Gen. Math. Notes, Vol. 32, No. 1, January 2016, pp.21-31
c
ISSN 2219-7184; Copyright ICSRS
Publication, 2016
www.i-csrs.org
Available free online at http://www.geman.in
On ij − ∧sδ Sets in Bitopological Spaces
A. Edward Samuel1 and D. Balan2
1
Ramanujan Research Centre
PG & Research Department of Mathematics
Government Arts College (Autonomous)
Kumbakonam - 612002, Tamil Nadu, India
E-mail: aedward74 thrc@yahoo.co.in
2
Research scholar, Ramanujan Research Centre
PG & Research Department of Mathematics
Government Arts College (Autonomous)
Kumbakonam - 612002, Tamil Nadu, India
E-mail: dh.balan@yahoo.com
(Received: 29-10-15 / Accepted: 26-12-15)
Abstract
In this present paper, by utilizing ij −δ semi open sets, we introduce ij −∧sδ ,
ij − ∨sδ , g − ij − ∧sδ , g − ij − ∨sδ sets and investigate some of their properties in
s
bitopological spaces. Also, we present and study the notions of ij − T ∨δ space,
ij −∧sδ continuous, ij −∧sδ irresolute, ij −∧sδ open and ij −∧sδ homeomorphism
functions.
Keywords: ij − δ semi open, ij − ∧sδ sets, ij − ∨sδ sets, g − ij − ∧sδ sets,
s
g − ij − ∨sδ sets, ij − T ∨δ space, ij − ∧sδ continuous, ij − ∧sδ irresolute, ij − ∧sδ
open, ij − ∧sδ homeomorphism.
1
Introduction
The class of generalized ∧ - sets studied by H. Maki in [17]. Caldas et al.
([3],[6]) introduced the concept ∧sδ - sets (resp. ∨sδ - sets) in topological spaces,
which is the intersection of δ - semiopen (resp. union of δ - semiclosed) sets. F.
H. Khedr and H. S. Al-saadi[15] introduced and studied the concept of ij −s∧semi-θ-Closed and pairwise θ-generalized s∧-set in bitopological spaces, which
is an extension of the class of generalized ∧-sets. The aim of this paper is
22
A. Edward Samuel et al.
to introduce the notions of ij − ∧sδ and g − ij − ∧sδ sets and study some of
s
their fundamental properties. Also, we study the new notions of ij −T ∨δ space,
ij − ∧sδ continuous, ij − ∧sδ irresolute, ij − ∧sδ open and ij − ∧sδ homeomorphism
functions and its properties.
2
Preliminaries
Throughout the present paper, (X, τ1 , τ2 )(or briefly X) always mean a bitopological space. Also i, j = 1, 2 and i 6= j. Let A be a subset of (X, τ1 , τ2 ). By
i − int(A) and i − cl(A), we mean respectively the interior and the closure
of A in the topological space (X, τi ) for i = 1, 2. A subset A of X is called
ij - regular open [13] if A = i − int[j − cl(A)]). A point x of X is called an
ij − δ-cluster point of A if i − int(j − cl(U )) ∩ A 6= φ for every τi - open set U
containing x. The set of all ij − δ-cluster points of A is called the ij − δ-closure
of A and is denoted by ij − δcl(A).
Definition 2.1 A subset A is said to be ij − δ closed if ij − δcl(A) = A.
The complement of an ij − δ closed set is said to be ij − δ open. The set of
all ij − δ open (resp. ij − δ closed) sets of X will be denoted by ij − δO(X)(
resp. ij − δC(X)).
ij − ∧sδ Sets and Generalized ij − ∧sδ Sets
3
Definition 3.1 A subset A of a bitopological space (X, τ1 , τ2 )is called ij − δ
semi open if there exists an ij − δ open set U such that U ⊆ A ⊆ j − cl(U ).
Definition 3.2 For a subset A of a bitopological space (X, τ1 , τ2 ), we define
T
A
and Aδs∨ij as follows, Aδs∧ij = {U : A ⊆ U, U ∈ ij − δSO(X)} and
S
Aδs∨ij = {U : U ⊆ A, U C ∈ ij − δSO(X)}.
δs∧ij
Definition 3.3 A subset A of a bitopological space (X, τ1 , τ2 )is called,
(a) ij − ∧sδ set if A = Aδs∧ij .
(b) ij − ∨sδ set if A = Aδs∨ij .
The family of all ij − ∧sδ sets (resp. ij − ∨sδ ) is denoted by ij − ∧sδ (X, τ1 , τ2 )
(resp. ij − ∨sδ (X, τ1 , τ2 )).
Theorem 3.4 Let A be a subset of a bitopological space (X, τ1 , τ2 ). Then
A
= (Aδs∧ij )δs∧ij
δs∧ij
On ij − ∧sδ Sets in Bitopological Spaces
23
Proof: We have (Aδs∧ij )δs∧ij = {U : U ∈ ij − δSO(X, τ1 , τ2 ), Aδs∧ij ⊆
T
T
U } = {U : U ∈ ij − δSO(X, τ1 , τ2 ), ( {V : V ∈ ij − δSO(X, τ1 , τ2 ), A ⊆
T
V }) ⊆ U } ⊆ {U : U ∈ ij − δSO(X, τ1 , τ2 ), A ⊆ U } = Aδs∧ij . This means
(Aδs∧ij )δs∧ij ⊆ Aδs∧ij . On the other hand, A ⊆ Aδs∧ij for each subset A. Then
Aδs∧ij ⊆ (Aδs∧ij )δs∧ij . Therefore Aδs∧ij = (Aδs∧ij )δs∧ij .
T
Theorem 3.5 For any subsets A and B of a bitopological space (X, τ1 , τ2 ),
the following are hold:
(a) A ⊆ Aδs∧ij .
(b) If A ⊆ B, then Aδs∧ij ⊆ B δs∧ij .
(c) If A ∈ ij − δSO(X, τ1 , τ2 ), then A = Aδs∧ij .
(d) (AC )δs∧ij = (Aδs∧ij )C .
(e) Aδs∨ij ⊆ A.
(f ) If A ∈ ij − δSC(X, τ1 , τ2 ), then A = Aδs∨ij .
Proof: (a) Obviously. By the definition of Aδs∧ij , A ⊆ Aδs∧ij .
(b) Suppose that x ∈
/ B δs∧ij . Then there exists a subset U ∈ ij −δSO(X, τ1 , τ2 )
such that U ⊇ B with x ∈
/ U . Since B ⊇ A, then x ∈
/ Aδs∧ij and thus
Aδs∧ij ⊆ B δs∧ij .
(c) By the definition of Aδs∧ij and A ∈ ij − δSO(X, τ1 , τ2 ), we have Aδs∧ij ⊆ A.
By (a), we have A = Aδs∧ij .
(d) By the definition, (Aδs∨ij )C = {U C : U C ⊇ AC , U C ∈ ij−δSO(X, τ1 , τ2 ) =
(AC )δs∧ij .
T
(e) Obviously. Clear by the definition.
(f) If A ∈ ij − δSC(X, τ1 , τ2 ), then AC ∈ ij − δSO(X, τ1 , τ2 ). By (d) and
(e), we have AC = (AC )δs∧ij = (Aδs∧ij )C . Therefore A = Aδs∨ij .
Theorem 3.6 Let A and {Aα , α ∈ J} be the subsets of a bitopological space
(X, τ1 , τ2 ). Then the following are valid:
S
(a) [
α∈J
Aα ]δs∧ij =
S
α∈J (Aα )
δs∧ij
.
24
A. Edward Samuel et al.
(b) [
T
(c) [
S
α∈J
Aα ]δs∧ij ⊆
T
α∈J (Aα )
δs∧ij
.
α∈J
Aα ]δs∨ij ⊇
S
α∈J (Aα )
δs∨ij
.
Proof: (a) Suppose there exists a point x such that x ∈
/ [ α∈J Aα ]δs∧ij . Then
S
there exists a subset U ∈ ij − δSO(X, τ1 , τ2 ), such that α∈J Aα ⊆ U and
x ∈
/ U . Thus for each α ∈ J, we have x ∈
/ (Aα )δs∧ij . This implies that
S
x∈
/ α∈J (Aα )δs∧ij .
S
Conversely, Suppose there exists a point x such that x ∈
/ α∈J (Aα )δs∧ij .
Then by the definition, there exist subsets Uα ∈ ij − δSO(X, τ1 , τ2 ), for each
S
α ∈ J such that x ∈
/ Uα and Aα ⊆ Uα . Let U = α∈J Uα . Then x ∈
/
S
S
S
δs∧ij
.
/ [ α∈J Aα ]
α∈J Uα , α∈J Aα ⊆ U and U ∈ ij − δSO(X, τ1 , τ2 ). Thus x ∈
S
(b) Suppose there exists a point x such that x ∈
/ α∈J (Aα )δs∧ij , then there exists α ∈ J such that x ∈
/ (Aα )δs∧ij . Hence there exists U ∈ ij − δSO(X, τ1 , τ2 )
T
such that U ⊇ Aα and x ∈
/ U . Thus x ∈
/ [ α∈J Aα ]δs∧ij .
T
δs∧ij C
] .
(c) [ α∈J Aα ]δs∨ij = [[( α∈J Aα )C ]δs∧ij ]C = [[ α∈J AC
α]
T
T
δs∨ij C C
C δs∧ij C
⊇ [ α∈J (Aα )
] = [ α∈J (Aα ) ] , by theorem 3.5(d). By (b), we have
S
S
δs∨
δs∨ij
⊇ α∈J Aα ij .
[ α∈J Aα ]
S
S
T
Definition 3.7 A subset A of a bitopological space (X, τ1 , τ2 )is called,
(a) generalized ij − ∧sδ set (g − ij − ∧sδ set) if Aδs∧ij ⊆ U whenever A ⊆ U and
U ∈ ji − δSC(X, τ1 , τ2 ).
(b) generalized ij − ∨sδ set (g − ij − ∨sδ set) if AC is a g∧sδ - set.
s
The family of all g − ij − ∧sδ sets and g − ij − ∨sδ sets are denoted by ij − D∧δ
s
and ij − D∨δ .
Theorem 3.8 (a) The φ and X are ij − ∧sδ sets and ij − ∨sδ sets.
(b) Every union of ij − ∧sδ sets (resp. ij − ∨sδ ) is a ij − ∧sδ set (resp. ij − ∨sδ ).
(c) Every intersection of ij − ∧sδ sets (resp. ij − ∨sδ sets) is a ij − ∧sδ set
(ij − ∨sδ set).
(d) A subset A of (X, τ1 , τ2 )is a ij − ∧sδ set if and only if AC is a ij − ∨sδ
set.
Proof: (a) Obvious.
On ij − ∧sδ Sets in Bitopological Spaces
25
(b) Let {Aα , α ∈ J} be a family of ij − ∧sδ sets in a bitopological space
S
S
(X, τ1 , τ2 ). Then by theorem 3.5(a), we have α∈J Aα = α∈J (Aα )δs∧ij =
S
[ α∈J Aα ]δs∨ij .
(c) Let {Aα , α ∈ J} be a family of ij − ∧sδ sets in a bitopological space
T
T
(X, τ1 , τ2 ). By theorem 3.5(b), we have [ α∈J Aα ]δs∧ij ⊆ α∈J (Aα )δs∧ij =
T
T
T
δs∧ij
.
α∈J Aα . Hence by theorem 3.4(b), we have
α∈J Aα = [ α∈J Aα ]
(d) Obvious.
δs∧ij
Remark 3.9 (a) In general [A1 ∩ A2 ]δs∧ij 6= A1
δs∧ij
∩ A2
.
(b) The family of all ij − ∧sδ sets and ij − ∨sδ sets are the topologies on X
containing
all ij − δ semi open sand ij − δ semi closed sets respectively.
Let
∧sδ
∨δ
∧sδ
s
s
τij = ij − ∧δ (X, τ1 , τ2 ) and τij = ij − ∨δ (X, τ1 , τ2 ). Clearly (X, τij ) and
∨s
(X, τij δ ) are Alexandroff spaces, i.e. arbitrary intersection of open sets is open.
Theorem 3.10 Let (X, τ1 , τ2 )be a bitopological space. Then
(a) Every ij − ∧sδ set is a g − ij − ∧sδ set.
(b) Every ij − ∨sδ set is a g − ij − ∨sδ set.
s
s
(c) If Aα ∈ ij − D∧δ for all α ∈ J, then
S
s
T
(d) If Aα ∈ ij − D∧δ for all α ∈ J, then
α∈J
Aα ∈ ij − D∧δ .
s
α∈J
Aα ∈ ij − D∧δ .
Proof: (a) Obvious. Follows from definition.
(b) Let A be a ij − ∨sδ subset of (X, τ1 , τ2 ). Then A = Aδs∨ij . By theorem 3.4(d), [AC ]δs∧ij = [Aδs∨ij ]C = AC . Therefore by (a), A is a g − ij − ∨sδ
set.
s
(c) Let Aα ∈ ij − D∧δ for all α ∈ J. Then by theorem 3.5(a), [ α∈J Aα ]δs∧ij =
S
S
s
δs∧ij
. Hence by hypothesis and definition, α∈J (Aα ) ∈ ij − D∧δ .
α∈J (Aα )
S
(d) Obvious. Follows from (c) and definition 3.2.
Theorem 3.11 A subset A of a bitopological space (X, τ1 , τ2 )is a g −ij −∨sδ
set if and only if U ⊆ Aδs∨ij , whenever U ⊆ A and U ∈ ij − δSO(X, τ1 , τ2 ).
Proof: Let U be a ij − δ semi open subset of (X, τ1 , τ2 )such that U ⊆ A.
Then since U C is ij − δ semi closed and U C ⊇ AC , we have U C ⊆ Vδs (AC ), by
26
A. Edward Samuel et al.
definition 3.2. Hence by theorem 3.8(d), U C ⊇ [Vδs (A)]C . Thus U ⊆ Vδs (A).
Conversely, let U be a ij − δ semi closed subset of (X, τ1 , τ2 )such that AC ⊆ U .
Since U C is ij −δ semi open and U C ⊆ A, by assumption we have U C ⊆ Vδs (A).
Then U ⊇ [Vδs (A)]C = ∧sδ (AC ) by theorem 3.8(d). Therefore AC is a g −ij −∧sδ
set. Thus A is a g − ij − ∨sδ .
A bitopological space (X, τ1 , τ2 )is called ij − δs − T1 if for each distinct points
x, y ∈ X, there exist two ij − δ semi open sets U and V such that x ∈ U \V
and y ∈ V \U . If X is 12 − δs − T1 and 21 − δs − T1 , then it is called pairwise
δs − T1 (P − δs − T1 ). Also let (X, τ1 , τ2 )is ij − δs − T1 if and only if every
singleton {x} is ij − δ semi closed.
Theorem 3.12 Let ij − δSC(X, τ1 , τ2 ) be closed by unions. Then for a
bitopological space (X, τ1 , τ2 ), the following are equivalent,
(a) (X, τ1 , τ2 )is ij − δs − T1 .
(b) Every subset of (X, τ1 , τ2 )is a ij − ∧sδ set.
(c) Every subset of (X, τ1 , τ2 )is a ij − ∨sδ set.
Proof: (a) =⇒ (c) Suppose that (X, τ1 , τ2 )is ij − δs − T1 . Let A be any subset
S
of X. Since every singleton {x} is ij − δ semi closed and A = {{x}, x ∈ A},
then A is the union of ij − δ semi closed sets. Hence A is a ij − ∨sδ set.
(c) =⇒ (a) Since by (c), we have that every singleton is the union of ij − δ
semi closed sets, i.e., it is ij − δ semi closed, then (X, τ1 , τ2 )is a ij − δs − T1
space.
(b) =⇒ (c) Obvious.
Theorem 3.13 Let (X, τ1 , τ2 )be a bitopological space. Then
∧s
∨s
(a) (X, τij δ ) and (X, τij δ ) are always δ − T1/2 spaces.
∧s
∨s
(b) If (X, τ1 , τ2 )is ij − δs − T1 , then both (X, τij δ ) and (X, τij δ ) are discrete
spaces.
∧s
(c) The identity function id : (X, τij δ ) −→ (X, τi ) is δ - continuous.
∨s
(d) The identity function id : (X, τij δ ) −→ (X, τi ) is δ - contra continuous.
Proof: (a) Let x ∈ X. Then {x} is ij − δ open or ij − δ semi closed in X. If
∧s
{x} is ij − δ open, then it is ij − δ semi open. Therefore {x} ∈ τij δ . If {x}
On ij − ∧sδ Sets in Bitopological Spaces
27
∧s
is ij − δ semi closed then X − {x} is ij − δ semi open and so X − {x} ∈ τij δ ,
∧s
∧s
i.e., {x} is τij δ - closed. Hence (X, τij δ ) is δ − T1/2 . In similar manner, we can
∨s
prove (X, τij δ ) is δ − T1/2 .
(b) Obvious. The proof follows from theorem 3.8.
∧s
(c) If A is ij − δ open, then A is ij − δ semi open. Hence A ∈ τij δ
(d) If A is ij − δ open, then A is ij − δ semi open. This implies that X − A is
∨s
∨s
ij − δ semi closed and hence X − A is τij δ - open or A is τij δ - closed.
4
Applications
Definition 4.1 A bitopological space (X, τ1 , τ2 )is called an ij−δs−R0 space
if for every ij − δ semi open set U , x ∈ U implies ij − δscl({x}) ⊆ U .
s
Definition 4.2 A bitopological space (X, τ1 , τ2 )is called a ij − T ∨δ space if
∨s
τij = τij δ .
∧sδ
Theorem 4.3 For a bitopological space (X, τ1 , τ2 )the following are equivalent,
(a) (X, τ1 , τ2 )is ij − δs − R0 .
∧s
(b) (X, τij δ ) is discrete space.
∨s
(c) (X, τij δ ) is discrete space.
(d) For each x ∈ X, {x} is a ij − ∧sδ set of (X, τ1 , τ2 ).
(e) For each ij − δ semi open set U of X, U = U δs∨ij .
s
(f ) (X, τ1 , τ2 )is ij − T ∨δ space.
∧s
(g) (X, τij δ ) is R0 - space.
Proof: (a) =⇒ (b) For any x ∈ X we have {x}δs∧ij = {U : {x} ⊆
U, U isij −δsemiopen}. Since X is ij −δs−R0 space, then each ij −δ semi open
set U containing x contains ij − δscl({x}). Hence ij − δscl({x}) ⊆ {x}δs∧ij .
∧s
Then by theorem 3.13, (X, τij δ ) is discrete space.
T
28
A. Edward Samuel et al.
∧s
(b) =⇒ (c) Suppose that (X, τij δ ) is discrete space. By the definition of Aδs∨ij ,
Aδs∧ij = [(X − A)δs∨ij ]C . Therefore if X is ij − ∧sδ set, then X − A is ij − ∨sδ
∨s
set. Then (X, τij δ ) is discrete space.
s
(c) =⇒ (d) For each x ∈ X, {x} is τi j ∧δ - open and {x} is a ij − ∧sδ set
of (X, τ1 , τ2 ).
(d) =⇒ (e) Let U be a ij − δ semi open set. Let x ∈ U C . By assumpS
tion {x} = xδs∧ij and therefore xδs∧ij ⊆ U C . Hence U C ⊇ {{x}δs∧ij : x ∈
S
U C } = [ {x : x ∈ U C }]δs∧ij = [U C ]δs∧ij . This shows that U C = [U C ]δs∧ij and
By the definition of Aδs∨ij , Aδs∧ij = [(X − A)δs∨ij ]C , we have U = U δs∨ij .
∨s
∧s
∨s
(e) =⇒ (f) By (e) ij − δSO(X, τ1 , τ2 ) ⊆ τij δ . First we show that τij δ ⊆ τij δ .
T
Let A be any ij − ∧sδ of (X, τ1 , τ2 ). Then A = Aδs∧ij = {U : U ⊆ A, U ∈
s
∨
ij − δSO(X)}. Since ij − δSO(X, τ1 , τ2 ) ⊆ τij δ , by theorem 3.10 we have
∨s
∧s
∨s
∨s
∧s
∨s
A ∈ τij δ and τij δ ⊆ τij δ . Next, let A ∈ τij δ . Then X − A ∈ τij δ ⊆ τij δ . Theres
∨s
∧s
∨s
fore A ∈ τij δ and τij δ ⊆ τij δ . Hence (X, τ1 , τ2 )is a ij − T ∨δ space.
∧s
s
(f) =⇒ (g) Let U ∈ τij δ and x ∈ U . Since (X, τ1 , τ2 )is a ij − T ∨δ space,
∨s
∧s
∧s
U ∈ τij δ and U C ∈ τij δ . Since {x} ∩ U C = φ, τij δ − cl({x}) ∩ U C = φ and
∧s
∧s
τij δ − cl({x}) ⊆ U . Hence (X, τij δ − cl({x})) is R0 - space.
(g) =⇒ (a) Let U ∈ ij − δSO(X, τ1 , τ2 ) and x ∈ U . Since ij − δSO(X, τ1 , τ2 ) ⊆
∧s
∧s
∧s
∨s
∨s
τij δ , by (g), τij δ − cl({x}) ⊆ U . Since τij δ − cl({x}) ∈ τij δ − cl({x}), τij δ −
S
∧s
cl({x}) = {F : F ∈ τij δ − cl({x})andF ∈ ij − δSC(X, τ1 , τ2 )} and x ∈
∧s
τij δ − cl({x}). Therefore for some F ∈ ij − δSC(X, τ1 , τ2 ), x ∈ F and
∧s
hence ij − δscl({x}) ⊆ F ⊆ τij δ − cl({x}) ⊆ U . This shows that (X, τ1 , τ2 )is
ij − δs − R0 .
Definition 4.4 A function f : (X, τ1 , τ2 ) −→ (Y, σ1 , σ2 ) is called,
(a) ij − ∧sδ continuous, if f −1 (V ) is a ij − ∧sδ set in (X, τ1 , τ2 )for each σi
- open set V of (Y, σ1 , σ2 ).
(b) ij − ∧sδ irresolute, if f −1 (V ) is a ij − ∧sδ set in (X, τ1 , τ2 )for each ij − ∧sδ
set V of (Y, σ1 , σ2 ).
(c) ij − ∧sδ open if f (U ) is a ij − ∧sδ set in (Y, σ1 , σ2 ) for each ij − ∧sδ set
U of (X, τ1 , τ2 ).
Definition 4.5 A function f : (X, τ1 , τ2 ) −→ (Y, σ1 , σ2 ) is called a ij − ∧sδ
On ij − ∧sδ Sets in Bitopological Spaces
29
homeomorphism if f is a ij − ∧sδ irresolute, ij − ∧sδ open and bijective.
Theorem 4.6 Let f : (X, τ1 , τ2 ) −→ (Y, σ1 , σ2 ) be a function.
s
(a) If f is ij − ∧sδ irresolute, injection and (Y, σ1 , σ2 ) is a ij − T ∨δ space,
s
then (X, τ1 , τ2 )is a ij − T ∨δ space.
s
(b) If f is ij − ∧sδ open surjection and (X, τ1 , τ2 )is a ij − T ∨δ space, then
s
(Y, σ1 , σ2 ) is a ij − T ∨δ space.
s
(c) If f is ij − ∧sδ homeomorphism, then (X, τ1 , τ2 )is a ij − T ∨δ space if and
s
only if (Y, σ1 , σ2 ) is a ij − T ∨δ space.
s
Proof: (a) Since (Y, σ1 , σ2 ) is a ij − T ∨δ space, (Y, σ1 , σ2 ) is discrete by the∧s
orem 4.3. Then {f (x)} ∈ σijδ for every x ∈ X. Since f is ij − ∧sδ irresolute,
∧s
∧s
f −1 (f (x)) ∈ τij δ for every x ∈ X. This implies {x} ∈ τij δ for every x ∈ X, since
∧s
f is injective. Therefore (X, τij δ ) is discrete and by Theorem 4.3, (X, τ1 , τ2 )is
s
a ij − T ∨δ space.
∧s
(b) Let y ∈ Y . {f −1 (y)} 6= φ, since f is surjective. Since (X, τij δ ) is discrete,
∧s
∧s
{f −1 (y)} ∈ τij δ for every y ∈ Y . Since f is ij − ∧sδ open, f ({f −1 (y)}) ∈ σijδ
∧s
∧s
for every y ∈ Y . This implies {y} ∈ σijδ for every y ∈ Y or (Y, σijδ ) is discrete.
s
Hence (Y, σ1 , σ2 ) is a ij − T ∨δ space.
(c) Follows from (a) and (b).
Acknowledgements: The authors would like to thank the referees for the
useful comments and valuable suggestions for improvement of the paper.
References
[1] S.P. Arya and T.M. Nour, Separation axioms for bitopological spaces,
Indian J. Pure. Apple. Math., 19(3) (1988), 42-50.
[2] G.K. Banerjee, On pairwise almost strongly θ - continuous mapping, Bull.
Calcutta Math. Soc., 79(1987), 314-320.
[3] M. Caldas and J. Dontchev, G.∧s - sets and G.∨s - sets, Mem. Fac. Sci.
Kochi Univ. Math., 21(2000), 21-30.
[4] M. Caldas, M. Ganster, D.N. Georgiou, S. Jafari and S.P. Moshokoa,
δ-semi open sets in topology, Topology Proc., 29(2) (2005), 369-383.
30
A. Edward Samuel et al.
[5] M. Caldas, M. Ganster, S. Jafari and T. Noiri, On ∧p - sets and functions,
Mem. Fac. Sci. Kochi Univ. Math., 25(2003), 1-8.
[6] M. Caldas, D.N. Georgiou, S. Jafari and T. Noiri, More on δ - semiopen
sets, Note Mat., 22(2) (2003/04), 113-126.
[7] M. Caldas, S. Jafari, S.A. Ponmani and M.L. Thivagar, On some low
separation axioms in bitopological Spaces, Bol. Soc. Paran. Mat., (3s.)
(24)(1-2) (2006), 69-78.
[8] M.M. El-Sharkasy, On ∧α -sets and the associated topology T ∧α , Journal
of the Egyptian Mathematical Society, 23(2015), 371-376.
[9] M. Ganster, S. Jafar and T. Noiri, On pre-∧-sets and pre-∨-sets, Acta
Math. Hungar, 95(4) (2002), 337-343.
[10] A. Ghareeb and T. Noiri, ∧ - Generalized closed sets in bitopological
spaces, Journal of the Egyptian Mathematical Society, 19(2011), 142-145.
[11] M.J. Jeyanthi, A. Kilicman, S.P. Missier and P. Thangavelu, ∧r -Sets and
separation axioms, Malaysian Journal of Mathematics, 5(1) (2011), 45-60.
[12] A.B. Khalaf and A.M. Omer, Si - Open sets and Si - continuity in bitopological spaces, Tamkang Journal of Mathematics, 43(1) (2012), 81-97.
[13] F.H. Khedr, Properties of ij - delta open sets, Fasciculi Mathematici,
52(2014), 65-81.
[14] F.H. Khedr and K.M. Abdelhakiem, Generalized ∧ - sets and λ - sets
in bitopological spaces, International Mathematical Forum, 4(15) (2009),
705-715.
[15] F.H. Khedr and H.S. Al-saadi, On pairwise θ-semi-generalized closed sets,
Far East J. Mathematical Sciences, 28(2) (February) (2008), 381-394.
[16] F.H. Khedr, A.M. Al-Shibani and T. Noiri, On δ - continuity in bitopological spaces, J. Egypt. Math. Soc., 5(3) (1997), 57-63.
[17] H. Maki, Generalized ∧ - sets and the associated closure operator, The
Special Issue in Commemoration of Prof. Kazuada IKEDA’s Retirement,
(1986), 139-146.
[18] M. Mirmiran, Weak insertion of a perfectly continuous function between
two real-valued functions, Mathematical Sciences and Applications ENotes, 3(1) (2015), 103-107.
On ij − ∧sδ Sets in Bitopological Spaces
31
[19] T.M. Nour, A note on five separation axioms in bitopological spaces,
Indian J. Pure and Appl. Math., 26(7) (1995), 669-674.
[20] M. Pritha, V. Chandrasekar and A. Vadivel, Some aspects of pairwise
fuzzy semi preopen sets in fuzzy bitopological spaces, Gen. Math. Notes,
26(1) (January) (2015), 35-45.