SUPRA PAIRWISE CONNECTED AND PAIRWISE SEMI-CONNECTED SPACES

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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

R. Gowri and A.K.R. RajayalDefinition 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 isgiven 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 feeblydisconnected(briefly, -pf-disconnected) if it can be written as two nonempty disjoint subsetsU 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 theconditionU -cl(V) -cl(U) V and U -cl(V) -cl(U) V X.Therefore, the supra bitopological space (X,bitopological space (X,) is -p-connected.) is-pf-disconnected. But, the supra4. SUPRA PAIRWISE SEMI CONNECTED SPACESIn this section we introduce the concept of supra pairwise semi connectedness in suprabitopological spaces.Definition 4.1 Let (X,) be a supra bitopological space, A X, A is said to be suprapairwise semi open (briefly, -ps-open) set if it is-s-open set and-s-open set.The complement of -ps-open set is calledExample 4.2 Let X {a, b, c, d}, { , X, {a, c},{a, b, d}, {a, c, d}},-ps-closed set in X. { , 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 followingexample.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 byscl(A) {B : B is a s closed and A B, for i, j 1,2}.-scl(A) and defined as-Definition 4.6 The-semi-interior of a set A is denoted bysint(A) {B : B is a s open and A B, for i , j 1,2}.-sint(A) and defined eme.com

Supra Pairwise Connected and Pairwise Semi-Connected SpacesTheorem 4.7 Finite union of-s-open sets is-s-open set.Proof: Let C and D be to-s-open sets. Then C int(D)). By Proposition 2.11, implies that C D -cl(Therefore, C D is a-cl( -int(C)) and D -int(C D)).-cl(--s-open set.Remark 4.8 Finite intersection of-s-open sets may fail to be-s-open set as seen fromthe 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 notset.Theorem 4.10 Finite intersection of-s-closed sets is-s-open set.-s-openProof: Let C and D be to-s-closed sets. Then -int( -cl(C)) C and D. By Proposition 2.12, implies that C D -int( -cl(C D)).-cl(D))Therefore, C D is a-int(-s-closed set.Remark 4.11 Finite union of-s-closed set is not-s-closed set as from the followingexample.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 pairwisesemi-separated (briefly, -ps-separated) if and only if U -scl(V) and Sτ2-scl(U) V .If X U V such that U and V are -ps-separated sets, then U, V form a -ps-separation ofX and it is denoted by X U V.Definition 4.14 A supra bitopological space (X,) is said to be supra pairwise semidisconnected (briefly, -ps-disconnected ) if it can be written as the disjoint non-emptysubsets 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 semiconnected(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, aeme.com

R. Gowri and A.K.R. RajayalHere 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 EveryProof: Let A beHence, A is-closed set is-s-closed set in (X,-closed set, for i 1,2. Then A -int(), for i 1,2.-cl(A)).-s-closed set in X.Theorem 4.19 If A and B are -p-separated in (X,), then A, B are -ps-separated inX.Proof: Let A, B are -p-separated sets in supra bitopological space (X,). Thisimplies, A -cl(B) and-cl(A) B , by Proposition 4.18, since everyclosed 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 thatspace does not preserves hereditary property.Example 4.22 Let X {a, b, c, d},Sτ1 { , X,{ c}, {a, b}, {a, b, c}, {b, c, d}},Sτ2 { , 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.) be a supra bitopological space then the following conditions (X, 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) is-ps-connected.-s-open and-s-closed.Proof: The proof of the theorem is similar to proof of the Theorem 3.12.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 3.13.Definition 4.25 Let (X,) be a supra bitopological space and x X. The supra semicomponent of x (briefly, -sc(x)) is the union of all -ps-connected subset of X containingx.Further, if E X and if x E, then the union of all -ps-connected sets containing x andcontained 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 apoint p X does not belongs to -sc(x). Then -sc(x) {p} is not -ps-connected eme.com

Supra Pairwise Connected and Pairwise Semi-Connected Spaceshence there exist a-ps-separation U / V in X such that-sc(x) {p} U / V. ByTheorem 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, thatis 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 ofX, that is given two distinct points x and y of X there is a -ps-separations, X U / V suchthat x U, y V or y U and x V.Definition 4.29 A supra bitopological space (X,) is said to be supra pairwise weaklysemi-T2-space(briefly, -pws-T2-space) if for every pair of distinct points of X, atleast onebelongs to a-s-open set and the other be

bitopological space has been introduced by Pervin[10]. The study of supra topology was . Gowri and Jegadeesan[2] studied the concept of pairwise connectedness in soft biCechˇ closure spaces. The purpose of this article is to introduce and . Supra Pairwise Connected and Pairwise Semi-Connected Spaces

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