PROPERTIES OF -OPEN SETS IN IDEAL GENERALIZED TOPOLOGICAL SPACES S. JAFARI, S. P. MOSHOKOA AND N. RAJESH
Abstract.
The aim of this paper is to introduce and characterize
the concepts of
-open sets and their related notions in ideal gener-
alized topological spaces. Moreover, we obtain the decomposition of (
g; g )-continuity. 0
1. Introduction In [1], Csaszar introduced the notions of generalized neighborhood systems and generalized topological spaces. He also introduced the notions of continuous functions and associated interior and closure operators on generalized neighborhood systems and generalized topological spaces. In the same paper he investigated characterizations of generalized continuous functions ( = (g; g0 )-continuous functions). A subfamily g of the power set P (X ) of a nonempty set X is called a generalized topology [1] on X if and only if ; 2 g and Gi 2 g for i 2 I 6= ; implies G = [ Gi 2 g. We call the pair (X; g) a generalized topological space i2I (brie y GTS) on X . The members of g are called g-open sets [1] and the complement of a g-open set is called a g-closed set. The generalized closure of a set S of X , denoted by g Cl(A), is the intersection of all g-closed sets containing A and the generalized interior of A, denoted by g Int(A), is the union of g-open sets included in A. The concept of ideals in topological spaces has been introduced and studied by Kuratowski [7] and Vaidyanathasamy [9]. An ideal I on a topological space (X; ) is a nonempty collection of subsets of X which satis es (i) A 2 I and B A implies B 2 I and (ii) A 2 I and B 2 I implies A [ B 2 I . The aim of this paper is to introduced and characterized the concepts of g- -I -open sets and their related notions in ideal generalized topological spaces. 2.
Preliminaries
De nition 2.1. Given a generalized topological space (X; g) with an ideal I on X (for short, IGTS) and if P (X ) is the set of all subsets of X , the generalized local function of A with respect to g and I , is Mathematics Subject Classi cation. 54D10. Key words and phrases. Ideal generalized topological g- -I -closed sets. 2000
1
spaces,
g- -I -open
sets,
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S. JAFARI, S. P. MOSHOKOA AND N. RAJESH
de ned as follows: for A X , Ag (g; I ) = fx 2 X : A \ U 2= I for every g -open set containing xg. When there is no ambiguity, we will write Ag for Ag (g; I ). De nition 2.2. Let (X; g; I ) be an IGTS. The set operator g Cl is called a g--closure and is denoted as g Cl (A) = A [ Ag for A X . We will denote by g (I ; g) the generalized topology generated by g Cl , that is g (I ; g) = fU X : g Cl (X nU ) = X nU g. Clearly, g (I ; g) is ner than g. The elements of g (I ; g) are called g--open sets and the complement of g--open set is a called g--closed set. Also, g Int (A) denotes the interior of A in g (I ).
De nition 2.3. A subset A of an IGTS (X; g; I ) is said to be (i) g-R-I -open [2] if A = g Int(g Cl (A)). (ii) g-semi-I -open [4] if A g Cl (g Int(A)). (iii) g-pre-I -open [3] if A g Int(g Cl (A)). (iv) g-b-I -open [5] if A g Int(g Cl (A)) [ g Cl (g Int(A)) . (v) g- -I -open [6] if A g Cl(g Int(g Cl (A))). (vi) g--I -open [2] if g Int(g Cl (A)) g Cl (g Int(A)). The complement of a g-pre-I -open (resp. g-semi-I -open, g- -I -open) set is called a g-pre-I -closed (resp. g-semi-I -closed, g- -I -closed) set. Lemma 2.4. Let (X; g; I ) be an IGTS. Then (i) A subset A is g-pre-I -closed if and only if g Cl(g Int (A)) A
[3];
(ii) A subset A is g- -I -closed if and only if g Int(g Cl(g Int (A))) A [6].
De nition 2.5. The union of all g-pre-I -open (resp. g-semi-I -open) sets of (X; g; I ) containing A is called the g -pre-I -interior [3] (resp. g semi-I -interior [4]) of A and is denote by gpI Int(A) (resp. gsI Int(A)). De nition 2.6. The intersection of all g-pre-I -closed (resp. g-semiI -closed) sets of (X; g; I ) contained in A is called the g-pre-I -closure [3] (resp. g-semi-I -closure [4]) of A and is denote by gpI Cl(A) (resp. gsI Cl(A)). Lemma 2.7. Let (X; g; I ) be an IGTS and A X . Then the following hold.
gsI Int(A) = A \ g Cl (g Int(A)). gsI Cl(A) = A [ g Int (g Cl(A)). gpI Int(A) = A \ g Int(g Cl (A)). gpI Cl(A) = A [ g Cl(g Int (A)). De nition 2.8. A function f : (X; g; I ) ! (Y; g0 ) is said to be (i) (g; g0 )-pre-I -continuous [3] if the inverse image of every g0 -open set of Y is g -pre-I -open in X .
(i) (ii) (iii) (iv)
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(ii) (g; g0 )-semi-I -continuous [4] if the inverse image of every g0 open set of Y is g -semi-I -open in X .
(iii) (g; g0 )-b-I -continuous [5] if the inverse image of every g0 -open set of Y is g -b-I -open in X .
(iv) (g; g0 )- -I -continuous [6] if the inverse image of every g0 -open set of Y is g - -I -open in X .
(v) (g; g0 )--I -continuous [4] if the inverse image of every g0 -open set of Y is g - -I -open in X .
(vi) strongly (g; g0 )- -I -continuous [6] if the inverse image of every g0 -open set of Y is strongly g- -I -open in X .
3. (g; g0 )- -I -open sets De nition 3.1. A subset A of an IGTS (X; g; I ) is said to be g- -I open if and only if A g Int(g Cl (g Int(A))). The family of all g - -I -open sets of (X; g; I ) is denoted by I O(X; g ). Also, The family of all g - -I -open sets of (X; g; I ) containing x is denoted by I O(X; g; x). Remark 3.2. Let I and J be two ideals on (X; g). If I J , then J O(X; g) I O(X; g0 ). Proposition 3.3. (i) Every g-open set is g- -I -open. (ii) Every g- -I -open set is g-semi-I -open. (iii) Every g- -I -open set is g-pre-I -open. Proof. The proof follows from the de nitions. Corollary 3.4. (i) Every g- -I -open set is g-b-I -open. (ii) Every g- -I -open set is g- -I -open. The following examples show that the converses of Proposition 3.3 is not true in general. Example 3.5. Let X = fa; b; cg g = f;; fa; bg; fb; cg; X g and I = f;g. Then the set fa; cg is g - -I -open but not g -open. Also, the set fbg is g-pre-I -open but not g- -I -open.
Example 3.6. Let X = fa; b; cg g = f;; fag; fbg; fa; bg; X g and I = f;; fagg. Then the set fb; cg is g-semi-I -open but not g- -I -open. Proposition 3.7. Let A be a subset of an IGTS (X; g; I ). If B is a g-semi-I -open set of X such that B A g Int(g Cl (B )), then A is a g - -I -open set of X . Proof. Since B is a g -semi-I -open set of X , we have B g Cl (g Int(B )). Thus, A g Int(g Cl (B )) g Int(g Cl (g Cl (g Int(B )))) = g Int(g Cl (g Int(B ))) g Int(g Cl (g Int(A))), and so A is a g- -I -open set of X . Proposition 3.8. Let (X; g; I ) be an IGTS. Then a subset of X is g- -I -open if and only if it is both g- -I -open and g-pre-I -open.
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Let A be a g- -I -open set. Since every g- -I -open set is g-semiI -open, by Proposition 3.3 A is g--I -open. Now we prove that A g Int(g Cl (A)). Since A is a g- -I -open, we have A g Int(g Cl (g Int(A))) g Int(g Cl (A)). Hence A is g-pre-I -open. Conversely, let A be a g- -I -open and g-pre-I -open set. Then we have g Int(g Cl (A)) g Cl (g Int(A)) and hence g Int(g Cl (A)) g Int(g Cl (g Int(A))). Since A is g-pre-I -open, we have A g Int(g Cl (A)). Therefore, we obtain that A g Int(g Cl (g Int(A))); hence A is g- -I -open. Proof.
Theorem 3.9. A subset A is g- -I -open if and only if it is g-semi-I open and g -pre-I -open. Proof. Let A be g -semi-I -open and g -pre-I -open. Then, A g Int(g Cl (A)) g Int(g Cl (g Cl (g Int(A)))) = g Int(g Cl (g Int(A))). This shows that A is g- -I -open. The converse is obvious. Corollary 3.10. The following properties are equivalent for subsets of an IGTS (X; g; I ): (i) Every g-pre-I -open set is g-semi-I -open. (ii) A subset A of X is g- -I -open if and only if it is g-pre-I -open. Corollary 3.11. The following properties are equivalent for subsets of an IGTS (X; g; I ): (i) Every g-semi-I -open set is g-pre-I -open. (ii) A subset A of X is g- -I -open if and only if it is g-semi-I -open. Proposition 3.12. Let A be a subset of an IGTS (X; g; I ). If A is g-pre-I -closed and g- -I -open, then it is g-open. Proof. Suppose A is g -pre-I -closed and g - -I -open. Then by Lemma 2.4 g Cl(g Int (A)) A and A g Int(g Cl (g Int(A))). Now g Cl(g Int(A)) g Cl(g Int(A)) g Cl(g Int (A)) A and so A g Int(g Cl (g Int(A)) A g Int(A). Therefore, A is g-open. Lemma 3.13. [2] If A is any subset of an IGTS (X; g; I ), then g Int(g Cl (A)) is g -R-I -open. Proposition 3.14. Let A be a subset of an IGTS (X; g; I ). If A is g- -I -open and g- -I -closed, then it is g-R-I -open. Proof. Let A be g - -I -open and g - -I -closed. We have by Lemma 2.4, A g Int(g Cl (g Int(A))) and g Int(g Cl (g Int(A))) g Int(g Cl(g Int (A))) A; hence A = g Int(g Cl (g Int(A))). Thus, by Lemma 3.13, A is g-R-
I -open.
Lemma 3.15. Let A be a subset of an IGTS (X; g; I ). Then A is g-open and g-closed if and only if it is g- -I -open and g-pre-I -closed. Proof. Let A be g - -I -open and g -pre-I -closed. We have A g Int(g Cl (g Int(A))) and g Cl (g Int(A)) A and hence A g Int(g Cl (g Int(A))) g Int(g Cl(g Int (A))) g Cl(g Int (A)) A. Thus, A = g Int(g Cl (g Int(A))) = g Cl(g Int (A)); hence A is g-open and g-closed.
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Remark 3.16. The intersection of two g- -I -open sets need not be g- -I -open in general. Let (X; g; I ) as in Example 3.5. Then the sets fa; cg and fb; cg are g- -I -open sets of (X; g; I ) but their intersection fcg is not an g- -I -open set of (X; g; I ). Theorem 3.17. If fA g 2 be a family of g- -I -open sets in (X; g; I ), S then A is g- -I -open in (X; g; I ). 2
Since fA : 2 g g- I O(X ), then A g Int(g Cl (g Int(A ))) for every 2 . Thus, [2 A [2 g Int(g Cl (g Int(A ))) g Int(g Cl ( [2 g Int(A )) = g Int(g Cl (g Int( [2 A )). Therefore, we obtain [2 A g Int(g Cl (g Int( [2 A )). Hence any union of g- -I -open sets is g- -I -open. De nition 3.18. In an IGTS (X; g; I ), A X is said to be g- -I Proof.
closed if X nA is g - -I -open in X . The family of all g - -I -closed sets of (X; g; I ) is denoted by I C (X; g ).
Theorem 3.19. If A is a g- -I -closed set in an IGTS (X; g; I ) if and only if g Cl(g Int (g Cl(A))) A. Proof.
The proof follows from the de nitions.
Proof.
The proof follows from the de nitions.
Theorem 3.20. If A is a g- -I -closed set in an IGTS (X; g; I ), then g Cl(g Int(g Cl (A))) A. Proof. Since A 2 I C (X; g ), X nA 2 I O(X; g ). Hence, X nA g Int(g Cl (g Int(X nA)) g Int(g Cl(g Int(X nA))) = X ng Cl(g Int(g Cl(A))) X n(g Cl(g Int(g Cl (A))). Therefore, we obtain g Cl(g Int(g Cl (A)) A. Proposition 3.21. Let (X; g; I ) be an IGTS. If a subset of X is g- I -closed and g--I -open, then it is g- -I -closed. Theorem 3.22. Arbitrary intersection of g- -I -closed sets is always g- -I -closed.
Follows from Theorems 3.17 and 3.20. De nition 3.23. Let (X; g; I ) be an IGTS, S a subset of X and x be Proof.
a point of X . Then (i) x is called an g- -I -interior point of S if there exists V 2 I O(X; g) such that x 2 V S . ii) the set of all g- -I -interior points of S is called g- -I -interior of S and is denoted by g I Int(S ).
Theorem 3.24. Let A and B be subsets of (X; g; I ). Then the following properties hold: (i) g I Int(A) =
[fT : T A and A 2 I O(X; g)g.
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(ii) g I Int(A) is the largest g- -I -open subset of X contained in
A. A is g- -I -open if and only if A = g I Int(A). g I Int(g I Int(A)) = g I Int(A). If A B , then g I Int(A) g I Int(B ). g I Int(A \ B ) = g I Int(A) \ g I Int(B ). g I Int(A [ B ) g I Int(A) [ g I Int(B ). Proof. (i). Let x 2 [fT : T A and A 2 I O(X; g )g. Then, there exists T 2 I O(X; g; x) such that x 2 T A and hence x 2 g I Int(A). This shows that [fT : T A and A 2 I O(X; g)g g I Int(A). For the reverse inclusion, let x 2 g I Int(A). Then there exists T 2 I O(X; g; x) such that x 2 T A. we obtain x 2 [fT : T A and A 2 I O(X; g)g. This shows that g I Int(A) [fT : T A and A 2 I O(X; g)g. Therefore, we obtain g I Int(A) = [fT : T A and A 2 I O(X; g)g. The proof of (ii) (v) are obvious. (vi). By (v), we have g I Int(A) g I Int(A [ B ) and g I Int(B ) g I Int(A[B ). Then we obtain g I Int(A) [ g I Int(B ) g I Int(A[ B ) Since g I Int(A) A and g I Int(B ) B , we obtain g I Int(A [ B ) g I Int(A) [ g I Int(B ). It follows that g I Int(A \ B ) = g I Int(A) \ g I Int(B ). (vii). Since A \ B A and A \ B B , we have g I Int(A \ B ) g I Int(A) and g I Int(A \ B ) g I Int(B ) . Therefore, g I Int(A) [ g I Int(B ) g I Int(A \ B ). Theorem 3.25. If (X; g; I ) is an IGTS, then g I Int(A) = A \ g Int(g Cl (g Int(A))) holds for every subset A of X . Proof. Since A\g Int(g Cl (g Int(A))) g Int(g Cl (g Int(A))) = g Int(g Int(g Cl (g Int(A)))) = g Int(g Cl (g Int(A))\(g Int(g Cl (g Int(A)))) g Int(g Cl (g Int(A)\ g Int(g Cl (g Int(A))))) = g Int(g Cl (g Int(A \ g Int(g Cl (g Int(A)))))), A \ g Int(g Cl (g Int(A))) is an g- -I -open set contained in A and so A \ g Int(g Cl (g Int(A))) g I Int(A). Since g- I Int(A) is g- -I -open, g- I Int(A) g Int(g Cl (g Int(g I Int(A)))) g Int(g Cl (g Int(A))) and so g I Int(A) A \ g Int(g Cl (Int(A))). Hence g I Int(A) = A \ Int(g Cl (g Int(A))). Theorem 3.26. If (X; g; I ) is an IGTS, then g I Cl(A) = A\g Cl(g Int (g Cl(A))) holds for every subset A of X .
(iii) (iv) (v) (vi) (vii)
The proof follows from Theorem 3.25. Theorem 3.27. If (X; g; I ) is an IGTS, then g I Int(A) = gpI Int(A) Proof.
holds for every g - -I -open subset A of X .
Since every g- -I -open set is g-pre-I -open, g I Int(A) gpI Int(A). By Theorem 3.25, g I Int(A) = A \ g Int(g Cl (g Int(A))). Since A is g -I -open, g I Int(A) A\g Int(g Int(g Cl (A))) = A\g Int(g Cl (A)) =
Proof.
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gpI Int(A) by Lemma 2.7 and so g I Int(A) gpI Int(A). Therefore, g I Int(A) = gpI Int(A). De nition 3.28. Let (X; g; I ) be an IGTS, S a subset of X and x be a point of X . Then (i) x is called an g- -I -cluster point of S if V \ S 6= ; for every V 2 I O(X; g; x). (ii) the set of all g- -I -cluster points of S is called g- -I -closure of S and is denoted by g I Cl(S ). Theorem 3.29. Let A and B be subsets of (X; g; I ). Then the following properties hold: (i) g- I Cl(A) = \fF : A F and F 2 I C (X; g)g. (ii) g I Cl(A) is the smallest g- -I -closed subset of X containing A. (iii) A is g- -I -closed if and only if A = g I Cl(A). (iv) g I Cl(g I Cl(A) = g I Cl(A). (v) If A B , then g I Cl(A) g I Cl(B ). (vi) g I Cl(A [ B ) = g I Cl(A) [ g I Cl(B ). (vii) g I Cl(A \ B ) g I Cl(A) \ g I Cl(B ). Proof. (i).
We shall only proof (i) as the proofs of the other statements are obvious. Suppose that x 2= g I Cl(A). Then there exists V 2 I O(X; g)g such that V \ A 6= ;. Since X nV is g- -I -closed set containing A and x 2= X nV , we obtain x 2= \fF : A F and F 2 I C (X; g)g. Then there exists F 2 I C (X; g) such that A F and x 2= F . Since X nV is g- -I -closed set containing x, we obtain (X nF ) \ A = ;. This shows that x 2= g I Cl(A). Therefore, we obtain g I Cl(A) = \fF : A F and F 2 I C (X; g)g.
Theorem 3.30. Let (X; g; I ) be an IGTS and A X . A point x 2 g I Cl(A) if and only if U \ A 6= ; for every U 2 I O(X; g; x). Proof. Suppose that x 2 g I Cl(A). We shall show that U \ A = 6 ; for every U 2 I O(X; g; x). Suppose that there exists U 2 I O(X; g; x) such that U \ A = ;. Then A X nU and X nU is g- -I -closed. since A X nU , g I Cl(A) g I Cl(X nU ). Since x 2 g I Cl(A), we have x 2 g I Cl(X nU ). Since X nU is g- -I -closed, we have x 2 X nU ; hence x 2= U , which is a contradiction that x 2 U . Therefore, U \ A = 6 ;. Conversely, suppose that U \ A =6 ; for every U 2 I O(X; g; x). We shall show that x 2 g I Cl(A). Suppose that x 2= g I Cl(A). Then there exists U 2 I O(X; g; x) such that U \ A = ;. This is a contradiction to U \ A = 6 ;; hence x 2 g I Cl(A). Theorem 3.31. Let (X; g; I ) be an IGTS and A X . Then the following propeties hold:
(i) g I Int(X nA) = X ng I Cl(A);
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(i) g I Cl(X nA) = X ng I Int(A). Proof. (i). Let x 2 X ng I Cl(A). Then, there exists V 2 I O(X; g; x) such that V \ A 6= ;; hence we obtain x 2 g I Int(X nA). This shows that X ng I Cl(A) g- I Int(X nA). Let x 2 g I Int(X nA). Since g I Int(X nA) \ A = ;, we obtain x 2= g I Cl(A); hence x 2 X ng I Cl(A). Therefore, we obtain g I Int(X nA) = X n I Cl(A). (ii). Follows from (i). De nition 3.32. A subset Bx of an IGTS (X; g; I ) is said to be an
g- -I -neighbourhood of a point x 2 X if there exists an g- -I -open set U such that x 2 U Bx . Theorem 3.33. A subset of an IGTS (X; g; I ) is g- -I -open if and only if it is an g - -I -neighbourhood of each of its points. Proof. Let G be an g - -I -open set of X . Then by de nition, it is clear that G is an g- -I -neighbourhood of each of its points, since for every x 2 G, x 2 G G and G is g- -I -open. Conversely, suppose G is an g- -I -neighbourhood of each of its points. Then for each S x 2 G, there exists Sx 2 I O(X; g) such that Sx G. Then G = fSx : x 2 Gg. Since each Sx is g- -I -open, G is g- -I -open in (X; g; I ).
4. (g; g0 )- -I -continuous functions De nition 4.1. A function f : (X; g; I ) ! (Y; g0 ) is said to be (g; g0 )-
-I -continuous if the inverse image of every g0 -open set of Y is g- I -open in X . Proposition 4.2. (i) Every (g; g0 )-continuous function is (g; g0 ) -I -continuous but not conversely. (ii) Every (g; g0 )- -I -continuous function is (g; g0 )-semi-I -continuous but not conversely.
(iii) Every (g; g0 )- -I -continuous function is (g; g0 )-pre-I -continuous but not conversely.
Proof.
3.6.
The proof follows from Proposition 3.3 and Examples 3.5 and
Corollary 4.3. (i) Every (g; g0 )- -I -continuous function is (g; g0 )b-I -continuous but not conversely. (ii) Every (g; g0 )- -I -continuous function is (g; g0 )- -I -continuous but not conversely.
Theorem 4.4. A function f : (X; g; I ) ! (Y; g0 ) is (g; g0 )- -I -continuous if and only if it is (g; g 0 )-semi-I -continuous and (g; g 0 )-pre-I -continuous.
This is an immediate consequence of Lemma 3.9. Theorem 4.5. For a function f : (X; g; I ) ! (Y; g0 ), the following Proof.
statements are equivalent:
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(i) f is (g; g0 )- -I -continuous; (ii) For each point x in X and each g0 -open set F in Y such that f (x) 2 F , there is a g- -I -open set A in X such that x 2 A, f (A) F ; (iii) The inverse image of each g0 -closed set in Y is g- -I -closed in X;
For each subset A of X , f (g I Cl(A)) g 0 Cl(f (A)); For each subset B of Y , g I Cl(f 1 (B )) f 1 (g 0 Cl(B )); For each subset C of Y , f 1 (g 0 Int(C )) g I Int(f 1 (C )). g Cl(g Int (g Cl(f 1 (B )))) f 1 (g0 Cl(B )) for each subset B of Y. (viii) f (g Cl(g Int (g Cl(A)))) g0 Cl(f (A)) for each subset A of X . Proof. (i) ) (ii): Let x 2 X and F 2 g 0 containing f (x). By (i),
(iv) (v) (vi) (vii)
f 1 (F ) is g- -I -open in X . Let A = f 1 (F ). Then x 2 A and f (A) F . (ii) ) (i): Let F 2 g0 and let x 2 f 1 (F ). Then f (x) 2 F . By (ii), there is a g- -I -open set Ux in X such that x 2 Ux and f (Ux ) F . Then x 2 Ux f 1 (F ). Hence f 1 (F ) is g- -I -open in X . (i) , (iii): This follows due to the fact that for any subset B of Y , f 1 (Y nB ) = X nf 1 (B ). (iii) ) (iv): Let A be a subset of X . Since A f 1 (f (A)) we have A f 1 (g0 Cl(f (A))). Now, g0 Cl(f (A)) is g0 -closed in Y and hence g I Cl(A)) f 1 (g0 Cl(f (A))), for g I Cl(A) is the smallest g- -I closed set containing A. Then f (g I Cl(A)) g0 Cl(f (A)). (iv) ) (iii): Let F be any g0 -closed subset of Y . Then f (g I Cl(f 1 (F ))) g0 Cl(f (f 1(F ))) = g0 Cl(F ) = F . Therefore, g I Cl(f 1(F )) f 1 (F ). Consequently, f 1 (F ) is g- -I -closed in X . (iv) ) (v): Let B be any subset of Y . Now, f (g I Cl(f 1 (B ))) g0 Cl(f (f 1 (B ))) g0 Cl(B ). Consequently, g I Cl(f 1 (B )) f 1 (g0 Cl(B )). (v) ) (iv): Let B = f (A) where A is a subset of X . Then, g I Cl(A) g I Cl(f 1(B )) f 1(g0 Cl(B )) = f 1(g0 Cl(f (A))). This shows that f (g I Cl(A)) g0 Cl(f (A)). (i) ) (vi): Let B 2 g0 . Clearly, f 1 (g0 Int(B )) is g- -I -open and we have f 1 (g0 Int(B )) g I Int(f 1 (g0 Int(B ))) g I Int(f 1 B ). (vi) ) (i): Let B 2 g0 . Then g0 Int(B ) = B and f 1 (B )nf 1 (g0 Int(B )) g I Int(f 1(B )). Hence we have f 1(B ) = g I Int(f 1(B )). This shows that f 1 (B ) is g- -I -open in X . (iii) ) (vii): Let B be any subset os Y . Since g0 Cl(B ) is g0 -closed in Y , by (iii), f 1 (g0 Cl(B )) is g- -I -closed and X nf 1 (g0 Cl(B )) is g- I -open. Then X nf 1(g0 Cl(B )) g Int(g Cl(g Int(f 1(g0 Cl(B))))) = X ng Cl(g Int (g Cl(f 1 (g0 Cl(B ))))). Hence we obtain g Cl(g Int (g Cl(f 1 (B )))) f 1 (g0 Cl(B )). (vii) ) (viii): Let A be any ubset of X . By(iv), we have g Cl(g Int (g Cl(A)) g Cl(g Int (g Cl(f 1 (f (A))))) f 1 (g0 Cl(f (A))) and hence f (g Cl(g Int (g Cl(A)))) g0 Cl(f (A)).
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(viii) ) (i): Let V 2 g0 . Then by (v), f (g Cl(g Int (g Cl(f 1 (Y nV ))))) g0 Cl(f (f 1 (Y nV ))) g0 Cl(Y nV ) = Y nV . Therefore, we have g Cl(g Int (g Cl(f 1 (Y nV )))) f 1 (Y nV ) X nf 1 (V ). Consequently, we obtain f 1 (V ) g Int(g Cl (g Int(f 1 (V )))). This shows that f 1 (V ) is -I -open. Thus, f is -I -continuous. Corollary 4.6. Let f : (X; g; I ) ! (Y; g0 ; I ) be an g- -I -continuous function, then (i) f (g Cl (U )) g0 Cl(f (U )) for every g-pre-I -open set U of X , (ii) g Cl (f 1 (V )) f 1 (g0 Cl(V )) for every g0 -pre-I -open set V of Y. Proof. (1). Let U be any g -pre-I -open set of X , then U g Int(g Cl (U )). Therefore, by Theorem 4.5, we have f (g Cl (U )) f (g Cl(U )) f (g Cl(g Int(g Cl (U )))) f (g Cl(g Int (g Cl(U )))) g0 Cl(f (U )). (2). Let V be any g0 -pre-I -open set of Y . By Theorem 4.5, g Cl (f 1 (V )) g Cl(f 1 (V )) g Cl(f (g0 Int(g0 Cl (V )))) g Cl(g Int(g Cl (g Int(f 1 (g0 Int(g0 Cl( (V )))))))) g Cl(g Int(g Cl(f 1(g0 Int(g0 Cl((V ))))))) f 1(g0 Cl(g0 Int(g0 Cl( V )))) f 1 (g0 Cl(V )).
Theorem 4.7. Let f : (X; g; I ) ! (Y; g0 ) be a (g; g0 )- -I -continuous function. Then for each subset V of Y , f 1 (g 0 Int(V )) g Cl (f 1 (V )). Proof. Let V be any subset of Y . Then g 0 Int(V ) is g 0 -open in Y and so f 1 (g0 Int(V )) is g- -I -open in X . Hence f 1 (g0 Int(V )) g Int(g Cl (g Int(f 1 (g0 Int(V ))))) g Cl (f 1 (V )). Theorem 4.8. Let f : (X; g; I ) ! (Y; g0 ) be a bijective. Then f is (g; g0 )- -I -continuous if and only if g0 Int(f (U )) f (g I Int(U )) for each subset U of X . Proof. Let U be any subset of X . Then by Theorem 4.5, f 1 (g 0 Int(f (U ))) g I Int(f 1(f (U ))). Since f is bijection, g0 Int(f (U )) = f (f 1(g0 Int(f (U ))) f (g I Int(U )). Conversely, let V be any subset of Y . Then g0 Int(f (f 1 (V ))) f (g I Int(f 1 (V ))). Since f is bijection, g0 Int(V ) = g0 Int(f (f 1 (V ))) f (g I Int(f 1 (V ))); hence f 1 (g0 Int(V ) g I Int(f 1 (V )). Therefore, by Theorem 4.5, f is (g; g0 )- -I -continuous. De nition 4.9. A subset K of an IGTS (X; g; I ) is said to be g- I -compact relative to X , if for every cover fU : 2 g of K by g- -SI -open sets of X , there exists a nite subset 0 of such that K n fU : 2 0 g 2 I . The space (X; g; I ) is said to be g- -I compact if X is g - -I -compact subsets of X . De nition 4.10. A subset K of an IGTS (X; g; I ) is said to be countably g - -I -compact relative to X , if for every cover fU : 2 g of K by countable gS- -I -open sets of X , there exists a nite subset 0 of such that K n fU : 2 0 g 2 I . The space (X; gI ) is said to be countably g - -I -compact if X is countable g - -I -compact subset of X .
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De nition 4.11. A subset K of an IGTS (X; g; I ) is said to be g -I -Lindelof relative to X , if for every cover fU : 2 g of K by g- -SI -open sets of X , there exists a nite subset 0 of such that K n fU : 2 0 g 2 I . The space (X; g; I ) is said to be g- -I Lindelof if X is g - -I -Lindelof subset of X . Lemma 4.12. [8] For any function f : (X; ; I ) ! (Y; ), f (I ) is an ideal on Y . Theorem 4.13. If f : (X; g; I ) ! (Y; g0 ) is a (g; g0 )- -I -continuous surjection and (X; g; I ) is g - -I -compact,then (Y; g 0 ; f (I )) is g 0 -f (I )compact.
Let fV : 2 g be a g0 -open cover of Y . Then ff 1 (V ) : 2 g is a g- -I -open cover hence, there exist a nite subset Sff of1(XV )and 0 of such that X n : 2 0g 2 I . Since f is surjective, S S 1 Y n fV : 2 0 g = f (X n ff (V ) : 2 0 g) 2 I . Therefore, (Y; g0 ; f (I )) is g0 -f (I )-compact. Proof.
The proofs of the next two Theorems follows from that of Theorem 4.13 and therefore we omit them. Theorem 4.14. If f : (X; g; I ) ! (Y; g0 ) is a (g; g0 )- -I -continuous surjection and (X; g; I ) is g - -I -Lindelof, then (Y; g 0 ; f (I )) is g 0 -f (I )Lindelof.
Theorem 4.15. If f : (X; g; I ) ! (Y; g0 ) is a (g; g0 )- -I -continuous surjection and (X; g; I ) is countably g - -I -compact, then (Y; g 0 ; f (I )) is countably g 0 -f (I )-compact. De nition 4.16. An GTS (X; g) is said to be g-connected if it cannot be expressed as the union of two nonempty disjoint sets g -open sets. De nition 4.17. An IGTS (X; g; I ) is said to be g- -I -connected if it cannot be expressed as the union of two nonempty disjoint g - -I -open sets.
Theorem 4.18. Let f : (X; g; I ) ! (Y; g0 ) is (g; g0 )- -I -continuous surjection and (X; g; I ) is g - -I -connected, then (Y; g 0 ) is g 0 -connected. Proof. Suppose Y is not g 0 -connected, Then Y = A [ B where A \ B = ;, A 6= ;, B 6= ; and A; B 2 g0. Since f is (g; g0)- -I -continuous f 1 (A); f 1 (B ) 2 S I O(X; g) such that f 1 (A) 6= ;, f 1 (B ) 6= ; and f 1 (A) \ f 1 (B ) = ; and f 1 (A) [ f 1 (B ) = X , which implies that X is not g- -I -connected.
(g; g0 )-continuity De nition 5.1. For an IGTS (X; g; I ), we de ne (i) g-I -D( ; p) = fA X : g I Int(A) = gpI Int(A)g. 5.
Decomposition of
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(ii) g-I -D( ; s) = fA X : g I Int(A) = gsI Int(A)g. De nition 5.2. A subset A of (X; g; I ) is said to be g-I -D( ; p)-set if A 2 g -I -D( ; p) (resp. g -I -D( ; s)-set if A 2 g -I -D( ; s)) Proposition 5.3. [4] A subset A of an IGTS (X; g; I ) is g-semi-I open if and only if g Cl (A) = g Cl (g Int(A)). Theorem 5.4. For an IGTS (X; g; I ), the relation S I O(X; g) g-I D( ; p) holds. Proof. Let A 2 S I O(X; g ). Then by Proposition 5.3, g Cl (A) = g Cl (g Int(A)). Then g Int(g Cl (A)) = g Int(g Cl (g Int(A))). We have A \ g Int(g Cl (A)) = A \ g Int(g Cl (g Int(A))). Hence gpI Int(A) = g I Int(A) by Theorem 3.25 and Lemma 2.7. Therefore, A 2 g-I D( ; p). The following example shows that the reverse inclusion of Theorem 5.4 is not true in general. Example 5.5. Let X = fa; b; cg g = f;; fag; fbg; fa; bg; X g and I = f;g. Then the set fag 2 g-I -D( ; p) but fag 2= S I O(X; g). Theorem 5.6. For an IGTS (X; g; I ), I O(X; g) = P I O(X; g) \ gI -D( ; p) holds. Proof. Let A 2 P I O(X; g ) \ g -I -D( ; p). Then A = gpI Int(A) and g I Int(A) = gpI Int(A). Hence A = g I Int(A). Therefore, A 2 I O(X; g). Conversely, let A 2 I O(X; g). Then A = g I Int(A) gpI Int(A). But gpI Int(A) A. Therefore, A 2 gpI Int(A) and hence we have A = g I Int(A) = gpI Int(A). Hence A 2 g-I -D( ; p). Theorem 5.7. For an IGTS (X; g; I ), I O(X; g) = S I O(X; g) \ gI -D( ; s) holds. Proof. The proof is similar to the Theorem 5.6. De nition 5.8. For an IGTS (X; g; I ), we de ne (i) g-I -D(g; ) = fA X : g Int(A) = g I Int(A)g. (ii) g-I -D(g; s) = fA X : g Int(A) = gsI Int(A)g. (iii) g-I -D(g; p) = fA X : g Int(A) = gsI Int(A)g. Theorem 5.9. For a subset A of (X; g; I ), the following statements are equivalent: (i) A 2 g-I -D(g; p).
(ii) g Int(A) = A \ g Cl (g Int(A)). Proof. (i) ) (ii): Let A 2 g -I -D(g; p).Then we have g Int(A) = gsI Int(A) = A \ g Cl (g Int(A)) by Lemma 2.7. (ii) ) (i): Let g Int(A) = A \ g Cl (g Int(A)). Then by Lemma 2.7, g Int(A) = gsI Int(A). Therefore, A 2 g-I -D(g; p).
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Proposition 5.10. For each x 2 (X; g; I ), fxg 2 g-I -D(g; s). Proof. Let x 2 X and fxg 2 g . Then g Int(fxg) = fxg = fxg \ g Cl (fxg) = fxg\ g Cl (g Int(fxg)). Therefore, by Theorem 5.9, fxg 2 g-I -D(g; s). Now, let fxg 2= g. Then g Int(fxg) = ; = fxg \ ; = fxg \ g Cl (g Int(fxg)). Therefore, by Theorem 5.9, fxg 2 g-I -D(g; s). Theorem 5.11. Let (X; g; I ) be an IGTS and A X . If A 2 g, then A 2 g-I -D(g; s). Proof. Let A 2 g . Then g Int(A) = A = A\g Cl (A) = A\g Cl (g Int(A)) = gsI Int(A). Hence A 2 g-I -D(g; s) by Lemma 2.7. Theorem 5.12. Let (X; g; I ) be an IGTS and A X . If A 2 g, then A 2 g-I -D(g; ). Proof. Let A 2 g . Then g Int(A) = A = g Int(A \ g Cl (A)) = A \ g Int(g Cl (g Int(A))) = A \ g Int(g Cl (g Int(A))) = g I Int(A). Hence A 2 g-I -D(g; ). Theorem 5.13. For an IGTS (X; g; I ), the following relations hold: (i) g-I -D(g; s) g-I -D(g; ). (ii) g-I -D(g; p) g-I -D(g; ). holds.
Proof. (i). Let A 2 g -I -D(g; s). Then g Int(A) = gsI (A) g I (A). But g Int(A) g I (A) and hence we have g Int(A) = g I (A). Therefore, A 2 g-I -D(g; ). The proof of (ii) is similar to (i).
Remark 5.14. The converses of the Theorem 5.12 and 5.13 are not true in general. Let (X; g; I ) as in Example 3.5. Then fbg 2 g -I D(g; ) but fbg 2= g and fbg 2= g-I -D(g; p). Let (X; g; I ) as in Example 3.6. Then fb; cg 2 g -I -D(g; ) but fbg 2 = g-I -D(g; s). Example 5.15. g-semi-I -open sets and g-I -D(g; s) sets are independent of each other. Let (X; g; I ) as in Example 3.6. Then fa; cg 2 S I O(X; g) but fa; cg 2= g-I -D(g; s). Also fcg 2= g-I -D(g; s) but fa; cg 2 S I O(X; g) Example 5.16. g-pre-I -open sets and g-I -D(g; p) sets are independent of each other. Let (X; g; I ) as in Example 3.5. Then fbg 2 P I O(X; g) but fbg 2= g-I -D(g; p). Also fag 2= g-I -D(g; p) but fag 2 P I O(X; g) Example 5.17. g- -I -open sets and g-I -D(g; ) sets are independent of each other. Let X = fa; b; cg, g = f;; fcg; fb; cg; X g and I = f;g. Then fa; cg 2 I O(X; g ) but fa; cg 2 = g-I -D(g; ). Also fag 2 g-I D(g; ) but fag 2= I O(X; g). Theorem 5.18. For a subset A of (X; g; I ), the following statements are equivalent:
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(i) A 2 g. (ii) A is g- -I -open and a g-I -D(g; ) set. (iii) A is g-pre-I -open and a g-I -D(g; p) set. Proof. The implication (i) ) (ii) is obvious. (ii) ) (iii): Let A be g- -I -open and a g-I -D(g; ) set. Then A 2 P I O(X; g) \ S I O(X; g) and g I Int(A) = g Int(A). By Lemma 3.25, we have A \ g Int(g Cl (g Int(A))) = g Int(A) and subsequently A \ g Int(g Cl (A)) = g Int(A). Now by Lemma 2.7, we have gpI Int(A) = g Int(A). Therefore, A is a g-I -D(g; ) set. (iii) ) (i): Let A be g-pre-I -open and a g-I -D(g; p) set. The by De nition A = gpI Int(A) and g Int(A) = gpI Int(A); hence A 2 g. Theorem 5.19. For a subset A of (X; g; I ), the following statements
are equivalent: (i) A 2 g. (ii) A is g- -I -open and a g-I -D(g; s) set. (iii) A is g-semi-I -open and a g-I -D(g; s) set.
The proof is similar to the proof of Theorem 5.18. De nition 5.20. A function f : (X; g; I ) ! (Y; g0 ) is said to be: (i) (g; g0 )-D( ; p)-I -continuous if f 1 (V ) 2 g-I -D( ; p) for every Proof.
V 2 g0. (ii) (g; g0 )-D( ; s)-I -continuous if f 1 (V ) 2 g-I -D( ; s) for every V 2 g0. (iii) (g; g0 )-D(g; )-I -continuous if f 1 (V ) 2 g-I -D(g; ) for every V 2 g0. (iv) (g; g0 )-D(g; s)-I -continuous if f 1 (V ) 2 g-I -D(g; s) for every V 2 g0. (v) (g; g0 )-D(g; p)-I -continuous if f 1 (V ) 2 g-I -D(g; p) for every V 2 g0. Theorem 5.21. A function f : (X; g; I ) ! (Y; g0 ) is (g; g0 )-continuous if and only if (i) it is (g; g0 )- -I -continuous and (g; g0 )-D(g; )-I -continuous. (ii) it is (g; g0 )-pre-I -continuous and (g; g0 )-D(g; p)-I -continuous. (iii) it is (g; g0 )- -I -continuous and (g; g0 )-D(g; s)-I -continuous. (iv) it is (g; g0 )-semi-I -continuous and (g; g0 )-D(g; s)-I -continuous.
The proof follows from Theorems 5.18 and 5.19. Theorem 5.22. A function f : (X; g; I ) ! (Y; g0 ) is (g; g0 )- -I Proof.
continuous if and only if (i) it is (g; g0 )-semi-I -continuous and (g; g0 )-pre-I -continuous. (ii) it is (g; g0 )-pre-I -continuous and (g; g0 )-D( ; p)-I -continuous. (iii) it is (g; g0 )-semi-I -continuous and (g; g0 )-D( ; s)-I -continuous. Proof.
The proof follows from Theorems 2.4, 5.6 and 5.7.
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Acknowledgment. S. P. Moshokoa acknowledges the support by the South African National Research Foundation. References [1] A. Csaszar, Generalized topology, generalized continuity,
96 (2002), 351-357. [2] S. Jafari and N. Rajesh,
Some subsets of ideal generalized topological spaces
(under preparation). [3] S. Jafari, S. P. Moshokoa and N. Rajesh,
topological spaces
(submitted).
[4] S. Jafari, S. P. Moshokoa and N. Rajesh,
topological spaces
Preopen sets in ideal generalized Semiopen sets in ideal generalized
(submitted).
[5] S. Jafari and N. Rajesh,
spaces
Acta Math. hungar.,
Properties of b-open sets in ideal generalized topological
(under preparation).
[6] S. Jafari and N. Rajesh,
ical spaces
Properties of -open sets in ideal generalized topolog-
(submitted).
[7] K. Kuratowski, Topology,
Academic press, New York, (1966).
[8] R. L. Newcomb, Topologies which are compact modulo an ideal, Ph.D. Thesis, University of California, USA(1967). [9] R. Vaidyanathaswamy, The localisation theory in set topology,
Acad. Sci., 20(1945), 51-61.
Proc. Indian
College of Vestsjaelland South, Herrestraede, 11, 4200 Slagelse, Denmark
E-mail address :
[email protected] Department of Mathematical Sciences, University of South Africa, P.O. Box 392, Pretoria, 0003,, South Africa.
E-mail address :
[email protected]
Department of Mathematics, Rajah Serfoji Govt. College, Thanjavur613005, Tamilnadu, India.
E-mail address : nrajesh
[email protected] 