J. Semigroup Theory Appl. 2018, 2018:7 https://doi.org/10.28919/jsta/3668 ISSN: 2051-2937
ON PAIRWISE L-CLOSED SPACES IN BITOPOLOGICAL SPACES
EMAN M. ALMOHOR∗, HASAN Z. HDEIB
Department of Mathematics, University of Jordan, Amman, Jordan
Copyright c2018 Almohor and Hdeib. This is an open access article distributed under the Creative Commons Attribution License, which
permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract. 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.
Keywords:pairwise L-closed spaces; bitopological spaces. 2010 AMS Subject Classification:54A05.
1. 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.
∗Corresponding author
E-mail address: thefirststep1980@gmail.com Received February 1, 2018
We useRand Nto denote the set of all real and natural numbers respectively, p- to denote
pairwise,clto denote the closure of a set, andτs,τlto denote Sorgenfrey and left ray topologies
onRorN.
2. Pairwise L-Closed spaces
Definition 2.1:A bitopological space (X,τ1,τ2) is said to bepairwise L-closed spaceif each τ1-Lindl ¨of subset ofX isτ2-closed and eachτ2-Lindl ¨of subset ofX isτ1-closed.
Definition 2.2: [5] A bitopological space (X,τ1,τ2) is called pairwise T1spaceif for each
pair of distinct pointsx,yinX, there exists aτ1-neighbourhoodU ofxand aτ2-neighbourhood
V ofysuch thatx∈U,y∈/U andy∈V,x∈/V.
Definition 2.3: [4] A bitopological space (X,τ1,τ2) is called p-Hausdorff space if ∀x6=y
inX, there exists aτ1-neighbourhoodU ofxand aτ2-neighbourhoodV ofysuch thatx∈U,
y∈V, andU∩V=φ
Definition 2.4: [3] A bitopological space (X,τ1,τ2)is calledLindlof¨ (resp. compact) if it is τ1-Lindl ¨of (resp.τ1-compact) andτ2-Lindl ¨of (resp.τ2-compact).
Example 2.5:ConsiderRequipped with Sorgenfreyline topologyτsand the left ray topology τl,then the bitopological space is not pairwise L-closed space because ifAis aτs−open subset,
thenAisτs-Lindl ¨of. HoweverAis notτl-closed.
Proposition 2.6:If every countable subset of (X,τ1,τ2) is closed, then every countable subset
ofX is discrete.
Proof: LetAbe aτi−subset ofX such thatA=∪∞k=1{xk}wherexk∈X,k∈N, thenAis a countable subset ofX and hence closed by the assumption,see [6] .
Each point ofAis an isolated point. ThusAis discrete.
Definition 2.7:[3] A bitopological space (X,τ1,τ2)is calledfirst(second)countableif (X,τ1)
is first (second) countable and (X,τ2)is first (second) countable.
Definition 2.8: A topological space (X,τ)is calledFre0chet Urysohnif and only if for every
A⊆X, and everyx∈clA, there exists a sequence (xn) inAconverges to x∀n∈N.A
Definition 2.9: A topological space (X,τ)is calledsequential if every non closed subsetF
ofX contains a sequence converging to a point inX-F. A bitopological space (X,τ1,τ2)is said
to bepairwise sequentialif it isτ1-sequential andτ2-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 ofA such that x∈clB. A
bitopological space (X,τ1,τ2) is said to have apairwise countable tightness property if it has τ1-countable tightness andτ2-countable tightness property.
Proposition 2.11: For a pairwiseT1pairwise L-closed space, the following are equivalent: a. X is first countable.
b.X is pairwise Fre0chet Urysohn.
c. X is pairwise sequential.
d. X has pairwise countable tightness property.
e. X is discrete.
Proof: a→b) Suppose that (X,τ1,τ2)is a first countable space, letAbe a subset ofX such that
x∈cliA∀i=1,2.
Then there exists aτi−countable local baseβe={βn:n∈N} atxsuch that x1∈ β1 , x2∈
β2−β1 , . . . each of which must intersect A. Now choosexm such that xm∈βn∩A ∀m≥1
m∈N,soxmis aτi−sequence inA.
Given anyτi−neighborhoodU ofx,U must containβnfor somen∈N.Soxn belongs toU
∀n≥n◦andxnconverges tox.ThusX is pairwise Fre0chet Urysohn.
b→c) Suppose thatXis pairwise Fre0chet Urysohn. LetCbe aτi−sequentially closed subset
ofX such thatCis notτi−closed∀i=1,2. Letx∈cliCsuch thatx∈/C.
Since X is pairwise Fre0chet Urysohn, there exists a τi−sequence (xn)such that (xn)→ x
∀n∈N.
Cisτi−sequentially closed, hencex∈C.ThusCisτi−closedwhich is a contradiction.
c→d) Let X be a pairwise sequential space, let D be a τi−subset of X such that x∈cliD
∀i=1,2.
Suppose thatK=∪{cliE: E⊆D|E |≤ω◦}. Clearly, K⊂cliD,henceK isτi−sequentially
Let xn be aτi−sequence in D such that xn→x∀n∈N, x∈/D,there exists a τi−countable
subsetEnofD such thatxn∈ ∪∞
n=1cliEn, sox∈K.SinceX is pairwise sequential, K=cliD, i.e
Kisτi−closed.
ThusX is pairwise countable tightness.
d→e) LetX be a pairwise countable tightness, letBbe aτi−open subset ofX such that
x∈cliB ∀i=1,2, ∃ a τi−countable subset M ⊆B such that x∈cliM. But X is a pairwise
L-closed space,
soM=cliMand henceM isτi−discrete by proposition2.6. ThusX is discrete.
e→a) LetX be a discrete space,∀x∈X ∃aτi−open setU containingx. Butx∈{x}because
X is discrete, so{x}⊆U .
Hence{{x}: x∈X}is aτi−local base forX.
ThusX is aτi−first countable∀i=1,2, and henceX is first countable.
Definition 2.12:A topological space (X,τ) is said to besequentially compactif every infinite sequence has a convergent subsequence. A bitopological space (X,τ1,τ2) is called pairwise
sequentially compactif it isτ1-sequentially compact andτ1-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,τ1,τ2) is called
pairwise strongly countably compact if it is τ1-strongly countably compact and τ2-strongly
countably compact.
Definition 2.14: A topological space (X,τ) is said to be countably compact if every
count-able open cover of X has a finite subcover. A bitopological space (X,τ1,τ2)is called pairwise
countably compactif it isτ1-countably compact andτ2-countably compact.
Proposition 2.15: If (X,τ1,τ2) 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.
Proof: a→b) Let (xn) be aτi-sequence that has noτi-convergent subsequence∀i=1,2∀n∈N.
w.l.o.gletxk6=xl ∀k,l∈N,k6=l. Each term of theτi-sequence (xn) is aτi-isolated point,
since otherwise (xn) would have aτi-convergent subsequence. So∀i=1,2∃aτi-open setun
neighborhood ofxk such thatxl∈/un∀k6=lbecauseX is p-Hausdorff.
Let un0=X-{xn:n∈N} be a τi-open set, then it’s complement consists only of τi-isolated
points.
So it is τi-closed. Hence{un0}∪{un : n∈N} is a τi-open cover ofX that has no τi-finite
subcover because any finite subcollection of these sets would fail to include infinitely many
τi-terms from (xn) in its union.
ThusX is not compact.
b→d) LetX be a pairwise sequentially compact andAbe aτi- infinite countable subset of X
∀i=1,2.
Let (xn) be aτi-sequence inA. (xn) has aτi-convergent subsequence (xnk)∀n,k∈N.
There exists a pointx∈X such that(xnk)→x.
IfU is aτi-neighborhood of x, thenU contains a tail of (xnk).(U-{x})∩A6=φ.Thusx is a
τi-cluster point ofA.
HenceX is pairwise countably compact.
Definition 2.16: [8] If (X,τ) is a L-closed space, then X is called a L4-space if whenever
A⊆X is Lindl ¨of, there exists a Lindl ¨of Fσ−setF such thatA⊆F ⊆clA.
Definition 2.17: [8] If (X,τ1,τ2) is a pairwise L-closed space, thenX is called apairwise L4
-spaceif∀τi−Lindl ¨of subsetAof X, there exists aτ−Lindl ¨of Fσ-subsetFsuch thatA⊆F⊆cliA
∀i=1,2.
Proposition 2.18: If (X,τ1,τ2) is a pairwise L-closed space, thenX is pairwise L4-space.
Proposition 2.19: If (X,τ1,τ2)is a pairwise L-closed space, then every τi-Lindl ¨of Fσ-set is τj-closed∀i,j=1,2 i6=j.
Proposition 2.20: If everyτi-Lindl ¨of subset of (X,τ1,τ2) is a τi-Fσ-set, thenX is pairwise
T1and every countable closed subset ofX is discrete.
Proposition 2.21: In a p-Hausdorff space (X,τ1,τ2), the following are equivalent:
b. everyτi-Lindl ¨of Fσ-set isτj-closed∀i,j=1,2 i6=j andτi-closure of aτi-Lindl ¨of subset of
X isτi-Lindl ¨of.
Definition 2.22:[7] A topological space (X,τ) is calledhypercountably compactif the union
of every countable family of compact subsets ofX has a compact closure.
Definition 2.23: A bitopolgical space (X,τ1,τ2) is said to be pairwise hypercountably
com-pactif it isτi−hypercountably compact∀i=1,2.
Proposition 2.25: If a bitopolgical space (X,τ1,τ2) is p-Hausdorff pairwise L4 strongly
countably compact, thenX is pairwise hypercountably compact.
Proof: Let ˜U={ Ak: Ak ⊆X , Ak is τi−compact} ∀k∈N, ∀i=1,2.X is pairwise
strong-ly countabstrong-ly compact, hence cliAk is compact. Since X is pairwise L4,there exists a Lindl ¨of
Fσ−subsetFsuch thatAk⊆F⊆cliAk.
Now∪∞
k=1cliAk is compact, thusX is pairwise hypercountably compact.
Conflict of Interests
The authors declare that there is no conflict of interests.
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