SUPRA PAIRWISE CONNECTED AND PAIRWISE SEMI-CONNECTED SPACES

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1 International Journal of Computer Engineering & Technology (IJCET) Volume 9, Issue 4, July-August 2018, pp , Article ID: IJCET_09_04_003 Available online at Journal Impact Factor (2016): (Calculated by GISI) ISSN Print: and ISSN Online: IAEME Publication SUPRA PAIRWISE CONNECTED AND PAIRWISE SEMI-CONNECTED SPACES R. Gowri Assistant Professor, Department of Mathematics, Government College for Women (Autonomous), Kumbakonam, India A.K.R. Rajayal Research Scholar, Department of Mathematics, Government College for Women (Autonomous), Kumbakonam, India ABSTRACT The aim of this paper is to introduce the notion of supra pairwise separated sets (briefly, -p-separated) in supra bitopological spaces. Based on this notion we introduced the notion of -p-connected and -ps-connected spaces. Further, we study some of their characterizations and properties. Mathematics Subject Classification: 54D05, 54D10, 54D08, 54D20 Key words: Supra bitopology, S_τ-p-separated sets, S_τ-p-connected spaces, S_τ-psconnected spaces, S_τ-pf-disconnected, S_τ-pt-disconnected, S_τ-pwt-disconnected, S_τ-pts-disconnected, S_τ-pwts-disconnected. Cite this Article: R. Gowri and A.K.R. Rajayal, Supra Pairwise Connected and Pairwise Semi-Connected Spaces. International Journal of Computer Engineering and Technology, 9(4), 2018, pp INTRODUCTION The study of bitopology was initiated by Kelly[7] in The concept of connectedness in bitopological space has been introduced by Pervin[10]. The study of supra topology was introduced by Mashhour[8] in In topological space the arbitrary union condition is enough to have a supra topological space. Gowri and Rajayal[3] are introduced the notion of supra bitopological spaces. Gowri and Jegadeesan[2] studied the concept of pairwise connectedness in soft bicechˇ closure spaces. The purpose of this article is to introduce and exhibit some results of supra pairwise connectedness and supra pairwise semi-connectedness in supra bitopological spaces. 2. PRELIMINARIES Definition 2.1 [8] (X, ) is said to be a supra topological space if it is satisfying these conditions: 23 editor@iaeme.com

2 R. Gowri and A.K.R. Rajayal X, The union of any number of sets in belongs to. Definition 2.2 [8] Each element A is called a supra open set in (X, ), and its compliment is called a supra closed set in (X, ). Definition 2.3 [8] If (X, ) is a supra topological spaces, A X, A, is the class of all intersection of A with each element in then (A, ) is called a supra subspace of (X, ). Definition 2.4 [8] The supra closure of the set A is denoted by -cl(a) and is defined as - cl(a) = {B : B is a supra closed and A B}. Definition 2.5 [8] The supra interior of the set A is denoted by -int(a) and is defined as - int(a) = {B : B is a supra open and B A}. Definition 2.6 [3] If and are two supra topologies on a non-empty set X, then the triplet (X, ) is said to be a supra bitopological space. Definition 2.7 [3] Each element of is called a supra -open sets (briefly -open sets) in (X, ). Then the complement of -open sets are called a supra -closed sets (briefly -closed sets), for i = 1, 2. Definition 2.8 [3] If (X, ) is a supra bitopological space, Y X, Y then (Y, is a supra bitopological subspace of if (X, ) if = {U Y ; U is a -open in X} and ; V is a open in X}. Definition 2.9 [3] The -closure of the set A is denoted by -cl(a) and is defined as - cl(a) = {B : B is a closed and A B, for i= 1,2}. Definition 2.10 [3] The -interior of the set A is denoted by -int(a) and is defined as - int(a)= {B : B is a open and B A, for i = 1,2}. Proposition 2.11 [4] Let (X, ) be supra bitopological spaces, if U and V are -open sets then U V also -open, for i = 1,2. Proposition 2.12 [4] Let (X, ) be supra bitopological spaces, if U and V are -closed sets then U V also -closed, for i = 1,2. Definition 2.13 [5] A subset A of a supra bitopological space (X, ) is called an - semi-open (briefly, -s-open) if A -cl( -int(a)), Where i j, i,j = 1,2. The complement of -s-open set is said to be -s-closed set. The family of -s-open (resp. -s-closed) sets of X is denoted by -so(x) (resp. -sc(x)), Where i j, i, j = 1,2. Throughout this paper, For a subset A of (X, ), -scl(a) denote the semi closure of A with respect to the topology. 3. SUPRA PAIRWISE CONNECTED SPACES In this section we introduce the concept of supra pairwise connectedness in supra bitopological spaces. Definition 3.1 Two non-empty subsets U and V of a supra bitopological space (X, ) are said to be supra pairwise separated (briefly, -p-separated) if and only if U -cl(v) = and -cl(u) V = editor@iaeme.com

3 Supra Pairwise Connected and Pairwise Semi-Connected Spaces Remark 3.2 In other words two non-empty sets U and V of a supra bitopological space (X, ) are said to be -p-separated if and only if [U -cl(v)] [ -cl(u) V] =. Definition 3.3 A supra bitopological space (X, ) is said to be supra pairwise disconnected (briefly, -p-disconnected ) if it can be written as two disjoint nonempty subsets U and V such that -cl(u) -cl(v) = and -cl(u) cl(v) = X. Example 3.4 Let X = {a, b, c, d}, = {, X,{a, b},{b, c, d}}, = {, X,{c, d},{a, b, d}}. Here U = {a, b} is -open and V = {c, d} is -open. Then U V = {a, b} {c, d} = X and U V = {a, b} {c, d} =. Therefore (X, ) is -p-disconnected. Definition 3.5 A supra bitopological space (X, ) is said to be supra pairwise connected (briefly, -p-connected) if it is not -p-disconnected. Example 3.6 Let X = {a, b, c, d}, = {, X, {a}, {a, b}, {a, b, c}, {a, c, d},{a,b, d}}, = {, X, {c}, {b, d}, {b, c, d}, {a,c, d}}. Here U = {a} is -open and V = {a, c, d} is -open. Then U V = {a} {a, c, d} X and U V = {a} {a, c, d} = {a}. Therefore (X, ) is -p-connected. Theorem 3.7 In a supra bitopological space (X, ) every subsets of -p-separated sets are also -p-separated. Proof: Let (X, ) be a supra bitopological space. Let U and V are -p-separated sets. Let A U and B V. Therefore, U -cl(v) = and -cl(u) V =. (1) Since, A U = -cl(a) -cl(u) = -cl(a) B -cl(u) B = -cl(a) B -cl(u) V = -cl(a) B by (1) = -cl(a) B = Since, B V = -cl(b) -cl(v) = -cl(b) A -cl(v) A = -cl(b) A -cl(v) U = -cl(b) A by (1) = -cl(b) A = Hence, A and B are -p-separated sets. Definition 3.8 A subset Y of a supra bitopological space (X, ) is called -pconnected if the space (Y, ) is -p-connected. Theorem 3.9 Let (Y, be a supra bitopological subspace of a supra bitopological space (X, ) and let U, V Y, then U and V are -p-separated in X if and only if U and V are -p-separated in Y editor@iaeme.com

4 R. Gowri and A.K.R. Rajayal Proof: Let (X, ) be a supra bitopological space and (Y, be a supra bitopological subspace of (X, ). Let U, V Y. Assume that, U and V are -pseparated in X implies that U -cl(v) = and S τ2 -cl(u) V =. That is, (U -cl(v)) ( -cl(u) V) =. Now,[U -cl(v)] [ -cl(u) V] = [U ( -cl(v) Y)] [( -cl(u) Y) V] = [U Y ( -cl(v))] [( -cl(u)) Y V] = [U ( -cl(v))] [( -cl(u)) V] =. Therefore, U and V are -p-separated in Y. Conversely, let us assume that U and V are -p-separated in Y. Implies that U -cl(v) = and -cl(u) V =. That is, (U -cl(v)) ( -cl(u) V) =. Hence U and V are -p-separated in X. Remark 3.10 The following example shows that -p-connectedness in supra bitopological space does not preserves hereditary property. Example 3.11 In Example 3.6, the supra bitopological space (X, ) is -p-connected. Consider (Y, be the supra bitopological subspace of X, Such that Y = {a, b, c}. Taking U = {a, b} and V = {c}, -cl(u) -cl(v) = and -cl(u) -cl(v) = Y. Therefore, the supra bitopological subspace (Y, is -pdisconnected. Theorem 3.12 Let (X, ) be a supra bitopological space then the following conditions are equivalent. (X, ) is -p-connected. X cannot be expressed as the union of two non-empty disjoint sets U and V such that U is -open and V is S τ2 -open. X contains no non-empty proper subset which is both -open and S τ2 -closed. Proof: Case (i): (1) = (2) Let (X, ) be -p-connected. Assume that (2) is not true. Let X = U V, Where U and V are non-empty disjoint sets such that U is -open. U V = = U X V = -cl(u) -cl(x V ) Hence ( -cl(u)) V =. Similarly, U ( -cl(v)) =. Thus [U -cl(v)] [ -cl(u) V] =. -open and V is So we have a -p-separation of X. Which is a contradiction to (1). Hence (2) holds. Therefore (1) = (2). Case (ii): (2) = (3) If possible, N be a proper non-empty subset of X which is -open and -closed. Then X N is a proper non-empty subset of X which is -open editor@iaeme.com

5 Supra Pairwise Connected and Pairwise Semi-Connected Spaces Also, X = N (X N). Thus, X is expressed as the union of two nonempty disjoint sets such that one is -open and the other -open. Which is a contradiction to (2). Hence X does not have proper non-empty set which is both -open and S τ2 closed. Case (iii): (3) = (1) Suppose X is -p-disconnected. That is, X = U V, U -cl(v) =, -cl(u) V =. Therefore, U V =, That is U = X V, -cl(u) V =, Therefore -cl(u) = X V. Hence U = -cl(u). That is U is -closed. Similarly, V= -cl(v). That is -closed. This implies that U is -open. Then there exists a non-empty proper subset U which is - closed and -open. This is a contradiction to (3).Therefore, (3) = (1). Theorem 3.13 Let C be a -p-connected subset of a supra bitopological space (X, ). If X has a -p- separation X = U / V, then C U or C V. Proof: Let X = U / V. Then X = U V and [U -cl(v)] [ -cl(u) V] = (1). From (1) implies U V =. Thus, U = X V or V = X U. Then we have, [(C U) [ -cl(c U) (C V)] [(U -cl(v)] [ -cl(u) V] =. -cl(c V)] That is, C = (C U)/(C V) is a -p-separation of C. But C is -p-connected. Then we have C U = (or) C V =. C (X U) (or) C (X V). Therefore, C U (or) C V. Theorem 3.14 If C is -p-connected set and C F -cl(c) -cl(c) then F is -pconnected set. Proof: Assume that F is not -p-connected. Then we have a -p-separation F = U / V and F = U V with [(U -cl(v)] [ -cl(u) V] = (1). And U and V are non-empty set. (2) By Theorem 3.13, C U (or) C V. Suppose C U. Then V = V V V F V S τ1 - cl(c) -cl(u) =. Which is a contradiction to (1). Then V. Thus, V = is a contradiction to (2). Therefore, F is -p-connected. Remark 3.15 It is clear that the above Theorem 3.15, the relation of belonging to a -pconnected subset divides up any set into its disjoint maximal -p-connected subsets which we shall call the supra components of the set. The next theorem generalizes the fact that the supra components of a supra topological space are closed. Theorem 3.16 Any supra component D of a supra bitopological space (X, ) satisfies the equation D = -cl(d) -cl(d). Proof: Let D be a supra component and suppose that q D. Then D {q} is not connected and we have some separation D {q} = U / V. By Theorem 3.13, either D U and {q} V or D V and {q} U. Thus, D {q} = D / {q} or D {q} = {q} / D. Hence {q} is -open or {q} is -open. And so q -cl(d)or q - cl(d). This is equivalent to saying that q -cl(d) -cl(d). Then we have -cl(d) -cl(d) D. Clearly, D -cl(d) -cl(d) and the equation is satisfied. Definition 3.17 If (X, ) is supra pairwise totally disconnected(briefly, -ptdisconnected) if for each pair of points of X can be separated by a -p-separation of X, that is given two distinct points x and y of X there is a -p-separations, X = U / V such that x U, y V editor@iaeme.com

6 R. Gowri and A.K.R. Rajayal Definition 3.18 If (X, ) is supra pairwise weakly totally disconnected(briefly, -pwtdisconnected) if for each pair of points of X can be separated by a -p-separation of X, that is given two distinct points x and y of X there is a -p-separations, X = U / V such that x U, y V or y U and x V. Definition 3.19 A supra bitopological space (X, ) is said to be supra pairwise feebly disconnected(briefly, -pf-disconnected) if it can be written as two nonempty disjoint subsets U and V such that U -cl(v) = -cl(u) V = and U -cl(v) = -cl(u) V = X. Result 3.20 Every -p-disconnected in supra bitopological space (X, ) is -pfdisconnected but the following example proves that the converse is not true. Example 3.21 In Example 3.6, consider U = {a} and V = {b, c, d} which satisfies the condition U -cl(v) = -cl(u) V = and U -cl(v) = -cl(u) V = X. Therefore, the supra bitopological space (X, ) is -pf-disconnected. But, the supra bitopological space (X, ) is -p-connected. 4. SUPRA PAIRWISE SEMI CONNECTED SPACES In this section we introduce the concept of supra pairwise semi connectedness in supra bitopological spaces. Definition 4.1 Let (X, ) be a supra bitopological space, A X, A is said to be supra pairwise semi open (briefly, -ps-open) set if it is -s-open set and -s-open set. The complement of -ps-open set is called -ps-closed set in X. Example 4.2 Let X = {a, b, c, d}, = {, X, {a, c},{a, b, d}, {a, c, d}}, = {, X, {c, d},{a, b, c}, {b, c, d}}, -so(x) = {, X, {a, c}, {a, b, c}, {a, b, d}, {a, c, d}}, -so(x) = {, X, {c, d}, {a, b, c}, {a, c, d}, {b, c, d}}. Here A = {a, b, c} is -s-open set and it is -open set. Therefore A is -ps-open. Remark 4.3 Every -open set is -s-open but the converse is false in the following example. Example 4.4 Let X = {a, b, c, d}, = {, X, {a, b}, {a,b, c}, {b, c, d}}, = {, X, {b}, {a, b, d}, {a, c, d}}, -so(x) = {, X, {a, b}, {a, b, c}, {a, b, d}, {b, c, d}}, Here {a, b, d} is -s-open set but it is not -open set. Definition 4.5 The -semi-closure of a set A is denoted by -scl(a) and defined as - scl(a) = {B : B is a s closed and A B, for i, j = 1,2}. Definition 4.6 The -semi-interior of a set A is denoted by -sint(a) and defined as - sint(a) = {B : B is a s open and A B, for i, j = 1,2} editor@iaeme.com

7 Supra Pairwise Connected and Pairwise Semi-Connected Spaces Theorem 4.7 Finite union of -s-open sets is -s-open set. Proof: Let C and D be to -s-open sets. Then C -cl( -int(c)) and D -cl( - int(d)). By Proposition 2.11, implies that C D -cl( -int(c D)). Therefore, C D is a -s-open set. Remark 4.8 Finite intersection of -s-open sets may fail to be -s-open set as seen from the following example. Example 4.9 In Example 4.4, -so(x) = {, X, {a, b}, {a, b, c},{a, b, d},{b, c, d}}, Here {a, b, d} and {b, c, d} are -s-open sets but their intersection {b, d} is not -s-open set. Theorem 4.10 Finite intersection of -s-closed sets is -s-open set. Proof: Let C and D be to -s-closed sets. Then -int( -cl(c)) C and -int( -cl(d)) D. By Proposition 2.12, implies that C D -int( -cl(c D)). Therefore, C D is a -s-closed set. Remark 4.11 Finite union of -s-closed set is not -s-closed set as from the following example. Example 4.12 In Example 4.4, -sc(x) = {, X, {a}, {c},{ d}, {c, d}}, Here {a} and {c} are -s-closed sets but their intersection {a, c} is not -sclosed set. Definition 4.13 Two non-empty subsets U and V of (X, ) are said to be supra pairwise semi-separated (briefly, -ps-separated) if and only if U -scl(v) = and S τ2 -scl(u) V =. If X = U V such that U and V are -ps-separated sets, then U, V form a -ps-separation of X and it is denoted by X = U V. Definition 4.14 A supra bitopological space (X, ) is said to be supra pairwise semi disconnected (briefly, -ps-disconnected ) if it can be written as the disjoint non-empty subsets U and V such that -scl(u) -scl(v) = and -scl(u) -scl(v) = X. Example 4.15 Let X = {a, b, c}, = {, X, {a}, {a, c}, {b, c}}, = {, X, {a, b}, {b, c}}, -so(x) = {, X, {a}, {a, c},{b, c}}, -so(x) = {, X, {a,b}, {b, c}}. Here U = {a} is -s-open and V = {b, c} -s-open. Then U V = {a, b, c} = X and U V =. Therefore, (X, ) is -ps-disconnected. Definition 4.16 A supra bitopological space (X, ) is said to be supra pairwise semi connected(briefly, -ps-connected) if it is not -ps-disconnected. Example 4.17 Let X = {a, b, c, d}, = {, X, {a, b}, {a, d}, {a, b, d}, {b, c, d}}, = {, X, {b, c},{c, d}, {a, b, d}, {b, c, d}}, -so(x) = {, X, {a, b}, {a, d}, {a, b, c}, {a, b, d}, {a, c, d},{b,c, d}}, -so(x) = {, X, {b, c}, {c, d},{a, b, c}, {a, b, d}, {a, c, d}, {b, c, d}} editor@iaeme.com

8 R. Gowri and A.K.R. Rajayal Here U = {a, b} is -s-open and V = {b, c} -s-open. Then U V = {a, b, c} X and U V = {b}. Therefore, (X, ) is -ps-connected. Proposition 4.18 Every -closed set is -s-closed set in (X, ), for i =1,2. Proof: Let A be -closed set, for i = 1,2. Then A -int( -cl(a)). Hence, A is -s-closed set in X. Theorem 4.19 If A and B are -p-separated in (X, ), then A, B are -ps-separated in X. Proof: Let A, B are -p-separated sets in supra bitopological space (X, ). This implies, A -cl(b) = and -cl(a) B =, by Proposition 4.18, since every - closed set is -s-closed in X. Hence, A -scl(b) = and -scl(a) B =. Therefore, A and B are -ps-separated in X. Definition 4.20 A subset Y of a supra bitopological space (X, ) is called -psconnected if the space (Y, is -p-disconnected. Remark 4.21 The following example shows that space does not preserves hereditary property. Example 4.22 Let X = {a, b, c, d}, S τ1 = {, X,{ c}, {a, b}, {a, b, c}, {b, c, d}}, S τ2 = {, X, {b, c},{a, c, d}, {b, c,d}},,. -ps-connectedness in supra bitopological Here the supra bitopological space (X, ) is -ps-connected. Let Y = {b, c, d} X. Taking U = {b} is -s-open and V = {c,d} is -s-open. Thus, U V= Y and U V =. Therefore, the supra bitopological subspace (Y, is -ps-disconnected. Theorem 4.23 Let (X, are equivalent. (X, ) is -ps-connected. ) be a supra bitopological space then the following conditions X cannot be expressed as the union of two non-empty disjoint sets U and V such that U is -s-open and V is -s-open. X contains no non-empty proper subset which is both -s-open and -s-closed. Proof: The proof of the theorem is similar to proof of the Theorem Theorem 4.24 Let A be a S -ps-connected subset of a supra bitopological space (X, ). If X has a -ps- separation X = U / V, then A U or A V. Proof: The proof is similar to the proof of the Theorem Definition 4.25 Let (X, ) be a supra bitopological space and x X. The supra semi component of x (briefly, -sc(x)) is the union of all -ps-connected subset of X containing x. Further, if E X and if x E, then the union of all -ps-connected sets containing x and contained in E is called a -sc(e). Theorem 4.26 In a supra bitopological space (X, ) each -sc(x) satisfies the equation -sc(x) = -scl( -sc(x)) -scl( -sc(x)). Proof: Let x be any point in X and let -sc(x) be its supra semi component. Suppose that a point p X does not belongs to -sc(x). Then -sc(x) {p} is not -ps-connected and 30 editor@iaeme.com

9 Supra Pairwise Connected and Pairwise Semi-Connected Spaces hence there exist a -ps-separation U / V in X such that -sc(x) {p} = U / V. By Theorem 4.24, either -sc(x) U and {p} V or -sc(x) V and {p} U. Thus, -sc(x) {p} = -sc(x) / {p} or -sc(x) {p} = {p} / -sc(x). Hence p -scl( -sc(x)) or p -scl( -sc(x)). Hence p -scl( -sc(x)) -scl( -sc(x)) and so -scl( -sc(x)) -scl( -sc(x)) -sc(x). Hence the theorem. Definition 4.27 If (X, ) is supra pairwise totally semi disconnected (briefly, -ptsdisconnected) if for each pair of points of X can be separated by a -psseparation of X, that is given two distinct points x and y of X there is a -psseparations, X = U / V such that x U, y V. Definition 4.28 If (X, )is supra pairwise weakly totally semi disconnected(briefly, - pwts-disconnected) if for each pair of points of X can be separated by a -ps-separation of X, that is given two distinct points x and y of X there is a -ps-separations, X = U / V such that x U, y V or y U and x V. Definition 4.29 A supra bitopological space (X, ) is said to be supra pairwise weakly semi-t 2 -space(briefly, -pws-t 2 -space) if for every pair of distinct points of X, atleast one belongs to a -s-open set and the other belongs to a S τ2 -s-open set satisfying U V =. Theorem 4.30 Let (X, ) be a -pw-t 2 -space. If has a base whose members are also -s-closed or has a base whose members are also -s-closed, then the space X is -pwts-disconnected. Proof: Suppose has a base whose sets are also -s-closed. Consider two distinct points of X. Since the space is -pw-t 2 -space, one of the points x has -open neighbourhood G which does not contain the other point Y and y has a -open neighbourhood H which does not contain x. Then there exist a -basic open set V such that V is also -s-closed and x V G. Then X = V / (X V) is a -ps-separation of X such that x V and y X V. Similarly, we obtain a -ps-separation X = U / (X U) with x X U, y U. Thus (X, ) is -pwts-disconnected. Theorem 4.31 Let (X, ) be a -p-t 2 -space. If has a base whose members are also -s-closed or has a base whose members are also -s-closed, then the space X is -pts-disconnected. Proof: The proof is similar to the proof of the Theorem CONCLUSIONS In this paper, some results of supra pairwise connectedness and semi connectedness in supra bitopological spaces have been discussed. We plan to extend our research work to supra δ semi connectedness and compactness in supra bitopological space. REFERENCES [1] S. P. Arya and T. M. Nour, Pairwise locally semi-connected space, Indian Jr.Pure and Appl.Math.No.18(5)(1987), [2] R. Gowri and G. Jegadeesan, Pairwise connectedness in soft bicechˇ closure spaces, Jr.ofMath.and Informatics, Vol 6,(2016), 1-6. [3] R. Gowri and A. K. R. Rajayal, On supra bitopological spaces, IOSR-JM Vol 13, (2017), editor@iaeme.com

10 R. Gowri and A.K.R. Rajayal [4] R. Gowri and A. K. R. Rajayal, Higher separation axioms in supra bitopological spaces, IJMER, ISSN: , Vol 6, 11(3), (2017), [5] R. Gowri and A. K. R. Rajayal, Supra α-open sets and supra α separation axioms in supra Bitopological Spaces, International Journal of Mathematics Trends and Technology, Vol. 52, (2017), [6] A. Kandil, O. A. E. Tantway, S. A. El-Sheik and Shawqi A. Hazza, Pairwise soft connected in soft bitopological spaces, Int. Jr. of. Computer Applications, Vol. 169,(2017), [7] J. C. Kelly, Bitopological spaces, Proc. London Math. Soc.No.(3), 13(1963), [8] A. S. Mashhour, A.A. Allam, F.S. Mahmoud and F.H. Khedr, On Supra topological spaces, Indian Jr.Pure and Appl.Math.No.4(14)(1983), [9] M. N. Mukherjee, Pairwise semi-connectedness in bitopological spaces, Indian Jr.Pure and Appl.Math.No.14(9)(1983), [10] W. J. Pervin, Connectedness in bitopological spaces, Indag.Math., 29,(1967),