SUPRA PAIRWISE CONNECTED AND PAIRWISE SEMI-CONNECTED SPACES

International Journal of Computer Engineering & Technology (IJCET)Volume 9, Issue 4, July-August 2018, pp. 23–32, Article ID: IJCET 09 04 003Available online athttp://www.iaeme.com/ijcet/issues.asp?JType IJCET&VType 9&IType 4Journal Impact Factor (2016): 9.3590(Calculated by GISI) www.jifactor.comISSN Print: 0976-6367 and ISSN Online: 0976–6375 IAEME PublicationSUPRA PAIRWISE CONNECTED ANDPAIRWISE SEMI-CONNECTED SPACESR. GowriAssistant Professor, Department of Mathematics,Government College for Women (Autonomous), Kumbakonam, IndiaA.K.R. RajayalResearch Scholar, Department of Mathematics,Government College for Women (Autonomous), Kumbakonam, IndiaABSTRACTThe 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 weintroduced the notion of -p-connected and -ps-connected spaces. Further, westudy some of their characterizations and properties.Mathematics Subject Classification: 54D05, 54D10, 54D08, 54D20Key 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 andPairwise Semi-Connected Spaces. International Journal of Computer Engineering andTechnology, 9(4), 2018, pp. 23-32.http://www.iaeme.com/IJCET/issues.asp?JType IJCET&VType 9&IType 41. INTRODUCTIONThe study of bitopology was initiated by Kelly[7] in 1963. The concept of connectedness inbitopological space has been introduced by Pervin[10]. The study of supra topology wasintroduced by Mashhour[8] in 1983. In topological space the arbitrary union condition isenough to have a supra topological space. Gowri and Rajayal[3] are introduced the notion ofsupra bitopological spaces. Gowri and Jegadeesan[2] studied the concept of pairwiseconnectedness in soft biCechˇ closure spaces. The purpose of this article is to introduce andexhibit some results of supra pairwise connectedness and supra pairwise semi-connectednessin supra bitopological spaces.2. PRELIMINARIESDefinition 2.1 [8] (X, ) is said to be a supra topological space if it is satisfying sp23editor@iaeme.com

R. Gowri and A.K.R. Rajayal X, The union of any number of sets inbelongs to.Definition 2.2 [8] Each element A is called a supra open set in (X, ), and itscompliment 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 allintersection of A with each element inthen (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] Ifandare 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 ofis 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 inX} and; V is a open in X}.Definition 2.9 [3] The -closure of the set A is denoted bycl(A) {B : B is a closed and A B, for i 1,2}.-cl(A) and is defined as-Definition 2.10 [3] The -interior of the set A is denoted byint(A) {B : B is a open and B A, for i 1,2}.-int(A) and is defined as-Proposition 2.11 [4] Let (X,) be supra bitopological spaces, if U and V aresets then U V also -open, for i 1,2.-openProposition 2.12 [4] Let (X,) be supra bitopological spaces, if U and V aresets then U V also -closed, for i 1,2.-closedDefinition 2.13 [5] A subset A of a supra bitopological space (X,semi-open (briefly,-s-open) if A -cl( -int(A)), Where icomplement of-s-open set is said to be-s-closed set.The family of-s-open (resp.-sC(X)), Where i j, i, j 1,2.) is called anj, i,j 1,2. The-s-closed) sets of X is denoted byThroughout this paper, For a subset A of (X,of A with respect to the topology.),-sO(X) (resp.-scl(A) denote the semi closure3. SUPRA PAIRWISE CONNECTED SPACESIn this section we introduce the concept of supra pairwise connectedness in suprabitopological 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 e.com

Supra Pairwise Connected and Pairwise Semi-Connected SpacesRemark 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 pairwisedisconnected (briefly,-p-disconnected ) if it can be written as two disjoint nonemptysubsets 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} isand U V {a, b} {c, d} . Therefore (X,-open. Then U V {a, b} {c, d} 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 setsare 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(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(B) A -cl(V) A -cl(B) A -cl(V) U -cl(B) A by (1) -cl(B) A -cl(U)-cl(V)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 bitopologicalspace (X,) and let U, V Y, then U and V are -p-separated in X if and only if Uand V are -p-separated in me.com

R. Gowri and A.K.R. RajayalProof: Let (X,) be a supra bitopological space and (Y,be a suprabitopological subspace of (X,). Let U, V Y. Assume that, U and V are -pseparated in X implies that U -cl(V) and Sτ2-cl(U) V .That is, (U Now,[U -cl(V)] [ [U Y ( [U (-cl(U) V) .-cl(V)) (-cl(U) V] [U (-cl(U) Y) V]-cl(U)) Y V]-cl(V))] [(-cl(V))] [(-cl(V) Y)] [(-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 bitopologicalspace 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 conditionsare equivalent. (X, X cannot be expressed as the union of two non-empty disjoint sets U and V such that U is-open and V is Sτ2-open. X contains no non-empty proper subset which is both) is-p-connected.-open and Sτ2-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) Hence (Thus [U -open and V is-cl(X V )-cl(U)) V . Similarly, U (-cl(V)] [-cl(V)) .-cl(U) V] .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 or@iaeme.com

Supra Pairwise Connected and Pairwise Semi-Connected SpacesAlso, X N (X N). Thus, X is expressed as the union of two nonempty disjoint sets suchthat one is-open and the other-open. Which is a contradiction to (2). Hence X doesnot have proper non-empty set which is both-open and Sτ2closed.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 isclosed 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) impliesU 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 Sτ1cl(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 whichwe shall call the supra components of the set. The next theorem generalizes the fact that thesupra components of a supra topological space are closed.Theorem 3.16 Any supra component D of a supra bitopological space (X,) satisfiesthe equation D -cl(D) -cl(D).Proof: Let D be a supra component and suppose that q D. Then D {q} is not connectedand we have some separation D {q} U / V. By Theorem 3.13, either D U and {q} Vor 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 isgiven two distinct points x and y of X there is a -p-separations, X U / V such that x U, y me.com