L. ELVINA MARY 1, E.L MARY TENCY 2 1. INTRODUCTION. 6. a generalized α-closed set (briefly gα - closed) if

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1 A Study On (I,J) (Gs)* Closed Sets In Bitopological Spaces L. ELVINA MARY 1, E.L MARY TENCY 2 1 Assistant professor in Mathematics, Nirmala College for Women (Autonomous), 2 Department of Mathematics, PPG Institute of Technology, Coimbatore, India. Abstract--In this dissertation, we introduce new class of (i, j) (gs)*- sets in Bitopological spaces. Properties of these sets are investigated and we introduce four new spaces namely, (i, j) T (gs)* spaces, (i, j) - g T (gs)* spaces, (i, j) (gs) T (gs)* spaces, (i, j) ψ* T (gs)* spaces. Key words: (i, j) (gs)*- sets, (i, j) T (gs)*, (i, j) - g T (gs)* s, (i, j) (gs) T (gs)*, (i, j) Ψ* T (gs)* spaces. 1. INTRODUCTION Norman Levine introduced and studied generalized (briefly g-) sets [14] and semi-open sets [13] in 1963 and 1970 respectively. Bhattacharya and Lahiri [4]Introduced and investigated semi-generalized (briefly sg-) sets in Arya and Nour [3] defined generalized semi- (briefly gs-) sets for obtaining some characterizations of s- normal spaces in M.K.R.S Veerakumar [22] introduced the generalized g*- sets in In this paper we study the relationships of (i, j) - (gs) * - sets. We also obtain the basic properties of (i, j) - (gs) * - sets. Norman Levine [14], Bhattacharya and Lahiri [4] and R.Devi et. al.[8] introduced T 1/2 spaces, T b and T d spaces respectively. We now introduce and study four new classes of spaces, namely the class of (i, j) T (gs)* spaces, (i, j) g T (gs)* spaces, (i, j) - (gs) T (gs)* spaces, (i, j) - ψ* T (gs)* spaces. We also study some relationships of these spaces with (i, j) -T 1/2 spaces, (i, j) -T b space and (i, j) T ψ* spaces. 2. PRELIMINARIES Definition 2.1: A subset A of a topological space (X, τ) is called 1. a semi-open set if A cl (int (A)) and a semi set if int (cl(a)) A. 2. a pre-open set if A Int(cl(A)) and a pre- set if cl(int(a)) A. 3. an α-open set if A int(cl(int(a))) and an α set if cl(int(cl(a))) A. 4. a generalized set (briefly g-) if cl (A) U whenever A U and U is open in (X, τ ). 5. a generalized semi- set (briefly gs-) if scl(a) U whenever A U and U is open in (X, τ) 6. a generalized α- set (briefly gα - ) if αcl(a) U whenever A U and U is α-open in 7. a α -generalized set (briefly α g- if αcl(a) U whenever A U and U is open in 8. a generalized semi-pre set (briefly gsp) if spcl (A) U whenever A U and U is regular open in 9. a generalized pre set (briefly gp-) if pcl(a) U whenever A U and U is open in 10. a generalized star set (briefly g*-) if cl (A) U whenever A U and U is g-open in 11. a ψ- if scl(a) U whenever A U and U is sg-open in 12. a ψ*- if scl(a) U whenever A U and U is ψ-open in 13. a g**- if cl(a) U whenever A U and U is g*-open in 14. a α*- if cl(a) U whenever A U and U is α-open in 15. a ω- if cl(a) U whenever A U and U is semi-open in If A is a subset of X with topology τ, then the closure of A is denoted by τ- cl(a) or cl(a). The interior of A is denoted by τ- int(a) and the complement of A in X is denoted by A c. Definition 2.2:A subset A of a bitopological space (X, τ i, τ j) is called 1. a (i,j)-g- if τ j - cl(a) U whenever A U and U is open in τ i. 2. a (i,j)-ω- if τ j - cl(a) U whenever A U and U is semi open in τ i. 3. a (i,j)-g*- if τ j - cl(a) U whenever A U and U is g-open in τ i. 4. a (i,j)-gs- ifτ j -scl(a) U whenever A U and U is open in τ i. 5. a (i,j)-αg- if τ j - αcl(a) U whenever A U and U is open in τ i. 6. a (i,j)-g**- ifτ j - cl(a) U whenever A

2 U and U is g*-open in τ i. 7. a (i,j)-ψ- ifτ j - scl(a) U whenever A U and U is sg-open in τ i. Definition 2.3:A bitopological space (X, τ i, τ j) is called 1. a (i,j)- T 1/2 space if every (i,j)-g- set in it is τ j. 2. a (i,j)- T b space if every (i,j)-gs- set in it is τ j. 3. a (i,j)-t ψ* space if every (i,j)-ψ* set in it is τ j. Example 3.12: Let X = {a, b, c}, Ф X, {a, b}}, Ф X, {a}}.then the set {b} is (1, 2) αg but not (1, 2)-(gs)*. Proposition 3.13: Every (i,j)-(gs)*- set is (i,j)-gα 3. RELATIONSHIPS OF (i, j)-(gs)*- CLOSED SETS WITH SOME OTHER CLOSED SETS We introduce the following definition. Definition 3.1: A subset A of a bitopological space (X, τ i,τ j) is called a (i, j) - (gs)*- if τ j - cl(a) U whenever A U and U is gs-open in τ i. Remark 3.2: By setting τ 1 = τ 2 in definition 3.1 a (i, j) - (gs)*- set is a (gs)* Proposition 3.3: Every τ j subset of (X, τ i, τ j ) is (i, j)-(gs)*- The following example shows that the converse of the above proposition is not true. Example 3.4:Let X ={a, b, c}, Ф X, {c}, {a, c}}, Ф X, {a}}.then the set {b} is (1, 2) (gs)* set but not. Proposition 3.5: Every (i, j)-(gs)*- set is (i, j) g The following example shows that the converse of the above proposition is not true. Example 3.6: Let X = {a, b, c}, Ф X, {a}, {a, b}}, Ф X, {a}}.then the set {c} is (1, 2) g but not (1, 2) - (gs)*-. Proposition 3.7: Every space (i, j)-(gs)*- set is (i, j) - g* - Example 3.8: Let X = {a, b, c}, Ф X, {a}, {a, b}}, Ф X, {a}}.then the set A={c} is (1, 2) g* but not (1, 2)-(gs)* Proposition 3.9: Every (i,j)-(gs)*- set is (i,j)- g** - Example 3.10: Let X = {a, b, c}, Ф X, {a},{a, b}}, Ф X, {a}}. Then the set {c} is (1, 2) g** but not (1, 2)-(gs)*. Proposition 3.11: Every (i,j)-(gs)*- set is (i,j)-αg Example 3.14: Let X = {a,b,c}, Ф X, {a,b}}, Ф X, {a}}.then the set {b} is (1, 2) gα but not (1, 2)-(gs)*. Proposition 3.15: Every (i,j)-(gs)*- set is (i,j)-gs Example 3.16: Let X = {a,b,c}, Ф X, {a}, {a,b}}, Ф X, {a}}.then the set {a, c} is (1, 2) gs but not (1, 2)-(gs)*. Proposition 3.17: Every (i,j)-(gs)*- set is (i,j)-gsp Example 3.18: Let X = {a,b,c}, Ф X, {c}, {a,c}}, Ф X, {a}}.the set {c} is (1, 2) gsp but not (1, 2)-(gs)*. Proposition 3.19: Every (i,j)-(gs)*- set is (i,j)-gp Example 3.20: Let X = {a,b,c}, Ф X, {c}, {a,c}}, Ф X, {a}}.then the set {c} is(1, 2) gp but not (1, 2)-(gs)*. Proposition 3.21: Every (i,j)-(gs)*- set is (i,j)-ω Example 3.22: Let X = {a, b, c}, Ф X, {c}, {a, c}}, Ф X, {a}}.then the set {c} is (1, 2) ω but not (1, 2)-(gs)*. Proposition 3.23: Every (i,j)-(gs)*- set is (i,j)-ψ Example 3.24: Let X = {a,b,c}, Ф X, {a,b}}, Ф X, {a}}.then the set {b} is (1, 2) ψ but not (1, 2)-(gs)*.

3 Proposition 3.25: Every (i,j)-(gs)*- set is (i,j)-ψ* Example 3.26: Let X = {a,b,c}, Ф X, {a,b}}, Ф X, {a}}. Then the set {b} is (1, 2) ψ* but not (1, 2)-(gs)*. Proposition 3.27: Every (i,j)-(gs)*- set is (i,j)- α* All the above results can be represented by following diagram. (i,j)- g (i,j)- g* ז j - (i,j)-g** ( i,j)- αg (i,j)-gs (i,j)-(gs)* (i,j)-gα (i,j)-gsp (i,j)- α* (i,j)-gp ((i,j)- ω (i,j)- ψ* ((i,j)- ψ Where A B represents A implies B and A B represents A does not imply B 4. Basic properties of (i, j) - (gs) * sets Proposition 4.1: Union of any two (i,j)- (gs) * sets is again (i,j)-(gs) *. Proof: Let A and B be (i,j) (gs) * sets. Then A U and B U where U is gs open in τ i.this implies j - cl (A) U and j - cl (B) U. Now, j -cl (A B) = j - cl (A) j - cl (B) U. Hence j cl (A B) U, whenever (A B) U and U is gs open in τ i. Therefore A B is (i,j) (gs) * sets. Remark 4.2: The intersection of two (i,j) (gs) * set need not be (i,j) (gs) * set as seen from the Example 4.3: Let X = {a, b, c}, 1 = {X, Ф, {a}, {b,c}}, 2 = {X, Ф, {b}, {c}, {b,c},{a,c}}. Here A = {a, b} and B = {b, c} are (2, 1) (gs) * sets, but A B = {b} is not a (2, 1) (gs) * Remark 4.4: (1, 2) (gs) * is generally not equal to (2, 1) (gs) * as seen from the following example. Example 4.5: Let X = {a, b, c}, τ 1 = {X, Ф, {a}, {b,c}}, 2 = {X, Ф, {b}, {c}, {b,c},{a,c}}. Here A = {b} (2, 1) (gs) * set, but A = {b} (1,2) (gs) * Therefore (1,2) (gs) * (2, 1) - (gs) *. Proposition 4.6: If A is (i, j)-(gs)* and i gs open, then A is j. Proof: Let A be both (i, j)-(gs)* and i gs open. Then j - cl(a) U whenever A U and U is gs-open in i.but, since A is i gs open and A A, j - cl(a) A. Therefore j - cl(a). Hence A is j. Proposition 4.7: If A is (i, j)-(gs)*, then j - cl(a) \ A contains no nonempty i (gs) set Proof: Let A be (i, j)-(gs)* sets and let F be a i

4 (gs) Such that F j - cl(a) \ A. Therefore F c, since A ε (gs)* c (i, j). We have j - cl(a) F c Therefore F ( j - cl(a)) c. Hence F ( j - cl(a)) j - cl(a)) c = F = Proposition 4.8: If A is an (i, j)-(gs)* set of (X, i j ) such that A B j - cl(a), then B is also (i, j)- (gs)* set of (X, i j ). Definition 5.3: A subset A of a bitopological space (X, i j ) is called a (i, j) (gs) T (gs)* space, if every (i, j)- gs set in it is (i,j)-(gs)*. Definition 5.4: A subset A of a bitopological space (X, i j ) is called a (i, j) ψ* T (gs)* space, if every (i, j)- ψ* set in it is (i,j)- (gs)*. Proposition 5.5: Every (i, j) T b space is an (i, j) (gs)t Proof: Let A be (i, j)-(gs)*. Let U be j gs open set such that B U, then A B U. Therefore j - cl(a) U, since A is (i, j)-(gs)*. B j - cl(a). Hence j - cl(b) j - cl(a) U. Therefore j - cl(b) U. Hence is i, j)-(gs)* Proposition 4.9: For each element x of (X, i j ), {x} is either i gs or X-{x} is (i, j)-(gs)*. cl(a) \ A = which is i gs. Proof: Suppose {x} is not i gs. Then {x} c is not i gs open. Therefore he only i gs open set containing {x} c is X and j cl({x} c ) X. Hence {x} c is (i, j)-(gs)*. Proposition 4.10: If A is (i, j)-(gs)* in (X, i j ), then A is j if and only if j - cl(a) \ A is i gs. Proof: Necessity: If A is j, then j - cl(a) = A. (i.e) j - cl(a) \ A = which is i gs. Sufficiency: If j - cl(a) \ A is i gs. Since A is (i, j)-(gs)*, j - cl(a) \ A contains no non empty i gs Therefore j - cl(a) \ A = and hence A = j - cl(a).therefore A is j. 4. APPLICATIONS OF (i, j)-(gs)* CLOSED SETS Example 5.6: Let X = {a, b, c}, Ф X, {a, b}, {a}}, Ф X, {b, c}, {b}}. Then (X, i j ) is an (i, j) (gs) T (gs)* space. But {c} is (i, j) - (gs) set but not a j. Therefore (X, i j ) is not a (i, j) T b space. Proposition 5.7: Every (i, j) Tψ * space is an (i, j) ψ* T Example 5.8: Let X = {a, b, c}, Ф X, {a, b}, {a}}, Ф X, {b}, {b, c}}. Then (X, i j ) is an (i, j) ψ*t (gs)* space. But {c} is (1, 2) - ψ* set but not 2. Therefore (X, 1 2 ) is not a (i, j) T ψ* space. Proposition 5.9: Every (i, j) T 1/2 space is an (i, j) g T Example 5.10: X = {a, b, c}, Ф X, {c}, {a, c}}, Ф X, {a}}. Then (X, i j ) is an (i, j) g T (gs)* space. But {b} is (1, 2) - g set but not 2. Therefore (X, i j ) is not a (i, j) T 1/2 space. Hence every (i, j) g T (gs)* space need not be an (i, j) T 1/2 space. Proposition 5.11: Every (i, j) T 1/2 space is an (i, j) T As applications of (i, j)-(gs)* sets, four new spaces namely (i, j) - T (gs)* spaces, (i, j) - g T (gs)* spaces, (i, j) (gs) T (gs)* spaces, (i, j) ψ* T (gs)* spaces are introduced. We introduce the following definitions: Definition 5.1: A subset A of a bitopological space (X, i j ) is called a (i, j) - T (gs)* space, if every (i,j)-(gs)* set in it is j. Definition 5.2: A subset A of a bitopological space (X, i j ) is called a (i, j) - g T (gs)* space, if every (i, j)-g set in it is (i,j)-(gs)*. Example 5.12: Let X = {a, b, c}, Ф X, {a}, {a, b}}, Ф X, {a}}. Then (X, i j ) is an (i, j) T (gs)* space. But {c} is (1,2) g but not j -. Therefore (X, i j ) is not (i, j) - T 1/2 space. Proposition 5.13: Every (i, j) α T d space is an (i, j) T Example 5.14: Let X = {a, b, c}, Ф X, {a}, {a, b}}, Ф X, {a}}. Then (X, i j ) is an (i, j) T (gs)* space. But {a, c} is (1,2) αg but not j -. Therefore (X, i j ) is not (i, j) α T d space. All the above results can be represented by following diagram.

5 (i, j) g T (gs)* space (i, j) T ψ* space (i, j) ψ*t (gs)* space (i, j) T b space (i, j) T 1/2 space (i, j) (gs) T (gs)* space (i, j) T (gs)* space (i, j) αt d space Where A B represents A implies B and A B represents A does not imply B REFERENCES [1]. D.Andrijevic, Semi-preopen sets, Mat. Vesnik., 38(1)(1986), [2]. S.P.Arya and T.M.Nour, Characterizations of s-normal spaces, Indian J.Pure. Appl. Math., 21(8)(1990), [3]. P.Bhattacharya and B.K.Lahiri, Semi-generalized sets in topology, Indian J.Math., 29(3) (1987), [4]. N.Bourbaki, General Topology, Part I, Addision-Wesley, Reading, Mass., [5]. J.Dontchev, On generalizing semi-preopen sets, Mem. Fac. Sci. Kochi Univ. Ser.A. Math., 16(1995), [6]. J.Dontchev, On generalizing semi-preopen sets, Mem. Fac. Sci. Kochi Univ. Ser.A. Math., 16(1995), [7]. Elvina Mary.L, Myvizhi.R, A study on (gs)* sets in topological spaces, International journal off Mathematics Trends and Technology-Volume 7 Number 2-March 2014 [8]. Y.Gnanambal, On generalized preregular sets in topological spaces, Indian J. Pure. Appl. Math., 28(3)(1997), [9].J.L.Kelly, General topology, D.Van Nostrand company Inc., Princeton, New Jersey, [10].N.Levine, Semi-open sets and semi-continuity in topological spaces,70(1963), [11]. N.Levine, Generalized sets in topology, Rend. Circ Mat. Palermo, 19(2)(1970), [12]. H.Maki, R.Devi and K.Balachandran, Associated topologies of generalized - sets and -generalized sets, Mem. Fac. Sci. Kochi Univ. Ser.A. Math.,15(1994), [13]. H.Maki, R.Devi and K.Balachandran, Generalized - sets in topology, Bull.Fukuoka Univ. Ed. Part III, 42(1993), [14]. S.R.Malghan, Generalized maps, J.Karnatak Univ. Sci., 27(1982), [15]. O.Njåstad, On some classes of nearly open sets, Pacific J. Math.,15(1965), [16]. N.Palaniappan and K.C.Rao, Regular generalized sets, Kyungpook Math. J., 33(2)(1993), [17]. P.Sundaram, H.Maki and K.Balachandran, Semigeneralized continuous maps Semi- T 1/2 spaces, Bull. Fukuoka Univ. Ed. Part III, 40 (1991), [18]. M.K.R.S.Veera Kumar, ĝ- sets and G L C- functions, Indian J. Math.,Indian J.Math., 43(2) (2001), [19]. M.K.R.S.Veera Kumar, Between sets and g sets, Mem.Fac.Sci.Kochi Univ.Ser.A.Math., 17(1996), [20]. Veronica Viayan, P.Aparna, A study on (i, j)-α* sets in bitopological spaces. International Journal of Computer Application. International Journal of Computer Application- Volume 4 (July August-2013). pg [21]. Veronica Viayan, R. Nithya kalyani, A study on (i, j) ψ*,(i, j) ψ, i, j) ψ* sets in bitopological spaces; International Journal of Computer Application-Volume 4 (July August-2013). pg