α-OPEN SETS BASED ON BITOPOLOGICAL SEPARATION AXIOMS

SOOCHOW JOURNAL OF MATHEMATICS

Volume 33, No. 3, pp. 375-381, July 2007

(1, 2)α-OPEN SETS BASED ON BITOPOLOGICAL SEPARATION AXIOMS BY SAEID JAFARI, M. LELLIS THIVAGAR AND S. ATHISAYA PONMANI

Abstract. The aim of this paper is to present and study some new separation axioms using the (1, 2)α-open sets in bitopological spaces.

1. Introduction In 1961, Njastad [16] offered and to some extent investigated some classes of nearly open sets called α-open sets. In 1963, Levine [17] introduced semiopen sets in topological spaces and these semi-open sets were used to define three new separation axioms called semi-T0 , semi-T1 and semi-T2 by Maheswari and Prasad [11] in 1975. Another set of new separation axioms, αTi , i = 0, 1 were characterized by Maki et al. [12] in 1993. Davis [13], in his paper discussed the regularity axioms for topological spaces. In bitopological spaces pairwise-R0 and pairwise-R1 spaces were seen in [5] and pairwise-R0 spaces have been studied in Misra and Dube [14]. Reilly [5] proposed a weaker form of pairwise-R1 property in bitopological spaces. Recently Veerakumar [19] used α-open sets to define weakly α-R0 spaces. The purpose of this present paper is to define some new separation properties by using the (1, 2)α-open sets, (1, 2) semi-open sets and (1, 2) pre-open sets in bitopological spaces and to investigate the relationships between them. Received February 15, 2006. AMS Subject Classification. 54C55. Key words. ultra α-T0 , ultra α-T1 , ultra α-R0 , ultra α-R1 , The third author wishes to acknowledge the support of University Grants Commission, New Delhi, India, FIP-X Plan. 375

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2. Preliminaries Throughout this paper by X, we mean bitopological space (X, τ1 , τ2 ). A subset A of X is τ1 τ2 -open ([9]) if A ∈ τ1 ∪ τ2 and τ1 τ2 -closed if its complement in X is τ1 τ2 -open. The τ1 τ2 -closure of A is denoted by τ1 τ2 -cl (A) and ∩ τ1 τ2 -cl (A) = {F : A ⊂ F and F c is τ1 τ2 -open}. Definition 2.1.([3]) A topological space (X, τ ) is said to be R0 ([3]) if for each G ∈ τ , x ∈ G implies τ -cl ({x}) ⊂ G and it is R1 if for each pair of points x, y ∈ X such that τ -cl ({x}) 6= τ -cl ({y}) there are disjoint open sets U and V such that x ∈ U , y ∈ V . Definition 2.2.([15]) A bitopological space X is pairwise R0 ([15]) if for each G ∈ τi , x ∈ G implies τj -cl ({x}) ⊂ G for i, j = 1, 2, i 6= j and pairwise R1 (MN pairwise R1 ) if for x, y ∈ X such that τj -cl ({x}) 6= τi -cl ({y}), there is a τi -open set U and a τj -open set V such that x ∈ U , y ∈ V and U ∩ V = ∅, for i, j = 1, 2 and i 6= j. Definition 2.3.([5]) A bitopological space X is pairwise-R1 ([5]) if for x, y ∈ X such that x ∈ / τi -cl ({y}), there is a τi -open set U and a τj -open set V such that x ∈ U , y ∈ V and U ∩ V = ∅, i, j = 1, 2, i 6= j. Definition 2.4.([9]) A subset A of X is called (i) a (1, 2)α-open set if A ⊂ τ1 -int(τ1 τ2 -cl (τ1 -int(A))), (ii) a (1, 2) semi-open set if A ⊂ τ1 τ2 -cl (τ1 -int(A)), (iii) a (1, 2) pre-open set if A ⊂ τ1 -int(τ1 τ2 -cl (A)). The family of all (1, 2)α-open (resp. (1, 2) semi-open, (1, 2) pre-open) sets of X is denoted by (1, 2)αO(X) (resp. (1, 2)SO(X), (1, 2)P O(X)). The complement of a (1, 2)α-open (resp. (1, 2) semi-open, (1, 2) pre-open) set in X is called (1, 2)αclosed (resp. (1, 2) semi-closed, (1, 2) pre-closed) set in X. (1, 2)αcl (A) (resp. (1, 2)scl(A), (1, 2)pcl(A)) denotes (1, 2)α-closure (resp. (1, 2) semi-closure, (1, 2) pre-closure) of A in X. Theorem 2.5.([9]) A subset A of a bitopological space X is a (1, 2)α-open set if and only if A is a (1, 2) semi-open set and a (1, 2) pre-open set. Equivalently, (1, 2)αO(X) = (1, 2)SO(X) ∩ (1, 2)P O(X) ([9]).

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Definition 2.6.([10]) A bitopological space X is said to be ultra α-T2 (resp. ultra semi-T2 ) ([10]) if for x, y ∈ X, x 6= y, there exist U , V in (1, 2)αO(X) (resp. (1, 2)SO(X)) such that x ∈ U , y ∈ V , U ∩ V = ∅. ∩

Definition 2.7.([19]) A topological space (X, τ ) is weakly α-R0 ([9]) if α cl({x}) = ∅.

x∈X

3. Ultra α-T0 and Ultra α-T1 Spaces Definition 3.1. A bitopological space X is said to be ultra α-T0 (resp. ultra semi-T0 ) if for x, y ∈ X, x 6= y, there exists U ∈ (1, 2) αO(X) (resp. (1, 2)SO(X)) such that U contains only one of x and y but not the other. Example 3.2. Let X = {a, b, c}, τ1 = {∅, {b}, {c}, {b, c}, X} and τ2 = {∅, {b}, {a, b}, {b, c}, X}. Then (1, 2)αO(X) = {∅, {b}, {c}, {b, c}, X}, (1, 2)SO(X) = {∅, {b}, {c}, {a, b}, {b, c}, X}. It is clear that X is ultra α-T0 and ultra semi-T0 . Definition 3.3. A bitopological space X is said to be ultra α-T1 (resp. ultra semi-T1 ) if for x, y ∈ X, x 6= y, there exist U, V ∈ (1, 2)αO(X) (resp. (1, 2)SO(X)) such that x ∈ U , y ∈ V but y ∈ / U and x ∈ / V. Example 3.4. Let X= {a, b, c}, τ1 = {∅, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, X} and τ2 = {∅, {b, c}, X}. Then (1, 2)αO(X) = {∅, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, X}. Therefore, X is ultra α-T1 . Example 3.5. Let X = {a, b, c, d}, τ1 = {∅, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a, b, c}, X} and τ2 = {∅, {a}, X}. Then (1, 2)SO(X) = {∅, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a, d}, {b, d}, {c, d}, {a, b, c}, {a, b, d}, {b, c, d}, {a, c, d}, X}. The space X is ultra semi-T1 . Remark 3.6. (i) Every ultra α-T1 space is ultra α-T0 . (ii) Every ultra semi-T1 space is ultra semi-T0 . Example 3.2 shows that the converse of Remark 3.6 is not true. Remark 3.7. Every ultra α-Ti space is ultra semi-Ti , i = 0, 1. But the

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converse is not true because every (1, 2)α-open set is (1, 2) semi-open set but not the converse. Theorem 3.8. A bitopological space X is ultra α-T1 if and only if {x} is (1, 2)α-closed in X for every x ∈ X. Proof. If {x} is (1, 2)α-closed in X for every x in X, for x 6= y, X \ {x}, X \ {y} are (1, 2)α-open sets such that y ∈ X \ {x} and x ∈ X \ {y}. Therefore, X is ultra α-T1 . Conversely, if X is ultra α-T1 and if y ∈ X \ {x} then x 6= y. Therefore, there exist (1, 2)α-open sets Ux , Vy in X such that x ∈ Ux but y ∈ / Ux and y ∈ Vy but x ∈ / Vy . Let G be the union of all such Vy . Then G is a (1, 2)αopen set and G ⊂ X \ {x} ⊂ X. Therefore, X \ {x} is a (1, 2)α-open set in X. Remark 3.9. (i) Every ultra α-T2 space is ultra α-T1 and hence ultra α-T0 . (ii) Every ultra semi-T2 space is ultra semi-T1 and hence ultra semi-T0 . 4. Ultra α-Ri and Ultra Semi-Ri Spaces, i = 0, 1 Definition 4.1. A bitopological space X is said to be (i) Ultra α-R0 (resp. ultra semi-R0 ) if (1, 2)αcl ({x}) ⊂ U (resp. (1, 2)scl({x}) ⊆ U ) whenever x ∈ U ∈ (1, 2)αO(X) (resp. x ∈ U ∈ (1, 2)SO(X)). (ii) Ultra α-R1 (resp. ultra semi-R1 ) if for x, y ∈ X such that x ∈ / (1, 2)αcl ({y}) (resp. (1, 2)scl ({y})), there exist disjoint (1, 2)α-open (resp. (1, 2) semiopen) sets U , V in X such that x ∈ U and y ∈ V . (iii) Weakly ultra α-R0 (resp. weakly ultra semi-R0 ) if ∩ (1, 2)scl ({x})) = ∅. x∈X ∩ (iv) Weakly ultra pre-R0 if (1, 2)pcl ({x}) = ∅.

∩

(1, 2)αcl ({x}) (resp.

x∈X

x∈X

Example 4.2. Let X = {a, b, c, d}, τ1 = {∅, {a}, {b}, {a, b}, X} and τ2 = {∅, {a}, X}. (1, 2)αO(X) = {∅, {a}, {b}, {a, b}, {a, b, c}, {a, b, d}, X}. (1, 2)αcl({a}) = {a, c, d} 6⊂ {a} ∈ (1, 2)αO(X) for a ∈ {a}. In this example, X is weakly ultra α-R0 but not ultra α-R0 .

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Remark 4.3. In example 3.5, (1, 2)αO(X) = {∅, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a, b, c}, X}. Here the space X is ultra semi-R0 and weakly ultra semi-R0 , since (1, 2)scl ({a}) = {a}, (1, 2)scl ({b}) = {b}, (1, 2)scl ({c}) = {c} and (1, 2)scl ({d}) = {d}. This example shows that an ultra semi-R0 space need not be an ultra α-R0 space. But an ultra α-R0 space is ultra semi-R0 . Proposition 4.4. If X is ultra α-R1 , then it is ultra α-R0 . Proof. Let X be ultra α-R1 , U ∈ (1, 2)αO(X) and x ∈ U . For each y ∈ X \ U, x ∈ / (1, 2)αcl ({y}). Therefore, there exist (1, 2)α-open sets Ux , Vy in X such that x ∈ Ux and y ∈ Vy such that Ux ∩Vy = ∅. Let A = U {Vy : y ∈ X \U } then X \ U ⊂ A and x ∈ / A which is a (1, 2)α-open set so that (1, 2)αcl ({x}) ⊂ X \ A ⊂ U . Therefore, X is ultra α-R0 . Rremark 4.5. If X is ultra semi-R1 , then it is ultra semi-R0 . Theorem 4.6. A bitopological space X is ultra α-T2 if and only if it is ultra α-T1 and ultra α-R1 . Proof. If X is ultra α-T2 then it is ultra α-T1 by Remark 3.9. We prove X is ultra α-R1 . If x, y ∈ X such that x ∈ / (1, 2)αcl ({y}) then x 6= y. Therefore, there exist disjoint (1, 2)α-open sets U , V in X such that x ∈ U and y ∈ V . Therefore, X is ultra α-R1 . Conversely, if X is ultra α-T1 and ultra α-R1 and x, y ∈ X such that x ∈ / (1, 2)αcl ({y}) there exist disjoint (1, 2)α-open sets U , V in X such that x ∈ U and y ∈ U . Since X is ultra α-T1 , (1, 2)αcl ({y}) = {y} by Theorem 3.8. Thus for x 6= y and V ∈ (1, 2)αO(X) such that x ∈ U and y ∈ V , U ∩ V = ∅. Therefore, X is ultra α-T2 . Corollary 4.7. A bitopological space X is ultra semi-T2 if and only if it is ultra semi-T1 and ultra semi-R1 . Proposition 4.8. Every weakly ultra α-R0 space is weakly ultra semi-R0 and weakly ultra pre-R0 . Proof. If X is weakly ultra α-R0 ,

∩ x∈X

(1, 2)αcl ({x}) = ∅.

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SAEID JAFARI, M. LELLIS THIVAGAR AND S. ATHISAYA PONMANI

Therefore,

∩

∩

(1, 2)scl ({x}) = ∅ and

x∈X

(1, 2)pcl ({x}) = ∅.

x∈X

Theorem 4.9. A space X is weakly ultra α-R0 if and only if (1, 2)αker({x}) ∩ 6 X for each x ∈ X where (1, 2)αker ({x}) = {U : x ∈ U ∈ (1, 2)αO(X)}. = Proof. Necessity. If there is some xo ∈ X with (1, 2)αker ({xo }) = X, then X is the only (1, 2)α-open set containing xo . This implies that xo ∈ (1, 2)αcl ({x}) ∩ for every x ∈ X. Hence (1, 2)αcl ({x}) 6= ∅, a contradiction. x∈X

∩

Sufficiency.

If X is not weakly ultra α-R0 , then choose some xo ∈

(1, 2)αcl ({x}).

So xo ∈ (1, 2)αcl ({x}) for each x ∈ X.

This implies

x∈X

that every (1, 2)α-open set containing xo contains every point of X. Hence (1, 2)αker ({xo }) = X, a contradiction. 5. Comparison From Example 3.2, Remarks 3.6, 3.7, 3.9, 4.3, 4.5, Theorems 4.6 and Proposition 4.4 we have the following diagram.

Where A → B (resp. A 9 B) represents that A imlies B (resp. A does not imply B), more over 1. Ultra α-T2

2. Ultra α-T1

3. Ultra α-T0

4. Ultra α-R1

5. Ultra semi-T1

6. Ultra α-R0

7. Ultra semi-R1

8. Ultra semi-T0

9. Ultra semi-R0 .

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References [1] S. P. Arya and T. M. Nour, Characterizations of S-normal spaces, Indian J. Pure Appl. Math., 21(1996), 1083-1085. [2] C. E. Aull and W. J. Thron, Separation axioms between T0 and T1 , Indag. math., 24(1962), 26-37. [3] A. S. Davis, Indexed systems of neighborhoods for general topological spaces, Amer. Math. Monthly, 68(1961), 886-893. [4] K. K. Dube and B. N. Patel, Semi-open sets with the axioms between T0 and T1 , Jour. National Sci. (Korea), 3:1(1993), 1-10. [5] Ivan L. Reilly, On essentially pairwise Hausdorff spaces, Rendiconti del Circolo Math. Ann. 1976, Serie II - TOMO XXV. [6] D. S. Jankovic and I. L. Reilly, On semi-separation properties, Indian J. Pure and Appl. Math., 16:9(1985), 957-964. [7] Julian Dontchev, On some separation axioms associated with the α-topology, Mem. Fac. Sci. Kouchi. Univ. (Math), 18(1997), 31-35. [8] J. C. Kelly, Bitopological spaces, Proc. London Math. Soc., 13(1963), 71-89. [9] M. Lellis Thivagar and O. Ravi, On stronger forms of (1, 2) quotient mappings, Int. J. of Mathematics, Game theory and Algebra - accepted. [10] M. Lellis Thivagar and S. Athisaya Ponmani, Note on some new bitopological separation axiosm, Proc. of the National Conference in Pure and Applied Mathematics, January 2005, 28-32. [11] S. N. Maheswari and R. Prasad, Some new separation axioms, Ann. Soc. Sci. Bruxelles, 89(1975), 395-402. [12] H. Maki, R. Devi and K. Balachandran, Generalized α-closed sets in Topology, Bull. Fukuoka Univ. Ed. Part III, 42(1993), 13-21. [13] A. S. Mashhoor, M. E. Abd. El. Monsef and El. Deep, On pre-continuous mappings and weak pre-continuous mappings, Proc. Math. Phy. Soc. Egypt, 53(1982), 47-53. [14] D. N. Misra and K. K. Dube, Pairwise R0 space, Ann. De. La. Soc. Sci. de. Bruxelles, T-87, 1(1973), 3-15. [15] M. G. Murudeshwar and S. A. Naimpally, Quasi-Uniform Topological Spaces, Noordhoff, Groningen, 1966. [16] O. Njastad, On some classes of nearly open sets, Pacific, J. Math., 15(1965), 961-970 [17] Norman Levine, Semi-open sets and semi-continuity in topological spaces, Amer. math. Monthly, 70(1963), 36-41. [18] I. L. Reilly, On bitopological separation properties, Nanta Mathematica, 5(1972), 14-25. [19] M. K. R. S. Veera Kumar, Some separation properties using α-open sets, Indian J. Math., 40:2(1998), 143-147. College of Vestsjaelland South, Herrestraede 11, 4200, Slagelse, Denmark. E-mail: [email protected] Department of Mathematics, Arul Anandar College, Karumathur, Madurai Dt.-625 514, Tamil Nadu, India. E-mail: [email protected] Department of Mathematics, Jayaraj Annapackiam College for Women, Periyakulam, Theni Dt. PIN- 625 601, Tamil Nadu India. E-mail: [email protected]