topological space (X,τ)is called Fre chet Urysohn if and only if for every A ⊆ X, and every x ∈ clA, there exists a sequence (x n ) in A converges to x ∀n ∈ N. A bitopological space (X,τ 1 ,τ 2 ) is said to be pairwise Fre chet Urysohn if it is τ 1 -Fre chet Urysohn and τ 2 -Fre chet Urysohn.Definition 2.9: A topological space (X,τ) is called sequential if every non closed subset F of X contains a sequence converging to a point in X-F.A bitopological space (X,τ 1 ,τ 2 ) is said to be pairwise sequential if it is τ 1 -sequential and τ 2 -sequential.Definition 2.10: A topological space (X,τ) is said to have a countable tightness property if whenever A ⊆ X and x ∈ clA, there exists a countable subset B of A such that x ∈ clB.A bitopological space (X,τ 1 ,τ 2 ) is said to have a pairwise countable tightness property if it has τ 1 -countable tightness and τ 2 -countable tightness property.
Proposition 2.11: For a pairwise T 1 pairwise L-closed space, the following are equivalent: Then there exists a τ i −countable local base β = {β n : n ∈ N} at x such that Given any τ i −neighborhood U of x, U must contain β n for some n ∈ N.So x n belongs to U ∀n ≥ n • and x n converges to x.Thus X is pairwise Fre chet Urysohn.b→c) Suppose that X is pairwise Fre chet Urysohn.Let C be a Since X is pairwise Fre chet Urysohn, there exists a Suppose that Thus X is pairwise countable tightness.d→e) Let X be a pairwise countable tightness, let B be a Thus X is a τ i −first countable ∀i=1,2, and hence X is first countable.Definition 2.12: A topological space (X,τ) is said to be sequentially compact if every infinite sequence has a convergent subsequence.A bitopological space (X,τ 1 ,τ 2 ) is called pairwise sequentially compact if it is τ 1 -sequentially compact and τ 1 -sequentially compact.Definition 2.13: A topological space (X,τ) is said to be strongly countably compact if the closure of every countable subset of X is compact.A bitopological space (X,τ 1 ,τ 2 ) is called pairwise strongly countably compact if it is τ 1 -strongly countably compact and τ 2 -strongly countably compact.Definition 2.14: A topological space (X,τ) is said to be countably compact if every countable open cover of X has a finite subcover.A bitopological space (X,τ 1 ,τ 2 ) is called pairwise countably compact if it is τ 1 -countably compact and τ 2 -countably compact.Proof: a→b) Let (x n ) be a τ i -sequence that has no τ i -convergent subsequence ∀i=1,2 ∀n∈ N. w.l.o.g let x k = x l ∀k,l ∈ N, k = l.Each term of the τ i -sequence (x n ) is a τ i -isolated point, since otherwise (x n ) would have a τ i -convergent subsequence.So ∀i=1,2 ∃ a τ i -open set u n neighborhood of x k such that x l / ∈ u n ∀k = l because X is p-Hausdorff.
Let u n =X-{x n :n∈ N} be a τ i -open set, then it's complement consists only of τ i -isolated points.
So it is τ i -closed.Hence {u n }∪{u n : n ∈ N} is a τ i -open cover of X that has no τ i -finite subcover because any finite subcollection of these sets would fail to include infinitely many τ i -terms from (x n ) in its union.
Thus X is not compact.
b→d) Let X be a pairwise sequentially compact and A be a τ i -infinite countable subset of X ∀i=1,2.
There exists a point x ∈ X such that (x n k ) → x.
If U is a τ i -neighborhood of x, then U contains a tail of (x n k ).(U-{x})∩A = φ .Thus x is a τ i -cluster point of A.
Hence X is pairwise countably compact.
Definition 2.16: [8] If (X,τ) is a L-closed space, then X is called a L 4 -space if whenever A ⊆ X is Lindl öf, there exists a Lindl öf F σ −set F such that A ⊆ F ⊆ clA.
Proposition 2.20: If every τ i -Lindl öf subset of (X,τ 1 ,τ 2 ) is a τ i -F σ -set, then X is pairwise T 1 and every countable closed subset of X is discrete.

Proposition 2 .
15: If (X,τ 1 ,τ 2 ) is p-Hausedorff pairwise L-closed, then the following are equivalent: a. X is compact.b.X is pairwise sequentially compact.c.X is pairwise strongly countably compact.d.X is pairwise countably compact.e. X is finite.
Consider R equipped with Sorgenfreyline topology τ s and the left ray topology τ l ,then the bitopological space is not pairwise L-closed space because if A is a τ s −open subset, then A is τ s -Lindl öf.However A is not τ l -closed.