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Sequential Bitopological spaces K.Chandrasekhara Rao and T.Indra Key words: Pairwise sequential separation axioms, pairwise sequential weak Hausdorff space: Mathematics Subject Classification: 54E55 The present work is a continuation of [2]. Introduction The notion of a sequential set and a sequential topological space is introduced in [2] which we explain first. Let X be non empty set. Any sequence An of subsets of X is called a sequential set in X. For each fixed n, An is called the nth component of An . The sequential sets are denoted by A(s), B(s), C(s) and so on. The following operations hold A(s) B(s) = An Bn n 1 A(s) B(s) = An Bn n 1 An is the empty sequential set if An = n An is the universal sequential set if An = X n Department of Mathematics, Mohamed Sathak Engineering College, Kilakarai-623 806. (T.N), India. Department of Mathematics, Seethalakshmi Ramaswamy College, Tiruchirapalli - 620 001.(T.N), India. 1 The empty sequential set in denoted by (s) The universal sequential set is denoted by X(s) A(s) B(s) if An Bn n A(s) = B(s) where A(s) B(s) and B(s) A(s). The complement As c is the set X(s) – A(s) = X An n 1 Let N be the set of all positive integers. A sequential set p(s)= Pn is called a sequential point in X if a non empty set M N and point x X such that Pn = x for n M = for n N M. The point x is called the support and M is called the base of the sequential point. The sequential point with support x and base M is denoted by (x, M). If M is the singleton n, the sequential point (x, M) is called a simple sequential point and is denoted by (x, n). The sequential point (x, N) is called a complete sequential point. Let p = (x, M) p A(s) if (x, M) A(s). p wA (s), that is p belongs to weakly to A(s) if x An for atleast one n M. We extend the results of [2] to the bitopological space and discuss some other results. 2 Definition Let X be a non empty set. A sequential topology on X is a collection of sequential sets in X with the following properties. (i) (s) (ii) X(s) (iii) Arbitrary union of members of (iv) Finite intersection of members of is a member of (X, called is a member of ) is called a sequential topological space. The members of are - open sequential sets. (X, 1, 2), where 1 and 2 are sequential topologies on X, is called a sequential bitopological space. Any sequential open set G containing a sequential point p is called a neighbour hood of p. Any sequential set F(s) = Fn n 1 is called a sequential closed set if its complement X(s) F(s) is open. Let A(s) be a sequential set. The sequential set As = {F(s): F(s) is a closed sequential set with F(s) A(s)} is called the closure of A(s) A sequential open set A(s) is called a weak neighbour hood of a sequential point p if p wA(s). 3 Definition Let D be a topology on X. The collection of sequences of D – opensets forms a sequential topology on X. This is called the sequential topology generated by the topology D on X. It is denoted by (D). Definition is a sequential topology X. Let Dn ( ) be the collection of the nth component of the sequential sets in . Dn ( ) forms a topology on X. For each fixed n= 1, 2 … … Dn ( ) is called the nth component topology on X. Suppose Separation axioms are discussed in [1]. Main results Theorem 1 (X, D1, D2) is pairwise To (X, (D1) (D2)) is pairwise To Proof Suppose (X, D1, D2) is pairwise To Let p = (x, P) and q= ( y, Q) be two distinct sequential points Case (i) Suppose x y . V D1 such that x V and y V p V(s) and q w V(s) where V(s) = Vn and Vn = V, n = 1, 2, ……. (X, (D1), (D2)) is pairwise To 4 Case (ii) Suppose that x = y V D1 such that x V Choose i P-Q. Then p w V(s) but q w V(s) where V(s) = {Vn} and Vi = V Vn = n = 1, 2 … … (n i) {X, (D1), (D2)} is pairwise To Conversely suppose that (X, (D1), (D2) is pairwise To Let x, y X. Then V(s) (D1) such that (x,1) V(s) and (y, 1) V(s) where V(s) = Vn n 1 x V1 and y V1 (X, D1, D2) is pair wise To 5 Theorem 2 (X, 1 , 2) X , Dn 1 , Dn 2 is pair wise To its component space are also pairwise To Proof Let x and y be two distinct points of X. The for any n N, the simple sequential points p = (x, n) and q= ( y, n) are distinct since (X, 1, 2) is pairwise To, a open 1 – sequential set V(s) = Vn n 1 such that p w V (s) and q w V(s) x Vn and y Vn (X, Dn ( 1), Dn ( 2)) is pairwise To. Definition (X, 1, 2) is called pairwise To if for every pair of distinct sequential points p and q, a 1 – open sequential set V(s) containing weakly p but not q. 6 Definition 1 , 2)is pairwise T1 if for every pair of distinct sequential points p and q a 1 – open sequential set U(s) and a 2 – open sequential set V(s) such that p w U(s) and q w U(s) A sequential bitopological space (X, q w V(s) and p w V(s) Note Every pairwise T1 – space is pairwise To Theorem 3 1, 2) is pairwise T1 every pair of sequential points pand q in X such that p is 2 – closed and q is 1 – closed. (X, Proof Step 1 1, 2) is pairwise T1 – space. Let p and q be any pair of distinct sequential points. Then a 2 – open sequential set V(s) such that q Suppose that (X, w V(s) and p w V(s) q is not a 2 - limit point of p p is a 2 – closed sequential set similarly we can show that q is 1 – closed sequential set. 7 Step 2 Suppose that each pair of sequential points p and q is such that p is closed and q is 2- 1 closed in X is a 2 - closed sequential set. Let p and q be any two distinct sequential points in X. U(s) = X(s) - q and Consider V s X (s) p . Then p w U (s), q w U (s) and q w V S , p w V S Since q is 1 - closed U(s) in 1 - open Since p is 2 - closed V(s) in 2 - open Theorem 4 (X, D1, D2) in pair wise T1 (X, (D1), (D2) is pairwise T1 Proof is similar to that of Theorem 1. Theorem 5 (X, 1, 2) is T1 its component spaces (X, Dn ( 1), Dn ( 2)) are also pair wise T1 8 Definition A sequential bitopological space (X, 1 , 2) is said to be a Pairwise Hausdorff or T2 - space if for any two distinct sequential points p and q 1 open sequential set U(s) and a 2 - open sequential set V(s) such that p w U (s), q w V (s) and p w 1 Cl V s q w 2 cl U ( s) Note A pair wise T2 - space is pairwise T1 Theorem 6 The following conditions are equivalent. (6.1) The space (X, 1, 2) is pairwise Hausdorff (6.2) For any two distinct sequential points p and q sequential set G(s) and a a 1 - open 2 - open sequential set H(s) such that p G(s) , q w H (s) Gs H s s and a 1 - open sequential set D(s) and a 2 - open sequential set E(s) such that p w D(s), q E (s), D(s) E (s) (s) 9 Proof Step 1 Suppose that 6.1 is true X ,1, 2 is pairwise Hausdorff Let p and q be two distinct sequential points in X. Then a 1 - open sequential set U (s ) and a 2- open sequential set V(s) such that p w U S , q w V S , p w 1 clV S , q w 2 clU S p w 1 Cl V s p X (s) 1clV (s) G(s) say, Then G(s) and H(s) = V(s) and are the required for 6.2 Similarly D(s) = U(s) E(s) = X(s) - 2 cl U (s) have the properties required on (6.2) Hence (6.2) holds. Step 2 Suppose (6.2) holds consider U(S) = G(s) D(s) and V(s) = H(s) E(s) U(s) is 1 - open sequential set and V(s) is 2 - open sequential set such that p w U (s), q w V (s) and p w 1 Cl V s and q w 2 Cl U s (X , 1, 2 ) is pairwise Hausdorff. 10 Definition A space (X, 1 , 2 ) is said to be pairwise weak Hausdorff if for any two distinct sequential points p and q there exists 1 open sequential set U(s) and a 2 - open sequential set V(s) such that p w U (s), q w V (s) and U(s) V(s) = (s) Note ( X , 1 , 2 ) is pairwise Hausdorff ( X , 1, 2 ) is pairwise weak Hausdorff Theorem 7 The following assertions are equivalent. (7.1) ( X , 1, 2 ) is sequential pairwise Hausdorff (7.2) p 2 cl N (s), N (s) is a 1 - open neighbour hood of p } for each sequential point p in X. Proof Step 1 Suppose that (7.1) bolds, Then ( X , 1, 2 ) is pair wise Hausdorff.. Let p be any sequential point in X. Let q be a simple sequential point in X. Let q be a simple sequential point distinct from p. Since ( X , 1 , 2 ) is pairwise Hausdorff , a 2 open sequencial set V(s) such that q V(s) and p w 1 Cl V s p X (s) 1cl V (s) U (s) say 11 2 clU (s) 1 clV ( s) 2 clU (s) X (s) 1 clV (s) X (s) V (s) q 2 clU (s) p { 2 clN (s) : N (s) is a 1 - open n bd of p } (7. 2) holds Step 2 Suppose that (7.2) holds. Let pand q be two distinct sequential points in X. By (7.2) a 1 - open nbd S(s) of p such that q 2 clS (s) Put V S X S 2 clS s Then q w V(s), p w 2 clV (s) Also V(s) is 2 - open Similarly a 1 - open sequential set U(s) such that p w U (s) and q w 1 clU (s) X , 1 , 2 in pairwise Hausdorff 12 Theorem 8 (X, D1 ,D2) is pairwise Hausdorff (X, (D1) , 2 (D2) is paiwise Hausdorff. Proof is obvious. Theorem 9 ((X, 1 , 2) is pairwise weakly Hausdorff the component spaces (X, Dn (1), Dn ( 2)) are Hausdorff for n = 1, 2, … … Proof is obvious. Definition A sequential bitopological space (X, 1, 2) is said to be pairwise regular if for any sequential point p and a 1 - closed sequential set F(s) with p F(s) there exist 1 - open sequential set U(s) with p w U(s) and 2 – open sequential set V(s) with F(s) w V(s) such that p 1 – cl V(s) and F(s) X (s) 2 –cl U(s) 13 Theorem 10 (X, 1, 2 ) is pairwise regular its components (X, Dn (1), Dn(2)) are pairwise regular. Proof Let F be a Dn (1) - closed in (X, Dn (1), Dn (2)). Let x X be any point with x F . Put Fn = F n Then F(s) = Fn n 1 be a 1 – closed sequential set. Let p = (x, n) . Then p F (s ) (Since X, 1, 2) is regular, a 1 - open sequential set U(s) = { Un } such that p w U (s) and F (s) X (s) 2 clU (s) V (s) Vn say Then x U n , F Vn , U n Vn U n Dn (1 ) and Vn Dn ( 2 ) X , Dn 1 , Dn 2 are pairwise regular for n = 1, 2 … … Theorem 11 X , D1, D2 is pairwise regular the generated sequential bitopological space X , Dn 1 , Dn 2 is pairwise regular. 14 Definition A Sequential bitopological space X , 1, 2 is said to be a weak Hausdorff space, if for any two distinct Sequential points p and q a Sequential 1 - open weak neighbourhood U(S) of p and a sequential 2 open weak neighbourhood of V(s) of q such that U s V s = s Definition A sequence of sequential points xn is said to converge to a sequential point p if every weak neighbourhood of p contains x n eventually. This situation is represented by xn p . Theorem 12 Let X ,1, 2 be a pairwise sequentially weak Hausdorff space. If xn p w.r.t. 1 and xn q w.r.t. 2 then p = q. Proof Assume that p q . But X , 1, 2 is pairwise weakly Hausdorff Hence a sequential 1 - open weak neighbourhood U (s) of p and a sequential 2 - open weak neighbourhood V(s) of q such that U(s) V(s) = (s) Now xn p w.r.t. 1 Hence xn U (s) n n1 for some n1 15 xn q w.r.to 2 . So xn V s , n n2 , for some n2 . Take m = max (n1,n2). Then xm U (s) and xm V(s), a contradiction. This contradiction shows that p = q. Theorem: 13 Every pairwise sequentially weak Hausdorff space is bi- T1 Proof: Let X ,1, 2 be a pairwise sequentially weak be a pair of distinct sequential points in X. Take with respect to 1 . Consequently Hausdorff space. Let p,q pn p n . Then pn p But X is pairwise sequentially Hausdorff. So, pn q. a 2 - weak nbd U s of p such that q U s . Similarly by considering the sequence qn q n , a 1 - weak nbd V(s) of q such that p V s . Hence X ,1, 2 is a bi- T1 space. Theorem 14 The property of being a pairwise sequential weak Hausdorff space is a hereditary property. Proof Let ( Y , 1 y , 2 y ) be a subspace of X , 1, 2 . Let p q in Y. Then p q in X . 16 But X , 1, 2 is pairwise sequential Hausdorff space. Hence a weak 1 - open neighbourhood U(s) of p and weak 2 open neighbourhood V(s) of q such that U(s) V(s) = (s). Take G(s) = U (s) Y and H(s) = VS Y. Then G(s) is weakly 1 y - open neighbourhood of p and H(s) is a weakly 2y open neighbourhood of q such that G (S) H(s) = (s). Hence ( Y , 1 y , 2 y ) is pairwise sequential Hausdorff space. Theorem 15 X ,1, 2 is weakly 1 regular with respect to 2 if for any sequential point p and any 1 – closed sequential set F(s) with p F(s) there exists 1 – open set U(s) with p w U(s) and a 2 open set V(s) such that F(s) w V(s) and U(s) V(s) = (s). Similarly pairwise weakly 2 - regular with respect to 1 is defined. If X is weakly 1 - regular with respect to 2 and vice versa, X is called pairwise weakly regular. A pairwise weakly regular, pairwise T1 space is called a pair wise weak T3 – space. 17 Definition X ,1, 2 is pairwise weakly normal if for any 1 - open sequential set A(s) and 2 - open sequential set B(s) with A(s) B(s)= (s). U( s) with A (s) w U (s) and 1 -open set 2 - open set V(s) with B (s) w V(s) such that U(s) V(s) = (s). A pair wise weakly normal, pairwise T1 - space is called a pairwise weakly T4 - space. Note: Pairwise weakly T4 space pairwise weakly T3 – space. Theorem:16 Product of an arbitrary family of pairwise sequentially weak Hausdorff spaces is a sequentially weak Hausdorff space. Proof: Let I be an index set. Let X i ,1i , 2i :i I be a family of pairwise weak Hausdorff spaces. Let and X ,1, 2 be their product space. Let pn p with respect to 1 pn q with respect to 2 . Then pni pi with respect to 1i and pni qi with respect to 2i for each i I . 18 But Therefore X i ,1i , 2i is a pairwise weak Hausdorff space for each iI . pi qi i . Consequently p q . Hence X ,1, 2 is a pairwise weak Hausdorff space. Theorem 17 The property of being a sequentially weak Hausdorff space is preserved under one-to-one, onto and pairwise open maps. Proof: Let f : X ,1, 2 Y , 1, 2 have the properties of the hypothesis. Let qn q with respect to 1 and q n s with respect to 2 . So But f is onto. pn in X such that f pn qn n . Also p and r in X such that that f p q , f r s . Thus f pn f p with respect to 1 and f pn f r with respect to 2 . But f is pairwise open and one-to-one . Hence p n p with respect to 1 , p n r with respect to 2 . X ,1, 2 is pairwise weakly Hausdorff. Hence p r . Consequently f p f r . Therefore q s . Thus, Y , 1, 2 is a pairwise But weak Hausdorff space. 19 Theorem 18 Let X ,1, 2 be a bitopological space. respect to 2 for each point 1 - open set Then 1 is weakly regular with x X , and every 1 open set G containing x, a H such that x H 2 clH G . Proof: Let G s be a 1 open set and let x Gs . Then x X G s . Also X G is 1 closed. By hypothesis, a 1 open set U s and 2 V s such that x U s , X G V s ,U s V s s . Thus U s X V s so open set that 2 clU s 2 cl X V s X V s Gs . Hence U s is a 1 open set such that x U s 2 cl U s Gs . Theorem 19 Every subspace of pairwise weakly regular space is pairwise weakly regular. Proof: X ,1, 2 be a pairwise weakly regular space and let Y , 1, 2 be a subspace of X , 1, 2 . Let y Y and let U s 2 such that y U . . Let 20 Then U s U * s Y , where pairwise weakly regular U * 2 . Since y U * and X , 1 , 2 is a 2 - open set V * such that y V * s 1 clV * s U * . Let V s V Then * s Y . V s 2 and y V s 1 cl V s 1 cl V * s Y Y 1 clV * s 1 cl Y Y 1 cl V * s Y U * s Y U Thus a V s 2 such that y V s 1 cl V s U . Y , 1, 2 is pairwise weakly regular. 21 Theorem 20 Product of an arbitrary family of pairwise weakly regular spaces is pairwise weakly regular. Proof: Let X ,1 , 2 : be a family of pairwise weakly regular spaces. Let X , 1 , 2 Let x x X . be their producct space. Let U s be a 1 open set containing x. Then U s U , where U 1 for each and U s X for all except finitely ’s say 1,...., n . Since x U for each , a 1 - open set V such that x V 2 cl V U . Let V V , such that V X except for 1 ,...., n . Then V 1 and x V 2 cl V 2 cl V U U . Thus there exists a 1 -open set V such that x V 2 cl V U . Hence X ,1, 2 is pairwise weakly regular. 22 Theorem 21 Every pairwise closed, pairwise continuous image of a pairwise normal space is pairwise normal. Proof: X ,1, 2 be a pairwise f : X ,1, 2 Y , 1, 2 be a pairwise Let mapping. is 2 normal space. closed, pairwise Let A and B be two disjoint subsets of Y where A is - closed. Then Let continuous 1 -closed and B f 1 A is 1 - closed and f 1 B is 2 -closed. Also A B f 1 A B f 1 f 1 A f 1 B . But X is pairwise normal. Hence disjoint sets G A and GB such that f 1 A G A , f 1 B GB , G A is 1 - open GB is 2 - open. Let * * Then G A GB and , A G *A , B GB* . Also G*A Y f X G A GB* Y f X GB . Hence Hence G*A y : f 1 y G A and GB* y : f 1 y GB . Y , 1, 2 is pairwise G *A is 2 - open and G B* is 1 -open. normal. 23 Theorem 22 Every bi-closed subspace of a pairwise normal space is pairwise normal. Proof: Y , 1, 2 be a bi-closed subspace of a pairwise normal space X ,1, 2 . Let A be a 1 - closed set and B a 2 - closed set, disjoint from A. Let But the space is bi-closed. Therefore A is 1 - closed and B is pairwise normality of X , 1 , 2 , 2 -closed. By a 2 - open set U and a 1 - open set V such that A U , B V , U V . Thus A A Y U Y B B Y V Y with U Y V Y . Also U Y is 2 -open, and V Y is 1 - open . Thus A U Y a 1 - open set V Y and B V Y , with Y , 1, 2 is pairwise 2 -open U Y such that U Y V Y . Hence and a normal. 24 REFERENCES 1. K.Chandrasekhara Rao and K.S.Narayanan, General Topology, S.Viswanathan (Printers & Publishers) Private Ltd., Chennai, 1986. 2. M.K.Bose and Indrajit Lahiri, Sequential Topological Spaces and Separation Axioms, Bulletin of the Allahabad Mathematical Society, Volume 17, 2002, p.23 - 37. 25