Weak Open Sets in Ideal Bitopological Spaces

Weak Open Sets in Ideal Bitopological Spaces

Weak Open Sets in Ideal Bitopological Spaces

Nirmala Rebecca Paul

Department of Mathematics, Lady Doak College, Madurai – 625 002, Tamil Nadu, India,

Abstract

The paper introduces (1,2)-semi-I-open sets, (1,2)-pre-I-open sets,(1,2)–I-open sets and (1,2)–I-open sets in ideal bitopological spaces. The relationship between them are established. Some of their basic properties are discussed. As an application some new types of sets are introduced and the relationship between them is derived.

AMS subject classifcation : 54C08,54A05

Keywords and phrases : (1,2)semi-I-open,(1,2)pre-I-open and(1,2)-I-open sets.

1. Introduction

   kelly[ 8 ] has introduced the concept of bitopological spaces by defning two topologies on a set. Lellis Thivagar et.al.[6] have defned(1,2)semi-open,(1,2)pre-open,(1,2)-open and (1,2)-open sets in bitopological spaces.Ideals play an important role in topology. Jankovic and Hamlet[7] have introduced the notion of I-open sets in topological spaces. Kuratowski[10] has introduced local function of a set with respect to a topology and an ideal and its properties are investigated. Hatir and Noiri [5] introduced -I-open sets, semi-I-open sets and -I-open sets and derived a decomposition of continuity.In this paper a new closure operator using 12-open set is defned and some of its basic properties are discussed.(1,2)-semi-I-open sets, (1,2)-pre-I-open sets,(1,2)–I-open sets and (1,2)–I-open sets are defned in ideal bitopological spaces. The relationship between them and other existing sets are derived. Some properties of the sets are discussed.

2. Preliminaries

   We list some defnitions which are useful in the following sections. The interior and the closure of a subset A of (X, ) are denoted by Int(A) and Cl(A), respectively. Throughout the present paper (X, ) and (Y, )(or X and Y ) represent non-empty topological spaces on which no separation axiom is defned, unless otherwise mentioned.

   Defnition 2.1 A subset A of a space X is called

   1. a semi-open set [3] if A Cl(Int(A))

   2. a pre-open set [4] if A Int(Cl(A))

   3. an -open set [11] if A Int(Cl(Int(A)))

   4. a -open set[1] if A Cl(Int(Cl(A)))

   5. a -set[5] if Int(A) = Int(Cl(A))

   6. a C-set[5] if A = U V where U is open and V is an -set

   7. a t-set[12] if Int(A) = Int(Cl(A))

   8. a B-set[12] if A = U T where U is an open set and T is a t-set

      The complement of a semi-open (resp. pre open, – open and ) set is called a semi-closed (resp. pre closed, -closed and -closed) set.

      Defnition 2.2 (5) An ideal I on a topological space (X, ) is a non empty collection of subsets of X which satisfes the following conditions. i)A I and B A implies B I. ii)A I and B I implies A B I.

      An ideal topological space (X, ) with an ideal I on X is denoted by (X, , I).

      Defnition 2.3 (5) Let (X, , I) be an ideal topological space and A X. A(I, ) = {x

      X : A U / I forevery U (X, x)} is called the local function of A with respect to I

      and . For every ideal topological space (X, , I) there exists a topology (I) fner than

      defned as (I) = {U X : Cl(X U ) = X U }generated by the base (I, J ) = {U J :

      U and J I} and Cl(A) = A A.

      Defnition 2.4 (5) A subset A of an ideal topological space (X, , I) is called

      1. semi-I-open if A Cl(Int(A))

      2. pre-I-open if A Int(Cl(A))

      3. -I-open if A Int(Cl(Int(A)))

      4. -I-open if A Cl(Int(Cl(A)))

      Defnition 2.5 A subset A of a bitopological space (X, 1, 2) is called

      1. 12-open[6] if A 1 2

      2. 12-closed[6] if Ac 1 2.

      Defnition 2.6 (6) Let A be a subset of (X, 1, 2). Then 12-Cl(A) denotes the 12- closure of A and is defned as the intersection of all 12-closed sets containing A. Also 12-Int(A) denotes the 12-interior of A and is defned as the union of all 12-open sets contained in A.

      Defnition 2.7 A subset A of (X, 1, 2) is said to be (i)(1, 2)-open[6] if A 1-Int(12-Cl(1-Int(A))).

      (ii)(1,2)semi-open[6] if A 12-Cl(1-Int(A))

      (iii)(1,2)pre-open[6] if A 1-Int(12-Cl(A)) and

      (iv)(1,2)semi-pre-open[6](briefy (1,2)sp-open) if

      A 12-Cl(1-Int(12-Cl(A)))

3. New closure operator

   We defne a new closure operator in terms of 12-open sets.

   1 2

   1 2

   { }

   { }

   Defnition 3.1 Let (X, 1, 2) be a bitopological space. A bitopological space together with an ideal is defned to be an ideal bitopological space and is denoted as (X, 1, 2, I). Let A X the local function with respect to the 12-open sets is defned as A(X, I, 1, 2) = x X : A U / I forevery U 12O(X, x) is called the local function of A with respect to I and the two topologies 1 and 2.In short it will be denoted as A . 12O(X, x) denotes

   the collection of all 12-open sets containing the point x.

   1 2

   1 2

   Proposition 3.2 Let (X, 1, 2, I) be an ideal bitopological space and A be a subset of X, then A A(I, 1) for every subset A of X.

   Proof.Let x / A(I, 1). Then there is a 1-open set U containing x such that A U I.

   Since every 1-open set is 12-open then x / A(I, 12).

   Remark 3.3 The converse of the proposition 3.2 need not be true.

   Example 3.4 Let X = {a, b, c, d}, 1 = {, {a}, X}, 2 = {, X{b}}, I = {, {c}}

   12O(X) = {, X, {a}, {b}}.A = {{c}}, A = X, A12 = {a, c, d}

   Remark 3.5 Neither A12 A nor A A12 .

   Example 3.6 Let X = {a, b, c, d}, 1 = {, {a}, {b}, {a, b}, X}, 2 = {, X, {c}}, I = {, {b}}. If A = {a, b, c}, A12 = {a, c, d}, if A = {b, d}, A12 = {d}

   Theorem 3.7 For subsets A and B of an ideal bitopological space (X, 1, 2) the following statements are true.

   (i)If A B A12 B12 .

   (ii) (A B)12 = A12 B12 . (iii)(A B)12 A12 B12 .

   (iv) A12 B12 (A B)12 .

   (v) If A I A12 = .

   (vi) B I (A B)12 = A12 = (A B)12 .

   (vii) If (A B), (B A) I then A12 = B12 .

   (viii) A12 (A12 )12 (A A12 )12 .

   (ix) (A12 )12 A12

   (x)A12 = 12-Cl(A12 ).

   (xi) If U 1 then U A12 = U (U A)12 (U A)12 . (xii)If I1 Ithen (A I1)12 A12 = (A I1)12 .

   Proof.

   (i)Let A B and x / B12 .There exists U 12O(X, x) such that U B I. Since

   A B, U A I and hence x / A12 .Hence A12 B12 .

   (ii) A A B, B A B, A12 (A B)12 , B12 (A B)12 . Thus A12 B12 (A B)12 . Claim (A B)12 A12 B12 . Let x (A B)12 . Then for every 12-open set U of x such that (U A)(U B) = U (A B) / I. Therefore U A / I or U B / I. This implies that x A12 or x B12 . That is x A12 B12 . Therefore we have (A B)12 A12 B12 .Thus we get (A B)12 = A12 B12 .

   (iii)A B A, A B B,by (i)(A B)12 A12 B12 .

   1. For every subset A and B of X, A = (A B) (A B). By(ii) A12 = (A B)12 (A B)12 and hence A12 B12 = A12 (X B12 ) = [(A B)12 (B A)12 (X B12 )] = [(A B)12 (X B12 )] [(A B)12 (X B12 )] (A B)12 B12 +] (A B)12 .

   2. By defnition if A I then A12 = .

   3. Let B I and by (ii),(iv) and (v) (A B)12 = A12 B12 = A12 = (A B)12 .

   4. Let E = (A B) (B A) I then A = (A B) (A B), B = (A B) (B A),By

      1 2 1 2 12

      1 2 1 2 12

      (ii) and (vi)A = (A B) = B .

      1 2 1 2 1 2 1 2 1 2

      1 2 1 2 1 2 1 2 1 2

   5. By(iv) A (A ) (A A ) .

   6. Let x (A12 )12 . Then for every U 12O(X, x), U A12 / I and hence

      U A12 6= . Let y U A12 . Then U 12O(X, y) and y A12 . Hence

      U A / I and x A12 . This implies that (A12 )12 A12 .

      1 2 1 2 1 2 1 2

      1 2 1 2 1 2 1 2

   7. In general A 12-Cl(A ).Let x 12 Cl(A )). Then A U 6= for every U 12O(X, x). Therefore there exist some y A12 U and U 12O(X, x).

      Since y A12 , A U / I and hence x A12 . Hence 12-Cl(A12 ) (A12 ).

      1 2

      1 2

      1 2

      1 2

      Hence A = 12-Cl(A ).

   8. If U 1O(X) and x U A12 . Then x U and x A12 . Let V be any 12-open set containing x. Then V U 12O(X, x) and V (U A) = (V U ) A / I. This shows that x (U A)12 and hence U A12 (U A)12 . Hence U A12 U (U A)12 . By(i) (U A)12 A12 and U A12 U (U A)A12 . Therefore U A12 = U (U A)12 (U A)12 .

   9. Since A I1 A.By (i) (A I1)12 A12 .Also (A I1)12 = A12 (I1)12 =

   A

   A

   1 2 = A1 2 . Hence (A I1)1 2 A1 2 = (A I1)1 2 . Since (I1)1 2 = .

   1 2

   1 2

   Defnition 3.8 For a subset A of an ideal topological space (X, , I),we defne Cl (A) =

   A A12 .

   1 2

   1 2

   Theorem 3.9 Cl satisfes Kuratowskis closure axioms.

   Proof.

   12 12

   12 12

   1. Cl () = , A Cl (A)A X

      12 1 2 12 12 12 12

      12 1 2 12 12 12 12

   2. Cl (A B) = A B (A B) = A B A B = Cl (A) Cl (B).

   3. For any A X, Cl12 (Cl12 (A)) = Cl12 (A A12 ) = A A12 (A A12 )12 =

   A A12 (A12 )12 = A A12 = Cl12 (A)( since (A12 )12 A12 ).

   1 2

   1 2

   1 2

   1 2

   Defnition 3.10 The topology generated by Cl is denoted by (I) and is defned as

   .

   .

   12 (I) = {U X : Cl1 2 (X U ) = X U }. Without ambiguity it will be denoted as

   1 2

   (i) X, Cl12 (X ) = Cl12 (X) = X and Cl12 (X X) = Cl12 () = . Hence

   , X 12 (I).

   1. Let {Ui}iI 12 (I) then Cl12 (X Ui) = X Uii. i.e (X Ui)(X Ui)12 = X

      12

      12

      Uii. Theref ore (X Ui) 1 2 X Uii. Claim Cl 1 2 ( X i Ui) = Cl ( i(X

      Ui)) = X i Ui = i(X Ui). By defniti on Cl12 ( i(X Ui)) = i(X Ui)

      ( i(X Ui))12 Cl12 ( i(X Ui)) i(X Ui).Also by hy pothesis i(X

      Cl (X

      12

      i Ui) = X

      Ui.

      Cl (X

      12

      i Ui) = X

      Ui.

      Ui) ( i(X Ui))12 i(X Ui).Hence Cl12 ( (X Ui)) i(X Ui).Thus

      i Ui) = X

      Ui.

      i Ui) = X

      Ui.

      i Ui) = X

      Ui.

      i Ui) = X

      Ui.

      Cl (

      12

      Cl (

      12

      i(X Ui)) = Cl12 (X U1) Cl12 (X U2) = (X U1) (X U2). Hence

      i(X Ui)) = Cl12 (X U1) Cl12 (X U2) = (X U1) (X U2). Hence

   2. Let U1, U2 12 (I) then Cl12 (X Ui) = X Ui, i = 1, 2. Cl12 (X (U1 U2)) =

   U1 U2 12 (I). Hence 12 (I) is a topology.

   U1 U2 12 (I). Hence 12 (I) is a topology.

   U1 U2 12 (I). Hence 12 (I) is a topology.

   U1 U2 12 (I). Hence 12 (I) is a topology.

   1 2

   1 2

   Proposition 3.11 Every is fner than .

   Proof.Since every 1-open set is 12-open. Therefore 12 .

   Example 3.12 Let X = {a, b, c}, 1 = {, {a}, {b}, {a, b}, X}, 2 = {, X, {b}}, I = {, {c}}.

   = {, {a}, {a, b, d}, X}.(with respect to 1)

   12 = {, {a}, {b}, {a, b}, {a, b, d}, X}. Hence 12 .

   Remark 3.13 Using the new closure operator some open sets are defned in an ideal bitopological space here Cl(A) represent the closure with respect to the topology 1.

   Defnition 3.14 A subset A of an ideal topological space (X, , I) is said to be

   1. (1,2)semi-I-open if A Cl12 (1-Int(A))

   2. (1,2)pre-I-open if A 1-Int(Cl12 (A))

   3. (1,2)-I-open ifA 1-Int(Cl12 (1-Int(A)))

   (iv)(1,2)-I-open if A 1-Cl(1-Int(Cl12 (A)))

   Remark 3.15 (2,1)semi-I-open,(2,1)pre-I-open,(2,1)-I-open and -I-open sets are defned by replacing 1 by 2.

   Proposition 3.16 (i)Every (1,2)semi-I-open set is semi-I-open and hence semi-open. (ii)Every (1,2)pre-I-open set is pre-I-open and hence pre-open.

   (iii) Every (1,2)-I-open set is -I-open and hence -open. (iv)(1,2)-I-open set is -I-open and hence -open.

   Proof.

   12

   12

   (i) Let A be (1,2)semi-I-open. i.e A Cl12 1-Int(A) Cl(1-Int(A)) 1-Cl(1- Int(A)). Since Cl (A) Cl(A) 1-Cl(A).

   (ii)A 1-Int(Cl12 (A)) 1-Int(Cl(A)) 1-Int(1-Cl(A)).

   (iii) A 1-Int(Cl12 (1-Int(A))) 1-Int(Cl(1-Int(A))) 1-Int(1-Cl(1-Int(A))). (iv)A 1-Cl(1-Int(Cl12 (A))) 1-Cl(1-Int(Cl(A))) 1-Cl(1-Int(1-Cl(A)))

   Remark 3.17 The converse of the proposition 3.16 is not true.

   Remark 3.18 The collection of all semi-I-open, pre-I-open, -I-open and -I-open sets are denoted as SIO(X), PIO(X), IO(X) and IO(X) respectively. The collection of all (1,2)semi-I-open,(1,2)pre-I-open,(1,2)-I-open and (1,2)-I-open sets are denoted as (1,2)SIO(X),(1,2)PIO(X),(1,2)IO(X) and (1,2)IO(X) respectively.

   Example 3.19 Let X = {a, b, c, d}, 1 = {, X, {a}, {a, b, c}}, 2 = {, X, {b, c}}, I = {, {b}}.SIO(X) = {, X, {a}, {a, b}, {a, c}, {a, d}, {a, b, c}, {a, b, d}, {a, c, d}} (1,2)SIO(X)={, X, {a}, {a, d}, {a, b, c}}.

   O(X) = {, X, {a}, {a, b}, {a, c}, {a, d}{a, b, c}, {a, b, d}, {a, c, d}}.

   IO(X) = {, X, {a}, {a, b}, {a, c}, {a, d}{a, b, c}, {a, b, d}}.

   (1,2)IO(X) = {, X, {a}, {b}, {a, b}}.

   PO(X) = {, X, {a}, {a, b}, {a, c}, {a, d}, {a, b, c}, {a, b, d}, {a, c, d}}.

   PIO(X) = {, X, {a}, {a, b}, {a, c}, {a, d}, {a, b, c}, {a, b, d}}.

   (1, 2)PIO(X) = {, X, {a}, {b}, {a, c}, {a, b, c}, {a, c, d}}.

   O(X) = {, X, {a}, {a, b}, {a, c}, {a, d}, {a, b, c}, {a, b, d}, {a, c, d}}.

   IO(X) = {, X, {a}, {a, b}, {a, c}, {a, d}, {a, b, c}, {a, b, d}, {a, c, d}}.

   (1, 2)IO(X) = {, X, {a}, {a, c}, {a, b, c}, {a, b, d}, {a, c, d}}.

   Proposition 3.20 Every 1-open set is

   1. (1,2)semi-I-open.

   2. (1,2)pre-I-open. (iii)(1,2)-I-open.

   (iv) (1,2)-I-open.

   Proof.Let A be an 1-open set.

   (i)A = 1-Int(A) 1-Int(Cl12 (A))

   (ii)A = 1-Int(A) Cl12 (1-Int(A))

   1. A = 1-Int(A) 1-Int(Cl12 (A))

   2. A = 1-Int(A) 1-Cl(1-Int(A)) 1-Cl(1-Int(Cl12 (A)))

   Remark 3.21 The converse of the proposition 3.20 is not true. It follows from the example 3.19

   { } { { }} { { }} { { }}

   { } { { }} { { }} { { }}

   Example 3.22 Let X = a, b, c, d , 1 = , X, a ,2 = , X, a, b , I = , c

   (1,2)SIO(X)=(1,2)IO(X) = (1, 2)PIO(X) = (1, 2)O(X) =

   <>{, X, {a}, {a, c}, {a, b}, {a, d}, {a, b, c}, {a, c, d}, {a, b, d}}.

   Proposition 3.23 A subset A of an ideal bitopological space (X, 1, 2, I) is (1, 2)-I-open if and only if A is (1,2)semi-I-open and (1,2)pre-I-open.

   Proof.Let A be (1, 2)-I-open. A 1-Int(Cl12 (1-Int(A))) Cl12 (1-Int(A)). Thus A is (1,2)semi-I-open.A 1-Int(Cl12 1-(Int(A))) 1-Int(Cl12 (A)). Thus A is (1,2)pre- I-open. Let A be (1,2)semi-I-open and (1,2)pre-I-open.i.e.A Cl12 (1-Int(A)) and A 1-Int(Cl12 (A)). A 1-Int(Cl12 (A) 1-Int(Cl12 (1-Int(A)). Hence A is (1,2)-I-

   open.

   Proposition 3.24 For any ideal topological space (X, , I), (1, 2)SIO(X) (1, 2)PIO(X) (1, 2)IO(X)

   Proof.Let A be (1,2)semi-I-open then A Cl12 (1-Int(A)) 1-Cl(1-Int(A)

   1 2

   1 2

   1-Cl(1-Int(Cl (A)). Hence A is (1, 2)-I-open. Or if A be (1,2)pre-I-open. Then

   A 1-Int(Cl12 ((A)) 1-Cl(1-Int(Cl12 (A)).Hence A is (1,2)-I-open.

   Remark 3.25 The converse of the proposition 3.24 is not true.

   Example 3.26 Let X = {a, b, c, d}, 1 = {, X, {a}, {b}, {a, b}}, 2 = {, X, {c}}.

   I = {, {b}}, (1, 2)SIO(X) = {, X, {a}, {b}, {a, b}}, (1, 2)PIO(X) = {, X, {a}, {b},

   {a, b}, {a, b, c}, {a, b, d}}, (1, 2)O(X) = {, X{a}, {b}, {a, b}, {a, c}, {a, d}, {b, c},

   {b, d}, {a, b, c}, {a, b, d}, {a, c, d}, {b, c, d}}

   { }

   { }

   Proposition 3.27 Let (X, 1, 2, I) be an ideal topological space and A : J be a family of subsets of X where J is an arbitrary index set.

   1. If {A : J } (1, 2)SIO(X) then {A : J } (1, 2)SIO(X).

   2. If A (1, 2)SIO(X) and B 1 then A U (1, 2)SIO(X)

   3. If {A : J } (1, 2)PIO(X) then {A : J } (1, 2)PIO(X).

   4. If A (1, 2)PIO(X) and B 1 then A U (1, 2)PIO(X)

   (v) If {A : J } (1, 2)IO(X) then {A : J } (1, 2)IO(X). (vi)If A (1, 2)IO(X) and B 1 then A U (1, 2)IO(X).

   (vii)If {A : J } (1, 2)IO(X) then {A : J } (1, 2)IO(X). (viii)If A (1, 2)IO(X) and B 1 then A U (1, 2)IO(X).

   Proof.

   1. Since U (1 , 2)SIO(X) for each J .U Cl12 (1-Int(U). U J Cl 12 (1-

      (1-Int

      (1-Int

      J U)12 (1-Int(

      J U)12 (1-Int(

      J U)) = Cl12 (1-Int(

      J U)) = Cl12 (1-Int(

      J U). Thus

      J U). Thus

      J U

      J U

      Int(U)) J (1-Int(U)) 12 (1-Int(U) J (1 -Int(U))12 1 -Int( J U)

      (1, 2)SIO(X).

   2. Let A (1, 2)SIO(X), B 1. A Cl12 1-Int(A), B = 1-Int(B). AB Cl12 1- Int(A) 1-Int(B) (1-Int(A))12 1-Int(A) 1-Int(B) (1-Int(A))12 1- Int(B) 1-Int(A) 1-Int(B) (1-Int(A) 1-Int(B))12 1-Int(A B) (1- Int(A B))12 1-Int(A B) = Cl12 1-Int(A B)). Hence A B (1, 2)SIO(X).

      IntCl (

      12

      IntCl (

      12

      U 1-Int(Cl12 (

      U 1-Int(Cl12 (

   3. Since U (1, 2)P IO(X) for each J .U 1-Int(Cl12 (U). U 1-

      U).

      J U)). Thus

      J U (1, 2)PIO(X).

      U).

      J U)). Thus

      J U (1, 2)PIO(X).

      U).

      J U)). Thus

      J U (1, 2)PIO(X).

      U).

      J U)). Thus

      J U (1, 2)PIO(X).

      12 12 1 2

      12 12 1 2

   4. Let A (1, 2)PIO(X), B 1. A 1-Int(Cl12 (A)), B = 1-Int(B). A B 1- Int(Cl (A)) 1-Int(B) = 1-Int[(Cl (A) B)] 1-Int[(A A ) B)] 1- Int[(A B) (A B)12 ] 1-IntCl12 (A B)). HenceA B (1, 2)PIO(X).

      12

      12

   5. Since U (1, 2 )IO(X ) for each J .U 1-Int(Cl12 1-Int(U)) 1-

      Thus

      J U (1, 2)IO(X).

      Thus

      J U (1, 2)IO(X).

      Int(Cl 1-Int( U)). U 1-Int(Cl12 1-Int( U)).

      Thus

      J U (1, 2)IO(X).

      Thus

      J U (1, 2)IO(X).

      Thus

      J U (1, 2)IO(X).

      Thus

      J U (1, 2)IO(X).

   6. Let A be (1, 2)I-open. Then A is (1,2)semi-I-open and (1,2)pre-I-open. By (ii) and

      (iv) A B (1, 2)SIO(X) and A B (1, 2)PIO(X). Hence A B (1, 2)IO(X).

   7. Since U (1, 2)IO(X) for each J .U 1-Cl(1-Int(Cl12 (U). U

      1-Cl(1-Int(Cl

      12

      1-Cl(1-Int(Cl

      12

      J (U)). Thus

      J (U)). Thus

      J U (1, 2)IO(X).

      J U (1, 2)IO(X).

      J 1-Cl(1-Int(C l12 (U))) 1-C l(1-Int( J Cl12 ((U))

   8. Let A (1, 2)IO(X), B 1. A 1-Cl(1-Int(Cl12 (A)), B = 1-Int(B).

   A B 1-Cl(1-Int(Cl12 (A))) 1-Int(B) 1-Cl(1-Int(A A12 ) B)) =

   1-Cl(1-Int((A12 B) (A B)) 1-Cl(1-Int((A B)12 (A B)) =

   1-Cl(1-Int(Cl12 (A B))). HenceA B (1, 2)IO(X).

   Defnition 3.28 The largest (1,2)semi-I-open, (1,2)pre-I-open, (1,2)-I-open and (1,2)- I-open sets contained in A are defned as (1,2)semi-I-int(A),(1,2) pre-I-int(A),(1,2)I-

   int(A) and (1,2)I-int(A) respectively. They are denoted as (1,2)sI-Int(A),(1,2)pI-Int(A),(1,2)I- Int(A),I-Int(A) respectively.

   Proposition 3.29 For any subset A of an ideal topological space the following holds. (i)Cl12 (A) 1-Int(Cl12 (A) Cl12 [A 1-Int(Cl12 (A))]

   (ii) Cl12 1-(Int(A)) 1-Int(Cl12 (1-Int(A))

   Cl12 [1-Int(A) 1-Int(Cl12 (1-Int(A))].

   Proof.(i) Let x Cl12 (A)1-Int(Cl12 (A) then x Cl12 (A) and x 1-Int(Cl12 (A). x Cl12 (A) x A or x A12 . If x A then x A 1-Int(Cl12 (A)) x Cl12 (A 1-Int(Cl12 (A)). Hence Cl12 (A) 1-Int(Cl12 (A) Cl12 (A 1- Int(Cl12 (A)). On the other hand if x A12 then x A12 1-Int(Cl12 (A) (1- Int(Cl12 (A) A)12 Cl12 (A 1-Int(Cl12 (A)). Thus Cl12 (A) 1-Int(Cl12 (A) Cl12 (A 1-Int(Cl12 (A)).

   Int(A)))]

   Int(A)))]

   1. Let x Cl12 (1-Int(A)) and x 1-Int(Cl12 (1-Int(A)). If x Cl12 (1-Int(A)) x 1-Int(A) (1-Int(A))12 . If x 1-Int(A) then x 1-Int(A) 1-Int(Cl12 (1- Int(A)) Cl12 (1-Int(A) 1-Int(Cl12 (1-Int(A)). On the other hand if x (1- Int(A))12 then x Int(Cl12 (1-Int(A))(1-Int(A))12 [1-Int(A)1-Int(Cl12 (1-

      1 2

      Cl12 [1-Int(A)1-Int(Cl12 (1-Int(A)))]. Hence Cl12 (1-Int(A))1-Int(Cl12 (1-

      Int(A)) Cl12 (1-Int(A) 1-Int(Cl12 (1-Int(A)).

      Lemma 3.30 Let A be a subset of a space (X, 1, 2, I). If O is an 1-open subset of X then

      O Cl12 (A) Cl12 (O A).

      Proof.By Theorem 3.7 O A12 (O A)12 . O Cl12 (A) = O (A A12 ) = (O A) (O A12 ) (O A) (O A)12 Cl12 (O A).

      Proposition 3.31 If A is any subset of an ideal topological space (X, 1, 2, I) then the following holds.

      (i) (1, 2)sI-Int(A) = A (Cl12 (1-Int((A)))

      1. (1, 2)pI-Int(A) = A 1-Int(Cl12 (A)).

      2. (1, 2)I-Int(A) = A 1-Int(Cl12 (1-Int(A))

      3. (1, 2)I-Int(A) = A 1-Cl(1-Int(Cl12 (A))

   Proof.

   1. (1, 2)sI-Int(A) A. If S be any (1,2)semi-I-open set contained in A then S Cl12 (1- Int(S)) Cl12 (1-Int(A)). Therefore it is true for all (1,2)semi-I-open set S. Therefore (1, 2)sI-Int(A) Cl12 (1-Int(A)),

      12 12 12 12

      12 12 12 12

      (1, 2)sI-Int(A) ACl12 (1-Int(A))(1). claim: ACl12 (1-Int(A)) is (1,2)semi- I-open. Cl (1-Int(ACl (1-In(A)) Cl (1-Int(A1-Int(A))) Cl (1- Int(1-Int(A))) Cl12 (1-Int(A)) A Cl12 (1-Int(A)). Thus A Cl12 (1- Int(A)) is (1,2)semi-I-open set contained in A. A Cl12 (1-Int(A)) (1, 2)sI- Int(A).Hence (1,2)sI-Int(A)=A (Cl12 (1-Int(A)).

      12 12

      12 12

      (ii)(1, 2)pI-Int(A) A. If S be any (1,2)pre-I-open set contained in A then S 1- Int(Cl (S)) 1-Int(Cl (A). This is true for all (1,2)pre-I-open set S. Therefore (1, 2)pI-Int(A) A 1-Int(Cl12 (A). claim:A 1-Int(Cl12 (A) is (1,2)preI- open.A 1-Int(Cl12 (A)) 1-Int(1-Int(Cl12 (A))

      Int(Cl (A)).

      Int(Cl (A)).

      12 12 12 12

      12 12 12 12

      = 1-Int(Cl (A)1-Int(Cl (A) 1-Int(Cl (A1-Int(Cl (A)). Therefore A 1-Int(Cl12 (A) is (1,2)pre-I-open. A 1-Int(Cl12 (A) is a (1,2)pre-I-open set contained in A. A 1-Int(Cl12 (A) (1, 2)pI-Int(A). Thus (1, 2)pI-Int(A) = A 1-

      1 2

      1. If S is any (1, 2)-I-open set contained in A then S 1-Int(Cl12 (1-Int(S))

         12 12

         12 12

         1-Int(Cl (1-Int(A)). Therefore (1, 2)I-Int(A) 1-Int(Cl (1-Int(A))A.

         (1). A 1-Int(Cl12 (1-Int(A)) 1-Int(Cl12 (1-Int(A)) =

         12 12 12

         12 12 12

         1-Int(1-Int(Cl (1-Int(A)))) = 1-Int(Cl (1Int(A))1-Int(Cl (1-Int(A))

         1-Int(Cl12 (1-Int(A) 1-Int(Cl12 (1-Int(A)))) =

         12 12

         12 12

         1-Int(Cl (1-Int(A 1-Int(Cl 1-Int(A))).

         Therefore A 1-Int(Cl12 (1-Int(A)) is (1,2)-I-open set contained in A.

         A 1-Int(Cl12 (1-Int(A)) (1, 2)I-Int(A).-(2). Therefore from (1) and (2)

         A 1-Int(Cl12 (1-Int(A)) = (1, 2)I-Int(A).

      2. If S is a (1, 2)I-open set contained in A then S -Cl(1-Int(Cl12 (S)))

      12

      12

      1-Cl(1-Int(Cl (A))). Also (1, 2)I-Int(A) A.Therefore (1, 2)I-Int(A)

      A 1-Cl(1-Int(Cl12 (A))). (1) Also A 1-Cl(1-Int(Cl12 (A)

      1-Cl(1-Int(Cl (A))) = 1-Cl(1-Int(1-Int(Cl (A)))) =

      1 2 1 2

      12 12

      12 12

      1-Cl(1-Int[1-Int(Cl (A) Cl (A)]

      12 12 12

      12 12 12

      1-Cl(1-Int(Cl [A1-Cl(1-Int(Cl (A))])). Hence A1-Cl(1-Int(Cl A)))

      is a (1, 2)I-open set contained in A. Therefore A 1-Cl(1-Int(Cl12 (A)))

      (1, 2)I-Int(A). Thus (1, 2)I-Int(A) = A 1-Cl(1-Int(Cl12 (A))

      1 2

      1 2

      Theorem 3.32 A subset A of a space (X, 1, 2, I) is (1,2)semi-I-open if and only if Cl (A) =

      1 2

      1 2

      Cl (1-Int(A)).

      Proof.Let A be (1,2)semi-I-open then A Cl12 (1-Int(A)) Cl12 (A)

      Cl (1-Int(A)).

      Cl (1-Int(A)).

      Cl12 (1-Int(A)), 1-Int(A) A Cl12 (1-Int(A)) Cl12 (A). Therefore Cl12 (A) =

      1 2

      Conversely Cl12 (A) Cl12 (1-Int(A)) A Cl12 (1-Int(A)). Hence A is (1,2)semi- I-open.

      Proposition 3.33 A subset A of a space (X, 1, 2, I) is (1,2)semi-I-open if and only if there exists an 1-open set U such that U A Cl12 (U ).

      Proof.Let A be (1,2)semi-I-open. Then A Cl12 (1-Int(A)). Let 1-Int(A) = U . Therefore U A Cl12 (U ). Conversely let U A Cl12 (U ) for some 1-open set U. Since U A U 1-Int(A) and hence Cl12 (U ) Cl12 (1-Int(A)). Therefore A Cl12 (1-Int(A)).

      1 2

      1 2

      Proposition 3.34 If a subset A of a space (X, 1, 2, , I) is (1,2)semi-I-open and A B Cl (A) then B is (1,2)semi-I-open in (X, 1, 2, I).

      12 12 12

      12 12 12

      Proof.Since A is (1,2)semi-I-open there exists an 1-open set U such that U A Cl (U ). Therefore U A B Cl (A) Cl (U ).Hence by the Proposition

      3.33 B is (1,2)semi-I-open.

      1 2

      1 2

      Proposition 3.35 A subset A of an ideal topological space (X, 1, 2, I) is (1, 2)-I-open if and only if 1-Cl(A) = 1-Cl(1-Int(Cl (A)).

      12 12

      12 12

      12 12

      12 12

      Proof.Let A be (1, 2)-I-open. Then A 1-Cl(1-Int(Cl12 (A)). 1-Cl(A) 1-Cl(1- Int(Cl (A))) 1-Cl(A). Hence 1-Cl(A) = 1-Cl(1-Int(Cl (A). Conversely 1- Cl(A) = 1-Cl(1-Int(Cl (A)). Then A 1-Cl(A) = 1-Cl(1-Int(Cl (A)).

      Proposition 3.36 (i)If V (1, 2)SIO(X) and A (1, 2)IO(X) then

      V A (1, 2)SIO(X).

   2. If V (1, 2)PIO(X) and A (1, 2)IO(X) then

   V A (1, 2)PIO(X).

   Proof.(i)Let V (1, 2)SIO(X) and A (1, 2)IO(X). V A

   12 12 12 12

   12 12 12 12

   Cl (1-Int(V ) 1-Int(Cl (1-Int(A)) Cl (1-Int(V ) Cl (1-Int(A))

   12 12 12 12

   12 12 12 12

   Cl (Cl (1-Int(V )1-Int(A)) Cl (1-Int(V )1-Int(A)) Cl 1-Int(V A). Hence VA (1, 2)SIO(X).

   (ii) V (1, 2)PIO(X) and A (1, 2)IO(X). V A 1-Int(Cl12 (V ))

   12 12 12

   12 12 12

   1-Int(Cl (1-Int(A)) = 1-Int(1-Int(Cl (V )) Cl (1-Int(A))

   12 12 12 12

   12 12 12 12

   1-Int(Cl (1-Int(Cl (V ) 1-Int(A)) 1-Int(Cl (Cl (V ) 1-Int(A)))

   1-Int[(Cl12 (Cl12 (V Int(A))] 1-Int(Cl12 (V A). Hence V A (1, 2)PIO(X).

4. Applications

As an application new types of sets are defned and their some of their properties are derived.

Defnition 4.1 A subset A of an ideal bitopological space X, 1, 2, I is called a

1 2

1 2

1. (1,2)t-I-set if 1-Int(Cl (A) = 1-Int(A).

   1 2

   1 2

2. (1,2)-I-set if 1-Int(Cl (1-Int(A)) = 1-Int(A).

   1 2

   1 2

3. (1,2)S-I-set if Cl (1-Int(A) = 1-Int(A).

Defnition 4.2 A subset A of an ideal bitopological space (X, 1, 2, I) is called a

1. (1,2)BI-set if A = U V where U is 1-open and V is a (1,2)t-I-set.

2. (1,2)CI-set if A = U V , where U is is an 1-open and V is a (1,2)-I-set.

3. (1,2)SI-set if A = U V where U is 1-open and V is a (1,2)S-I-set.

Example 4.3 Let X = {a, b, c, d}, 1 = {, X, {a}, {a, b, c}}, 2 = {, X, {b, c}}, I = {, {a}}

12O(X) = {, X, {a}, {b, c}, {a, b, c}}. The (1,2)t-I- sets are ,X,{a}, {b}, {c}, {d}, {a, d}, {b, c}, {b, d}, {c, d}, {

(1,2)-I-sets are ,X,{a}, {b}, {c}, {d}, {a, b}, {a, c}, {a, d}, {b, c}, {b, d}, {c, d}, {a, b, d}, {a, c, d}, {b, c, d}}

(1,2)S-I-sets are ,X,{a}, {b}, {c}, {d}, {a, b}, {a, c}, {a, d}, {b, c}, {b, d}, {c, d}, {a, b, d}, {a, c, d}, {b, c, d}}

(1,2)BI- sets are ,X,{a}, {b}, {c}, {d}, {a, d}, {b, c}, {b, d}, {c, d}, {a, b, c}, {b, c, d}}

(1,2)CI-sets are ,X,{a}, {b}, {c}, {d}, {a, b}, {a, c}, {a, d}, {b, c}, {b, d}, {c, d}, {a, b, c}, {a, b, d}, {a, c, d}, {b, c, d}

(1,2)SI-sets are ,X,{a}, {b}, {c}, {d}, {a, b}, {a, c}, {a, d}, {b, c}, {b, d}, {c, d}, {a, b, c}, {a, b, d}, {a, c, d}, {b, c, d}

Proposition 4.4 For a subset A of an ideal bitopological space (X, 1, 2, I) the following hold.

1. If A is a t set with respect to 1, then A is a (1,2)t-I-set.

2. If A is a (1,2)t-I-set then A is a (1,2)BI-set.

1 2 1 2

1 2 1 2

Proof.(i) Let A be a t-set. 1-Int(A)=1-Int(1-Cl(A)). 1-Int(Cl (A) = 1-Int(A A)

1-Int(1-Cl(A)A)) = 1-Int(1-Cl(A)) = 1-Int(A)(1).Cl12 (A) A and

12

12

1-Int(Cl (A)) 1-Int(A).

(ii)Let A be a (1,2)t-I-set . If U = X 1 then A = U A then A is a (1,2)BI-set.

Remark 4.5 The converse of the proposition 4.4 is not true.

Example 4.6 (i)In Example 4.3 the set {a} is a (1,2)t-I-set but not a t-set. (ii)The set {a,b,c,} is a (1,2)BI-set but not a (1,2)t-I-set./p>

Proposition 4.7 Let (X, 1, 2, I) be an ideal bitopological space and A a subset of X. Then the following hold.

1. If A is an -set with respect to 1 then A is an (1, 2)-I-set.

2. If A is a (1,2)t-I-set then A is a (1,2)-I-set.

3. If A is a (1,2)-set then A is a (1,2)CI-set.

1 2

1 2

Proof.(i)Let A be an -set with respect to 1. Then 1-Int(Cl (1-Int(A)) =

1 2

1 2

1-Int(1-Int(A)) 1-Int(A)) 1-Int(1-Cl(1-Int(A) 1-Int(A)) =

12

12

1-Int(1-Cl( 1-Int(A)) = 1-Int(A)-(1). Also Cl (1-Int(A)) 1-Int(A).

12 12

12 12

12 12

12 12

Hence 1-Int(Cl (1-Int(A)) 1-Int(A).Therefore 1-Int(Cl (1-Int(A)) = 1-Int(A). (ii)Let A be a (1,2)t-I-set then 1-Int(A) = 1-Int(Cl (A)) 1-Int(Cl (1-Int(A))

12 12

12 12

1-Int(A). Also 1-Int(Cl (1-Int(A)) 1-Int(A). Hence 1-Int(Cl (1-Int(A)) =

1-Int(A)

(iii) Let A be an (1,2)-I-set and U=X. Then A = U A. Hence A is a (1,2)CI-set.

Remark 4.8 The converse of the proposition 4.7 is not true.

Example 4.9 In Example 4.3 the set {a,b} is a (1,2)-I-set but not a and a(1,2)t-I-set. The set {a,b,c} is a (1,2)CI-set but not a (1,2)-I-set.

Proposition 4.10 Let(X, 1, 2, I) be an ideal bitopological space.A be a subset of X. Then the following hold.

1. If A is (1,2)CI-set then (1, 2)I-Int(A)= 1-Int(A).

2. If A is A (1,2)BI-set then (1,2)pI-Int(A)= 1-Int(A). (iii)If A is a (1,2)SI-set then (1,2)pI-Int(A)= 1-Int(A).

Proof.

1. (1, 2)I-Int(A) 1-Int(A). Since A is a (1,2)CI-set, A=UV , where U is an 1-open and V is a (1,2)-I-set.A V 1-Int(Cl12 (1-Int(A)) 1-Int(Cl12 (1-Int(V )) = 1-Int(V ). By the Proposition 3.31(iii)(1, 2)I-Int(A)=A1-Int(Cl12 (1-Int(A))) A 1-Int(V ) = U 1-Int(V ) = 1-Int(U V ) = 1-Int(A). Hence (1, 2)I-Int(A)= 1-Int(A).

   (ii)(1,2)pI-Int(A) 1-Int(A). Since A is a (1,2)BI-set, A=UV , where U is an 1-open and V is a (1,2)t-I-set. A V 1-Int(Cl12 (A)) 1-Int(Cl12 (V )) = 1-Int(V ). By the Proposition 3.31(ii)(1, 2)pI-Int(A)=A1-Int(Cl12 (A))) A 1-Int(V ) = U 1- Int(V ) = 1-Int(U V ) = 1-Int(A). Hence (1,2)pI-Int(A)= 1-Int(A).

   (iii)(1,2)sI-Int(A) 1-Int(A). Since A is a (1,2)SI-set, A=UV , where U is an 1-open and V is a (1,2)S-I-set. A V (Cl12 (1-Int(A)) (Cl12 (1-Int(V )) = 1-Int(V ). By the Proposition 3.31(i)(1, 2)pI-Int(A)=A(Cl12 (1-Int(A))) A 1-Int(V ) = U 1- Int(V ) = 1-Int(U V ) = 1-Int(A). Hence (1, 2)pI-Int(A)= 1-Int(A).

   References

   1. Abd el-monsef M.E.,El-Deeb S.N,Mahmoud R.A.:-open sets and -continuous mappings, Bull.Fac.Sc.Assuit Univ.,12(1983),77-90.

   2. Acikgoz A,Noiri T,Yuksel S,On -I-continuous and -I-open functions,Acta Math Hungar,105(1-2)(2004),27-37.

   3. M.Caldas,M.Ganster,D.N.Georgiou,S.Jafari,On -Semi open sets and separation axioms in topological spaces,Carpathian J.Math.,24(2008),13-22.

   4. Chandan Chattopadhyay,On strongly pre-open sets and a decomposition of continuity,Matematnykn Bechnk,57(2005),121-125.

   5. Hatir E and Noiri T,On decomposition of continuity via idealization, Acta Math. Hungar., 96(2002),341-349.

   6. Jafari S, Lellis Thivagar M, Athisaya Ponmani S,(1,2)-open Sets Based on Bitopological Separation Axioms,Soochow Journal of Mathematics, Vol 33, No.3 July(2007),375-381.

   7. Jankovic D and Hamlet T.R.,New topologies from old via ideals,Amer.Math.Monthly,97(1990),295-310.

   8. Kelly J.C.,Bitopological spaces,Proc. London Math. Soc.,13(1963),71-89.

   9. Khan M,Noiri T, Semi-local function in ideal topological spaces, J.Adv.Res.Pure Math.2010,2(1),36-42.

   10. Kuratowski K,Topology, Vol I,Academic Press,New York,1966.

   11. Njastad O,On some classes of nearly open sets,Pacifc J.Math.,15(1965),961-970.

   12. Tong J,On decomposition of continuity in topological spaces,Acta. Math.Hungar.,54(1989),51-55.