EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 6, No. 2, 2013, 247-255
ISSN 1307-5543 – www.ejpam.com

Some Fundamental Properties of β-Open Sets in Ideal
Bitopological Spaces

M. Caldas1, S. Jafari2 and N. Rajesh3,∗

1 Departamento de Matematica Aplicada, Universidade Federal Fluminense, Rua Mario Santos
Braga, S/n, 24020-140, Niteroi, RJ Brasil
2 College of Vestsjaelland South, Herrestraede 11, 4200 Slagelse, Denmark
3 Department of Mathematics, Rajah Serfoji Govt. College, Thanjavur-613005, Tamilnadu, India

Abstract. In this paper we introduce and characterize the concepts of β-open sets and their related
notions in ideal bitopological spaces.

2010 Mathematics Subject Classifications: 54D10

Key Words and Phrases: Ideal bitopological spaces, (i, j)−β−I -open sets, (i, j)−β−I -closed sets.

1. Introduction

Kuratowski [7] and Vaidyanathasamy [9] introduced and investigated the concept of ide-
als in topological spaces. An ideal I on a topological space (X ,τ) is a nonempty collection
of subsets of X which satisfies (i) A ∈ I and B ⊂ A implies B ∈ I and (ii) A ∈ I and B ∈ I
implies A∪B ∈ I . Given a bitopological space (X ,τ1,τ2) with an ideal I on X and if P (X ) is
the set of all subsets of X , a set operator (.)∗i : P (X )→P (X ), called the local function [9] of
A with respect to τi and I , is defined as follows: for A⊂ X , A∗i (τi ,I ) = {x ∈ X |U ∩A /∈ I for
every U ∈ τi(x)}, where τi(x) = {U ∈ τi|x ∈ U}. For every ideal topological space (X ,τ,I ),
there exists a topology τ∗(I ), finer than τ, generated by the base
β(I ,τ) = {U\I |U ∈ τ and I ∈ I }, but in general β(I ,τ) is not always a topology [4]. Ob-
serve additionally that τi − Cl∗(A) = A∪ A∗i (τi ,I ) defines a Kuratowski closure operator for
τ∗(I ), when there is no chance of confusion, A∗i (I ) is denoted by A∗i and τi− Int∗(A) denotes
the interior of A in τ∗i (I ). In this paper we introduce and characterize the concepts of β-open
sets and their related notions in ideal bitopological spaces.

∗Corresponding author.

Email addresses: gmamccs@vm.uff.br (M. Caldas), jafari@stofanet.dk (S. Jafari),
nrajesh_topology@yahoo.co.in (N. Rajesh)

http://www.ejpam.com 247 c© 2013 EJPAM All rights reserved.



M. Caldas, S. Jafari, N. Rajesh / Eur. J. Pure Appl. Math, 6 (2013), 247-255 248

2. Preiliminaries

For a subset A of a bitopological space (X ,τ1,τ2), we denote the closure of A and the
interior of A with respect to τi by τi −Cl(A) and τi − Int(A), respectively.

Definition 1. A subset A of a bitopological space (X ,τ1,τ2) is said to be (i, j)-semiopen [5]
(resp. (i, j)-preopen [5], (i, j)-semi-preopen [6]) if A⊂ τ j −Cl(τi − Int(A)) (resp.
A⊂ τi − Int(τ j −Cl(A)), A⊂ τ j −Cl(τi − Int(τ j −Cl(A)))), where i, j = 1, 2 and i 6= j.

Definition 2. A subset A of an ideal bitopological space (X ,τ1,τ2,I ) is said to be

(i) (i, j)-semi-I -open [3] if A⊂ τ j −Cl∗(τi − Int(A)).

(ii) (i, j)-pre-I -open [2] if A⊂ τi − Int(τ j −Cl∗(A)).

(iii) (i, j)− b−I -open [3] if A⊂ τi − Int(τ j −Cl∗(A))∪τ j −Cl∗(τi − Int(A)) .

(iv) (i, j)−α−I -open [3] if A⊂ τi − Int(τ j −Cl∗(τi − Int(A))).

Definition 3. A function f : (X ,τ1,τ2,I )→ (Y,σ1,σ2) is said to be

(i) (i, j)-pre-I -continuous [2] if the inverse image of every σi-open set of Y is (i, j)-pre-I -
open in X , where i 6= j, i, j=1, 2.

(ii) (i, j)-semi-I -continuous [3] if the inverse image of every σi-open set of Y is (i, j)-semi-I -
open in X , where i 6= j, i, j=1, 2.

(iii) (i, j)− b−I -continuous [3] if the inverse image of every σi-open set of Y is (i, j)− b−I -
open in X , where i 6= j, i, j=1, 2.

(iv) (i, j)−α−I -continuous [3] if the inverse image of every σi-open set of Y is (i, j)−α−I -
open in X , where i 6= j, i, j=1, 2.

(v) pairwise semi-precontinuous [6] if the inverse image of every σi-open set in (Y,σ1,σ2) is
(i, j)-semi-preopen in (X ,τ1,τ2), where i 6= j, i, j=1, 2.

3. Properties of (i, j)− β −I -open Sets

Definition 4. A subset A of an ideal bitopological space (X ,τ1,τ2,I ) is said to be (i, j)−β−I -
open if A⊂ τ j −Cl(τi − Int(τ j −Cl∗(A))), where i, j = 1,2 and i 6= j.
The family of all (i, j) − β − I -open sets of (X ,τ1,τ2,I ) is denoted by βIO(X ,τ1,τ2) or
(i, j)−βIO(X ). Also, The family of all (i, j)−β −I -open sets of (X ,τ1,τ2,I ) containing x is
denoted by (i, j)− βIO(X , x).

Remark 1. Let I and J be two ideals on (X ,τ1,τ2). If I ⊂ J , then
βJ O(X ,τ1,τ2)⊂ βIO(X ,τ1,τ2).

Proposition 1. (i) Every (i, j)− b−I -open set is (i, j)− β −I -open.



M. Caldas, S. Jafari, N. Rajesh / Eur. J. Pure Appl. Math, 6 (2013), 247-255 249

(ii) Every (i, j)− β −I -open set is (i, j)-semi-preopen.

Proof. The proof follows from the definitions.

The following example shows that the converses of Proposition 1 is not true in general.

Example 1. Let X = {a, b, c}, τ1 = {;, {a}, X }, τ2 = {;, {a}, {a, b}, X } and I = {;, {a}}. Then
the set {a, c} is (i, j)− β −I -open but not (i, j)− b−I -open.

Corollary 1. (i) Every (i, j)−α−I -open set is (i, j)− β −I -open.

(ii) Every (i, j)-semi-I -open set is (i, j)− β −I -open.

(iii) Every (i, j)-pre-I -open set is (i, j)− β −I -open.

Proposition 2. For an ideal bitopological space (X ,τ1,τ2,I ) and A⊂ X , we have:

(i) If I = {;}, then A is (i, j)− β −I -open if and only if A is (i, j)-semi-preopen.

(ii) If I =P (X ), then A is (i, j)− β −I -open if and only if A is (i, j)-semiopen.

Proof. The proof follows from the fact that

(i) If I = {;}, then A∗ = Cl(A).

(ii) If I =P (X ), then A∗ = ; for every subset A of X .

Remark 2. The intersection of any two (i, j)−β −I -open sets is not an (i, j)−β −I -open set
as it can be seen from the following example.

Example 2. Let X = {a, b, c, d}, τ1 = {∅, {a}, {b}, {a, b}, {a, b, c}, X }, τ2 = {∅, X } and
I = {∅, {c}, {d}, {c, d}}. Then the sets {a, c} and {b, c} are (1, 2)−β−I -open sets of (X ,τ1,τ2,I )
but their intersection {c} is not an (1,2)− β −I -open set of (X ,τ1,τ2,I ).

Theorem 1. If {Aα}α∈Ω is a family of (i, j)− β −I -open sets in (X ,τ1,τ2,I ), then
⋃

α∈Ω
Aα is

(i, j)− β −I -open in (X ,τ1,τ2,I ).

Proof. Since {Aα : α ∈ Ω} ⊂ (i, j)−βIO(X ), then Aα ⊂ τ j −Cl(τi − Int(τ j −Cl∗(Aα))) for
every α ∈ Ω. Thus,

∪
α∈Ω

Aα ⊂ ∪
α∈Ω
τ j −Cl(τi − Int(τ j −Cl∗(Aα)))⊂ τ j −Cl(τi − Int( ∪

α∈Ω
τ j −Cl∗(Aα)))

= τ j −Cl(τi − Int(τ j −Cl∗( ∪
α∈Ω

Aα))).

Therefore, we obtain ∪
α∈Ω

Aα ⊂ τ j−Cl(τi− Int(τ j−Cl∗( ∪
α∈Ω

Aα))). Hence any union of (i, j)−
β −I -open sets is (i, j)− β −I -open.

Theorem 2. A subset A of an ideal bitopological space (X ,τ1,τ2,I ) is (i, j)−β−I -open if and
only if τ j −Cl(A) = τ j −Cl(τi − Int(τ j −Cl∗(A))).



M. Caldas, S. Jafari, N. Rajesh / Eur. J. Pure Appl. Math, 6 (2013), 247-255 250

Proof. Let A be an (i, j)− β −I -open subset of X . Then, we have
A⊂ τ j −Cl(τi − Int(τ j −Cl∗(A))) and hence

τ j −Cl(A)⊂ τ j −Cl(τi − Int(τ j −Cl∗(A)))⊂ τ j −Cl(τi − Int(τ j −Cl(A)))⊂ τ j −Cl(A).

Therefore, τ j −Cl(A) = τ j −Cl(τi − Int(τ j −Cl∗(A))). The converse is obvious.

Definition 5. A bitopological space (X ,τ1,τ2) is said to be pairwise extremally disconnected [1]
if τ j −Cl(A) ∈ τi for every A∈ τi .

Proposition 3. Let (X ,τ1,τ2,I ) be a pairwise extremally disconnected space. If A is (i, j)−β−
I -open, then it is (i, j)-preopen in X .

Proof. Let A be (i, j)− β −I -open set of X , we have A ⊂ τ j − Cl(τi − Int(τ j − Cl∗(A))).
Since X is pairwise extremally disconnected, for τi − Int(τ j −Cl∗(A)) ∈ τi , we have
τ j −Cl(τi − Int(τ j −Cl∗(A))) ∈ τi . So, we have

A⊂τ j −Cl(τi − Int(τ j −Cl∗(A)))⊂ τi − Int(τ j −Cl(τi − Int(τ j −Cl∗(A))))

⊂τi − Int(τ j −Cl(τ j −Cl∗(A)))⊂ τi − Int(τ j −Cl(A∪ A∗))

= τi − Int(τ j −Cl(A)∪τ j −Cl(A∗))⊂ τi − Int(τ j −Cl(A));

hence A is (i, j)-preopen in X .

An ideal bitopological space is said to satisfy the condition (A ) if
U ∩τ j −Cl∗(A)⊂ τ j −Cl∗(U ∩ A) for every U ∈ τi .

Theorem 3. Let (X ,τ1,τ2,I ) be a pairwise extremally disconnected space which satisfies the
condition A . If A is (i, j)-semi-I -open and B is (i, j)-pre-I -open, then A∩ B is (i, j)− β −I -
open.

Proof. Let A be (i, j)-semi-I -open and B an (i, j)-pre-I -open set of X . Then

A∩ B ⊂τ j −Cl∗(τi − Int(A))∩τi − Int(τ j −Cl∗(B))⊂ τ j −Cl∗(τi − Int(A)∩τi − Int(τ j −Cl∗(B))

= τ j −Cl∗(τi − Int(τi − Int(A))∩τ j −Cl∗(B)))⊂ τ j −Cl∗(τi − Int(τ j −Cl∗(τi − Int(A)∩ B)))

⊂ τ j −Cl∗(τi − Int(τ j −Cl∗(A∩ B)))⊂ τ j −Cl(τi − Int(τ j −Cl∗(A∩ B))).

Thus, A∩ B is (i, j)− β −I -open in X .

Definition 6. In an ideal bitopological space (X ,τ1,τ2,I ), A⊂ X is said to be (i, j)− β −I -
closed if X\A is (i, j)− β −I -open in X , i, j = 1,2 and i 6= j.

Theorem 4. If A is an (i, j)− β −I -closed set in an ideal bitopological space (X ,τ1,τ2,I ) if
and only if τ j − Int(τi −Cl(τ j − Int∗(A)))⊂ A.

Proof. The proof follows from the definitions.



M. Caldas, S. Jafari, N. Rajesh / Eur. J. Pure Appl. Math, 6 (2013), 247-255 251

Theorem 5. A subset A of an ideal bitopological space (X ,τ1,τ2,I ) is (i, j)−β−I -closed, then
τ j − Int(τi −Cl∗(τ j − Int(A)))⊂ A

Proof. The proof follows from the fact that Cl∗(A)⊂ Cl(A) for every subset A of X .

Theorem 6. Arbitrary intersection of (i, j)− β −I -closed sets is always (i, j)− β −I -closed.

Proof. Follows from Theorems 1 and 5.

Definition 7. Let (X ,τ1,τ2,I ) be an ideal bitopological space, S a subset of X and x be a point
of X . Then

(i) x is called an (i, j)− β −I -interior point of S if there exists V ∈ (i, j)− βIO(X ,τ1,τ2)
such that x ∈ V ⊂ S.

(ii) the set of all (i, j)− β −I -interior points of S is called (i, j)− β −I -interior of S and is
denoted by (i, j)− βI Int(S).

Theorem 7. Let A and B be subsets of (X ,τ1,τ2,I ). Then the following properties hold:

(i) (i, j)− βI Int(A) = ∪{T : T ⊂ A and T ∈ (i, j)− βIO(X )}.

(ii) (i, j)− βI Int(A) is the largest (i, j)− β −I -open subset of X contained in A.

(iii) A is (i, j)− β −I -open if and only if A= (i, j)− βI Int(A).

(iv) (i, j)− βI Int((i, j)− βI Int(A)) = (i, j)− βI Int(A).

(v) If A⊂ B, then (i, j)− βI Int(A)⊂ (i, j)− βI Int(B).

(vi) (i, j)− βI Int(A∩ B)⊂ (i, j)− βI Int(A)∩ (i, j)− βI Int(B).

(vii) (i, j)− βI Int(A∪ B)⊃ (i, j)− βI Int(A)∪ (i, j)− βI Int(B).

Proof. (vi). Since A∩ B ⊂ A and A∩ B ⊂ B, by (iv), we have
(i, j) − βI Int(A ∩ B) ⊂ (i, j) − βI Int(A) and (i, j) − βI Int(A ∩ B) ⊂ (i, j) − βI Int(B).
Therefore, (i, j)− βI Int(A∩ B)⊂ (i, j)− βI Int(A)∩ (i, j)− βI Int(B).
(vii). We have (i, j)− βI Int(A)⊂ (i, j)− βI Int(A∪ B) and
(i, j)− βI Int(B)⊂ (i, j)− βI Int(A∪ B). Then we obtain
(i, j)− βI Int(A)∪ (i, j)− βI Int(B)⊂ (i, j)− βI Int(A∪ B).
The other proofs are obvious.

Definition 8. Let (X ,τ1,τ2,I ) be an ideal bitopological space, S a subset of X and x be a point
of X . Then

(i) x is called an (i, j)−β−I -cluster point of S if V ∩S 6= ; for every V ∈ (i, j)−βIO(X , x).

(ii) the set of all (i, j)− β −I -cluster points of S is called (i, j)− β −I -closure of S and is
denoted by (i, j)− βI Cl(S).



M. Caldas, S. Jafari, N. Rajesh / Eur. J. Pure Appl. Math, 6 (2013), 247-255 252

Theorem 8. Let A and B be subsets of (X ,τ1,τ2,I ). Then the following properties hold:

(i) (i, j)− βI Cl(A) = ∩{F : A⊂ F and F ∈ (i, j)− βI C(X )}.

(ii) (i, j)− βI Cl(A) is the smallest (i, j)− β −I -closed subset of X containing A.

(iii) A is (i, j)− β −I -closed if and only if A= (i, j)− βI Cl(A).

(iv) (i, j)− βI Cl((i, j)− βI Cl(A) = (i, j)− βI Cl(A).

(v) If A⊂ B, then (i, j)− βI Cl(A)⊂ (i, j)− βI Cl(B).

(vi) (i, j)− βI Cl(A∪ B)⊃ (i, j)− βI Cl(A)∪ (i, j)− βI Cl(B).

(vii) ] (i, j)− βI Cl(A∩ B)⊂ (i, j)− βI Cl(A)∩ (i, j)− βI Cl(B).

Proof. The proofs follows from the definitions.

Theorem 9. Let (X ,τ1,τ2,I ) be an ideal bitopological space and A⊂ X . A point
x ∈ (i, j)− βI Cl(A) if and only if U ∩ A 6= ; for every U ∈ (i, j)− βIO(X , x).

Proof. Suppose that x ∈ (i, j)− βI Cl(A). We shall show that U ∩ A 6= ; for every
U ∈ (i, j)−βIO(X , x). Suppose that there exists U ∈ (i, j)−βIO(X , x) such that U ∩A= ;.
Then A⊂ X\U and X\U is (i, j)− β −I -closed. Since A⊂ X\U ,
(i, j)− βI Cl(A)⊂ (i, j)− βI Cl(X\U). Since x ∈ (i, j)− βI Cl(A), we have
x ∈ (i, j)− βI Cl(X\U). Since X\U is (i, j)− β −I -closed, we have x ∈ X\U; hence x /∈ U ,
which is a contradiction that x ∈ U . Therefore, U ∩A 6= ;. Conversely, suppose that U ∩ A 6= ;
for every U ∈ (i, j) − βIO(X , x). We shall show that x ∈ (i, j) − βI Cl(A). Suppose that
x /∈ (i, j)− βI Cl(A). Then there exists U ∈ (i, j)− βIO(X , x) such that U ∩ A= empt yset.
This is a contradiction to U ∩ A 6= ;; hence x ∈ (i, j)− βI Cl(A).

Theorem 10. Let (X ,τ1,τ2,I ) be an ideal bitopological space and A⊂ X . Then the following
propeties hold:

(i) (i, j)− βI Int(X\A) = X\(i, j)− βI Cl(A);

(ii) (i, j)− βI Cl(X\A) = X\(i, j)− βI Int(A).

Proof. (i). Let x ∈ (i, j)−βI Cl(A). There exists V ∈ (i, j)−βIO(X , x) such that V∩A 6= ;;
hence we obtain x ∈ (i, j)− βI Int(X\A). This shows that
X\(i, j)− βI Cl(A)⊂ (i, j)− βI Int(X\A). Let x ∈ (i, j)− βI Int(X\A). Since
(i, j)− βI Int(X\A)∩ A= ;, we obtain x /∈ (i, j)− βI Cl(A); hence x ∈ X\(i, j)− βI Cl(A).
Therefore, we obtain (i, j)− βI Int(X\A) = X\(i, j)− βI Cl(A).
(ii). Follows from (i).

Proposition 4. The product of two (i, j)− β −I -open sets is (i, j)− β −I -open.

Proof. The proof follows from Lemma 3.3 of [10].



M. Caldas, S. Jafari, N. Rajesh / Eur. J. Pure Appl. Math, 6 (2013), 247-255 253

4. (i, j)− β −I -continuous Functions

Definition 9. A function f : (X ,τ1,τ2,I )→ (Y,σ1,σ2) is said to be (i, j)−β −I -continuous
if the inverse image of every σi-open set of Y is (i, j)−β −I -open in X , where i 6= j, i, j = 1, 2.

Proposition 5. Every (i, j) − b − I -continuous function is (i, j) − β − I -continuous but not
conversely.

Proof. The proof follows from Proposition 1.

The following example shows that the converse of Proposition 5 is not true, in general.

Example 3. Let X = {a, b, c}, τ1 = {;, {a}, X }, τ2 = {;, {a}, {a, b}, X }, σ1 = {;, {a, c}, X },
σ2 = {;, {a}, X } and I = {;, {a}}. Then the identity function f : (X ,τ1,τ2,I )→ (Y,σ1,σ2) is
(1, 2)− β −I -continuous but not (1, 2)− b−I -continuous.

Corollary 2. (i) Every (i, j)−α−I -continuous function is (i, j)−β−I -continuous but not
conversely.

(ii) Every (i, j)-semi-I -continuous function is (i, j)− β-continuous but not conversely.

(iii) Every (i, j)-pre-I -continuous function is (i, j)− β −I -continuous but not conversely.

Theorem 11. For a function f : (X ,τ1,τ2,I ) → (Y,σ1,σ2), the following statements are
equivalent:

(i) f is (i, j)− β −I -continuous.

(ii) For each point x in X and each σi-open set F in Y such that f (x) ∈ F, there exists an
(i, j)− β −I -open set A in X such that x ∈ A, f (A)⊂ F.

(iii) The inverse image of each σi-closed set in Y is (i, j)− β −I -closed in X .

(iv) For each subset A of X , f ((i, j)− βI Cl(A))⊂ σi −Cl( f (A)).

(v) For each subset B of Y , (i, j)− βI Cl( f −1(B))⊂ f −1(σi −Cl(B)).

(vi) For each subset C of Y , f −1(σi − Int(C))⊂ (i, j)− βI Int( f −1(C)).

Proof. (i)⇒(ii): Let x ∈ X and F be a σi-open set of Y containing f (x). By (i), f −1(F) is
(i, j)− β −I -open in X . Let A= f −1(F). Then x ∈ A and f (A)⊂ F .
(ii)⇒(i): Let F be σi-open in Y and let x ∈ f −1(F). Then f (x) ∈ F . By (ii), there is an
(i, j)− β − I -open set Ux in X such that x ∈ Ux and f (Ux) ⊂ F . Then x ∈ Ux ⊂ f −1(F).
Hence f −1(F) is (i, j)− β −I -open in X .
(i)⇔(iii): This follows due to the fact that for any subset B of Y , f −1(Y \B) = X\ f −1(B).
(iii)⇒(iv): Let A be a subset of X . Since A ⊂ f −1( f (A)) we have A ⊂ f −1(σi − Cl( f (A))).
Now, σi − Cl( f (A)) is σi-closed in Y and hence (i, j)− βI Cl(A) ⊂ f −1(σi − Cl( f (A))) for
(i, j)− βI Cl(A) is the smallest (i, j)− β −I -closed set containing A. Then
f ((i, j)− βI Cl(A))⊂ σi −Cl( f (A)).



M. Caldas, S. Jafari, N. Rajesh / Eur. J. Pure Appl. Math, 6 (2013), 247-255 254

(iv)⇒(iii): Let F be any (i, j)− β −I -closed subset of Y . Then
f ((i, j)− βI Cl( f −1(F)))⊂ σi −Cl( f ( f −1(F))) = σi −Cl(F) = F . Therefore,
(i, j)− βI Cl( f −1(F))⊂ f −1(F). Consequently, f −1(F) is (i, j)− β −I -closed in X .
(iv)⇒(v): Let B be any subset of Y . Now,

f ((i, j)− βI Cl( f −1(B)))⊂ σi −Cl( f ( f −1(B)))⊂ σi −Cl(B).

Consequently, (i, j)− βI Cl( f −1(B))⊂ f −1(σi −Cl(B)).
(v)⇒(iv): Let B = f (A), where A is a subset of X . Then,

(i, j)− βI Cl(A)⊂ (i, j)− βI Cl( f −1(B))⊂ f −1(σi −Cl(B)) = f −1(σi −Cl( f (A))).

This shows that f ((i, j)− βI Cl(A))⊂ σi −Cl( f (A)).
(i)⇒(vi): Let B be a σi-open set in Y . Clearly, f −1(σi − Int(B)) is (i, j)−β −I -open and we
have f −1(σi − Int(B))⊂ (i, j)− βI Int( f −1(σi − Int(B)))⊂ (i, j)− βI Int( f −1(B)).
(vi)⇒(i): Let B be a σi-open set in Y . Then σi − Int(B) = B and
f −1(B)\ f −1(σi−Int(B))⊂ (i, j)−βI Int( f −1(B)). Hence we have f −1(B) = (i, j)−βI Int( f −1(B)).
This shows that f −1(B) is (i, j)− β −I -open in X .

If I = {;} in Theorem 11, we get the following

Corollary 3 ([6, Theorem 5.1]). For a function f : (X ,τ1,τ2) → (Y,σ1,σ2), the following
statements are equivalent:

(i) f is pairwise semi-precontinuous;

(ii) For each point x in X and each σi-open set F in Y such that f (x) ∈ F, there is an (i, j)-
semi-preopen set A in X such that x ∈ A, f (A)⊂ F;

(iii) The inverse image of each σi-closed set in Y is (i, j)-semi-preclosed in X ;

(iv) For each subset A of X , f ((i, j)− sp Cl(A))⊂ σi −Cl( f (A));

(v) For each subset B of Y , (i, j)− sp Cl( f −1(B))⊂ f −1(σi −Cl(B)).

Theorem 12. Let f : (X ,τ1,τ2,I )→ (Y,σ1,σ2) be a function. If
g : (X ,τ1,τ2,I )→ (X×Y,σ1×σ2) defined by g(x) = (x , f (x)) is an (i, j)−β−I -continuous
function, then f is (i, j)− β −I -continuous.

Proof. Let V be a σi-open set of Y . Then f −1(V ) = X ∩ f −1(V ) = g−1(X × V ). Since g is
an (i, j)− β −I -continuous function and X × V is a τi ×σi-open set of X × Y , f −1(V ) is an
(i, j)− β −I -open set of X . Hence f is (i, j)− β −I -continuous.

Definition 10. A bitopological space (X ,τ1,τ2) is said to be pairwise connected [8] if it cannot
be expressed as the union of two nonempty disjoint sets U and V such that U is τi-open and V is
τ j-open, where i, j = {1, 2}.



REFERENCES 255

Definition 11. An ideal bitopological space (X ,τ1,τ2,I ) is said to be (i, j)− β −I -connected
if it cannot be expressed as the union of two nonempty disjoint sets U and V such that U is
(i, j)− β −I -open and V is (i, j)− β −I -open.

Theorem 13. Let f : (X ,τ1,τ2,I ) → (Y,σ1,σ2) is (i, j)− β − I -continuous surjection and
(X ,τ1,τ2,I ) is (i, j)− β −I -connected, then (Y,σ1,σ2) is pairwise connected.

Proof. Suppose Y is not pairwise connected, Then Y = A∪ B where A∩ B = ;, A 6= ;,
B 6= ; and A ∈ σi , B ∈ σ j . Since f is (i, j)− β − I -continuous f −1(A) ∈ (i, j)− βIO(X )
and f −1(B) ∈ (i, j)− βIO(X ), such that f −1(A) 6= ;, f −1(B) 6= ;. f −1(A) ∩ f −1(B) = ; and
f −1(A)∪ f −1(B) = X , which implies that X is not (i, j)− β −I -connected.

References

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