225 American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) ISSN (Print) 2313-4410, ISSN (Online) 2313-4402 Β© Global Society of Scientific Research and Researchers http://asrjetsjournal.org/ On Pairwise 𝝀𝝀-Open Soft Sets and Pairwise Locally Closed Soft Sets Kandila, O. A. E. Tantawyb, S. A. El-Sheikhc , Shawqi. A. Hazzad* aDepartment of Mathematics, Faculty of Science, Helwan University, Helwan, Egypt bDepartment of Mathematics, Faculty of Science, Zagazig University, Zagazig, Egypt cDepartment of Mathematics, Faculty of Education, Ain Shams University, Cairo, Egypt dDepartment of Mathematics, Faculty of Education, Taiz University, Taiz, Yemen aEmail: dr.ali_kandil@yahoo.com bEmail: drosamat@yahoo.com cEmail: sobhyelsheikh@yahoo.com dEmail: shawqialalimi@yahoo.com Abstract Kandil and his colleagues [10], introduced the notion of Ξ»p -closed soft set by involving Ξ›p -soft set and p - closed soft set. In this paper, we give some additional properties of Ξ»p -closed soft sets. We also introduce and study a related new class of 𝑃𝑃𝑃𝑃𝑇𝑇1 4οΏ½ βˆ— -spaces which lies between π‘·π‘·π‘·π‘·π‘»π‘»πŸŽπŸŽβˆ— and 𝑃𝑃𝑃𝑃𝑇𝑇1 2οΏ½ βˆ— . Moreover, we show that there exists a very important relation between the notion of Ξ»p -closed soft sets and the 𝑃𝑃𝑃𝑃𝑇𝑇1 4οΏ½ βˆ— property, 𝑖𝑖 = 0, 1 4οΏ½ , 1 2οΏ½ . In addition, we offer the notion of p -locally closed soft sets and we investigate a related new pairwise soft separation axiom π‘ƒπ‘ƒπ‘ƒπ‘ƒπ‘‡π‘‡πΏπΏβˆ— which is independent from 𝑃𝑃𝑃𝑃𝑇𝑇1 4οΏ½ βˆ— . The relationships between the Ξ»p -closed soft sets and the p -locally closed soft sets are obtained. Furthermore, we introduce the notion of Ξ»p -open soft sets and we construct supra soft topology associated with the class of Ξ»p -open soft sets and we present pairwise soft separation axioms related to such soft sets, namely * Ξ»PST . We provide some illustrative examples to support the results. Keywords: Soft set; Soft topology; Soft bitopology; Soft bitopological spaces; Pairwise soft separation axioms; Ξ»p - closed soft sets; Pairwise Ξ»p -open soft sets; and p - locally closed soft sets. ------------------------------------------------------------------------ * Corresponding author. http://asrjetsjournal.org/ American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2017) Volume 28, No 1, pp 225-247 226 1. Introduction In 1999, Molodtsov [14] introduced the concept of soft set as a mathematical modeling for dealing with uncertainties inherent in many of real world problems. Shabir and Naz [20] introduced the concept of soft topological spaces and investigate some fundamental properties of such spaces. Many researches (see, for example, [1,3,4,5,6,7,8,15,16,18]) introduced and discuss new notions of soft topological spaces. Ittanagi [2] introduced the notion of soft bitopological space. He also offer some types of soft separation axioms in soft bitopological space. Kandil and his colleagues [10] introduced the notions of Ξ›p -soft sets and Ξ»p -closed soft sets in soft bitopological spaces. They the family of all Ξ›p -soft sets defines an Alexandroff soft topology. Kandil and his colleagues [9] introduced the concept of gp -closed soft sets and defined the associated pairwise soft separation axioms, namely, 𝑃𝑃𝑃𝑃𝑇𝑇1 2οΏ½ βˆ— and 𝑃𝑃𝑃𝑃𝑅𝑅0βˆ— . Recently, Kandil and his colleagues [11] introduced some types of pairwise soft separation axioms, namely, 𝑃𝑃𝑃𝑃𝑇𝑇0βˆ— , 𝑃𝑃𝑃𝑃𝑇𝑇1βˆ— , 𝑃𝑃𝑃𝑃𝑇𝑇2βˆ— and 𝑃𝑃𝑃𝑃𝑅𝑅1βˆ— . They studied the characterization and implications among these types of separation axioms. The motivation of the present paper is to introduce new classes of soft sets called p -locally closed soft sets and Ξ»p -open soft sets in soft bitopological spaces. It turn out that Ξ»p -closed soft sets, Ξ»p -open soft sets and p -locally closed soft sets are weaker forms of p -open and p -closed soft sets. We also conclude several important properties of such soft sets. Moreover, we introduce and study a related pairwise soft separation axioms, namely, 𝑃𝑃𝑃𝑃𝑇𝑇1 4οΏ½ βˆ— , π‘ƒπ‘ƒπ‘ƒπ‘ƒπ‘‡π‘‡πΏπΏβˆ— and π‘ƒπ‘ƒπ‘ƒπ‘ƒπ‘‡π‘‡πœ†πœ†βˆ— . We studied the relationships between these types of separation axioms and the other in [9,11]. 2. Preliminaries In this section, we briefly review some concepts and some related results of soft set, soft topological space and soft bitopological space which are needed to used in current paper. For more details about these concepts you can see [2,4,5,6,7,8,9,10,11,14,15,16,19,20,21,22]. Let X be an initial universe, E be a set of parameters and )(XP be the power set of X . Definition 2.1 [16] A pair ),( EF is called a soft set over X , where F is a mapping given by )(: XPEF β†’ . A soft set can also be defined by the set of ordered pairs )}(:,:))(,{(=),( XPEFEeeFeEF β†’βˆˆ . From now on, EXSS )( denotes the family of all soft sets over X with a fixed set of parameters E . For two soft sets EXSSEGEF )(),(),,( ∈ , ),( EF is called a soft subset of ),( EG , denoted by ),(~),( EGEF βŠ† , if )()( eGeF βŠ† , Eeβˆˆβˆ€ . In this case, ),( EG is called a soft superset of ),( EF . In American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2017) Volume 28, No 1, pp 225-247 227 addition, the union of soft sets ),( EF and ),( EG , denoted by ),(~),( EGEF βˆͺ , is the soft set ),( EH which defined as )()(=)( eGeFeH βˆͺ , Eeβˆˆβˆ€ . Moreover, the intersection of soft sets ),( EF and ),( EG , denoted by ),(~),( EGEF ∩ , is the soft set ),( EM which defined as, )()(=)( eGeFeM ∩ , Eeβˆˆβˆ€ . The complement of a soft set ),( EF , denoted by cEF ),( , is defined as, ),(=),( EFEF cc , where )(: XPEF c β†’ is a mapping given by )(\=)( eFXeF c , Eeβˆˆβˆ€ . The difference of soft sets ),( EF and ),( EG , denoted by ),(\),( EGEF , is the soft set ),( EH , which defined as, )(\)(=)( eGeFeH , Eeβˆˆβˆ€ . Clearly, cEGEFEGEF ),(~),(=),(\),( ∩ . A soft set ),( EF is called a null soft set, denoted by ),~( EΟ† , if Ο†=)(eF , Eeβˆˆβˆ€ . Moreover, a soft set ),( EF is called an absolute soft set, denoted by ),~( EX , if XeF =)( , Eeβˆˆβˆ€ . Clearly, we have ),~(=),~( EXE cΟ† and ),~(=),~( EEX c Ο† . Moreover, a soft set ),( EG is said to be a finite soft set if )(eG is a finite set for all Ee∈ . Otherwise, it is called an infinite soft set. Definition 2.2 ([15,17,21]) A soft set ),( EF over X is said to be a soft point in ),~( EX if there exist Xx∈ and Ee∈ such that }{=)( xeF and Ο†=)(eF β€² for each }{\ eEe βˆˆβ€² . This soft point is denoted by ),( Exe or ex , i.e., )(: XPExe β†’ is a mapping defined as ο£³ ο£² ο£± β‰  aeif aeifx axe Ο† ,=}{ =)( for all Ea∈ . A soft point ),( Exe is said to be belonging to the soft set ),( EG , denoted by ),(~ EGxe ∈ , if )()( eGexe βŠ† , i.e., )(}{ eGx βŠ† . Clearly, ),(~ EGxe ∈ if and only if ),(~),( EGExe βŠ† . In addition, two soft points 1e x , 2ey over X are said to be equal if yx = and 21 = ee . Thus, 21 ee yx β‰  iff yx β‰  or 21 ee β‰  . The family of all soft points in ),~( EX is denoted by EX )(ΞΎ . Proposition 2.1 [21] The union of any collection of soft points can be considered as a soft set and every soft set can be expressed as a union of all soft points belonging to it, i.e., )},(~:),{(=),( EGxExEG ee βˆˆο• . Proposition 2.2 [21] Let ),( EG , ),( EH be two soft sets over X . Then, 1) c ee EGxEGx ),(~),(~ ∈/β‡”βˆˆ . 2) ),(~),(~),(~ EGxEHEGx ee βˆˆβ‡”βˆͺ∈ or ),(~ EHxe ∈ . 3) ),(~),(~),(~ EGxEHEGx ee βˆˆβ‡”βˆ©βˆˆ and American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2017) Volume 28, No 1, pp 225-247 228 ),(~ EHxe ∈ . 4) )],(~),(~[),(~),( EHxEGxEHEG ee βˆˆβ‡’βˆˆβ‡”βŠ† . For more details for soft point you can see in [15,21,17]. Definition 2.3 [20] Let Ξ· be a collection of soft sets over a universe X with a fixed set of parameters E , i.e., EXSS )(βŠ†Ξ· . The collection Ξ· is called a soft topology on X if it satisfies the following axioms: 1) Ξ·Ο† ∈),~(),,~( EEX , 2) The union of any number of soft sets in Ξ· belongs to Ξ· , 3) The intersection of any two soft sets in Ξ· belongs to Ξ· . The triple ),,( EX Ξ· is called a soft topological space. Any member of Ξ· is said to be an open soft set in ),,( EX Ξ· . A soft set ),( EF over X is said to be a closed soft set in ),,( EX Ξ· , if its complement cEF ),( is an open soft set in ),,( EX Ξ· . We denote the family of all closed soft sets by cΞ· . Definition 2.4 [20] Let ),,( EX Ξ· be a soft topological space and EXSSEF )(),( ∈ . The soft closure of ),( EF , denoted by ),( EFsclΞ· , is the intersection of all closed soft super sets of ),( EF , i.e, )}),(~),(:)(),{(=),( EHEFXSCEHEFscl βŠ†βˆˆ Ξ·Ξ·  . Clearly ),( EFsclΞ· is the smallest closed soft set over X which contains ),( EF . Definition 2.5 [13] A soft set ),( EG in a soft topological space ),,( EX Ξ· is called a generalized closed soft set [briefly, g -closed soft set] if ),(~),( EHEGscl βŠ†Ξ· whenever ),(~),( EHEG βŠ† and η∈),( EH . Definition 2.6 [16] A soft topological space ),,( EX Ξ· is said to be a soft 0T [briefly, 0ST ] if for each EXyx )(, ΞΎΞ²Ξ± ∈ with Ξ²Ξ± yx β‰  , there exists η∈),( EG such that ),(~ EGx ∈α , ),(~ EGy ∈/Ξ² or ),(~ EGy ∈β , ),(~ EGx ∈/Ξ± . Definition 2.7 [13] A soft topological space ),,( EX Ξ· is called a soft 𝑇𝑇1 2οΏ½ [briefly,𝑃𝑃 𝑇𝑇1 2οΏ½ ] if every g -closed soft set is a closed soft set. Theorem 2.1 [9] A soft topological space ),,( EX Ξ· is a soft 𝑇𝑇1 2οΏ½ if and only if every soft point either open American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2017) Volume 28, No 1, pp 225-247 229 soft set or closed soft set. Definition 2.8 [4] Let Β΅ be a collection of soft sets over X [i.e., EXSS )(βŠ†Β΅ ]. The collection Β΅ is called a supra soft topology on X if it satisfies the following axioms: 1) ),~( EX , ¡φ ∈),~( E , 2) The union of any number of soft sets in Β΅ belongs to Β΅ . Definition 2.9 [2] A quadrable system ),,,( 21 EX Ξ·Ξ· is called a soft bitopological space [briefly, sbts], where 1Ξ· , 2Ξ· are arbitrary soft topologies on X with a fixed set of parameters E . Definition 2.10 [10] Let ),,,( 21 EX Ξ·Ξ· be a sbts. A soft set ),( EG over X is said to be a pairwise open soft set in ),,,( 21 EX Ξ·Ξ· [briefly, p -open soft set] if there exist an open soft set ),( 1 EG in 1Ξ· and an open soft set ),( 2 EG in 2Ξ· such that ),(~),(=),( 21 EGEGEG βˆͺ . A soft set ),( EG over X is said to be a pairwise closed soft set in ),,,( 21 EX Ξ·Ξ· [briefly, p -closed soft set] if its complement is a p -open soft set in ),,,( 21 EX Ξ·Ξ· . Clearly, a soft set ),( EF over X is a p -closed soft set in ),,,( 21 EX Ξ·Ξ· if there exist an 1Ξ· -closed soft set ),( 1 EF and an 2Ξ· -closed soft set ),( 2 EF such that ),(~),(=),( 21 EFEFEF ∩ . The family of all p -open ( p -closed) soft sets in a sbts ),,,( 21 EX Ξ·Ξ· is denoted by 12Ξ· ( c 12Ξ· ), respectively. Theorem 2.2 [10] Let ),,,( 21 EX Ξ·Ξ· be a sbts. The family of all p -open soft sets 12Ξ· is a supra soft topology on X , where 1,2}=,),(:),(~),(=),{(= 2112 iEGEGEGEG ii Ξ·Ξ· ∈βˆͺ . The triple ),,( 12 EX Ξ· is the supra soft topological space associated to the soft bitopological space ),,,( 21 EX Ξ·Ξ· . Definition 2.11 [10] Let ),,,( 21 EX Ξ·Ξ· be a sbts and let EXSSEG )(),( ∈ . The pairwise soft closure of ),( EG , denoted by ),(12 EGscl , is defined by )},(~),(:),{(=),( 1212 EFEGEFEGscl c βŠ†βˆˆΞ·ο‰ . Clearly, ),(12 EGscl is the smallest p -closed soft set contains ),( EG .For more details about the properties of pairwise soft closure operator see in [10]. American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2017) Volume 28, No 1, pp 225-247 230 Definition 2.12 [10] Let ),,,( 21 EX Ξ·Ξ· be a sbts and let EXSSEG )(),( ∈ . The pairwise soft interior of ),( EG , denoted by ),(12 EGsint , is defined by )},(~),(:),{(=),( 1212 EGEHEHEGsint βŠ†βˆˆΞ·ο• . Clearly, ),(12 EGsint is the largest p -open soft set contained in ),( EG . For more details about the properties of pairwise soft interior operator you can see see [10]. Definition 2.13 [10] Let ),,,( 21 EX Ξ·Ξ· be a sbts and let EXSSEG )(),( ∈ . The pairwise soft kernel of ),( EG [briefly, ),(12 EGsker ], is the intersection of all p -open soft supersets of ),( EG , i.e., )},(~),(:),{(=),( 1212 EHEGEHEGsker βŠ†βˆˆΞ·ο‰ . Definition 2.14 [10] A soft set ),( EG is said to be a pairwise Ξ› - soft set in a soft bitopological space ),,,( 21 EX Ξ·Ξ· [briefly, Ξ›p -soft set] if ),(=),(12 EGEGsker . Theorem 2.3 [10] Every p -open soft set is a Ξ›p -soft set. Theorem 2.4 [10] Let ),,,( 21 EX Ξ·Ξ· be a sbts. Then, the class of all Ξ›p -soft sets is an Alexandroff soft topology on X . This soft topology we denoted by Ξ›pΞ· . The triple ),,( EX pΛη is the soft topological space associated to the soft bitopological space ),,,( 21 EX Ξ·Ξ· , induced by the family of all Ξ›p -soft sets. Theorem 2.5 [10] Let ),,,( 21 EX Ξ·Ξ· be a sbts. Then, Ep XSS )(1221 βŠ†βŠ†βŠ†βˆͺ Ληηηη . Definition 2.15 [10] A soft set ),( EG is said to be a pairwise Ξ» -closed soft set in a sbts ),,,( 21 EX Ξ·Ξ· [briefly, Ξ»p -closed soft set] if ),(~),(=),( EHEFEG ∩ , where ),( EF is a p -closed soft set and ),( EH is a Ξ›p -soft set. The family of all Ξ»p -closed soft sets we denoted by EXCSP ),,( 21 Ξ·Ξ·Ξ» . Theorem 2.6 [10] Let ),,,( 21 EX Ξ·Ξ· be a sbts. Then, 1) Every p -closed soft set is a Ξ»p -closed soft set. 2) Every Ξ›p -soft set is a Ξ»p -closed soft set. Definition 2.16 [12] Let ),,,( 21 EX Ξ·Ξ· be a sbts and let EXSSEG )(),( ∈ . The pairwise soft sub Kernel of American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2017) Volume 28, No 1, pp 225-247 231 ),( EG [briefly, ),(* 12 EGsker ] is defined by )},(~),(:),{(=),( 12 * 12 EGEFEFEGsker c βŠ†βˆˆΞ·ο• . For more details you can see in [12]. Definition 2.17 [12] A soft set ),( EG is said to be a pairwise ∨ -soft set [briefly, ∨p -soft set] in a sbts ),,,( 21 EX Ξ·Ξ· if ),(=),(* 12 EGEGsker . We denote the family of all ∨p -soft sets by EXSP ),,( 21 ηη∨ .Clearly, ),( EG is a Ξ›p -soft set if and only if cEG ),( is a ∨p -soft set. Corollary 2.1 [12] Let ),,,( 21 EX Ξ·Ξ· be a sbts. The family of all ∨p -soft sets is an Alexandroff soft topology on X . This soft topology we denoted by ∨pΞ· . Theorem 2.7 [12] For any sbts ),,,( 21 EX Ξ·Ξ· , we have c pp βˆ¨Ξ› Ξ·Ξ· = . Definition 2.18 [11] A soft bitopological space ),,,( 21 EX Ξ·Ξ· is said to be a pairwise soft π‘»π‘»πŸŽπŸŽβˆ— [briefly, π‘·π‘·π‘·π‘·π‘»π‘»πŸŽπŸŽβˆ— ] if for each EXyx )(, ΞΎΞ²Ξ± ∈ with Ξ²Ξ± yx β‰  , there exists 12),( η∈EG such that ),(~ EGx ∈α , ),(~ EGy ∈/Ξ² or ),(~ EGy ∈β , ),(~ EGx ∈/Ξ± . Theorem 2.8 [11] Let ),,,( 21 EX Ξ·Ξ· be a sbts. Then ),,,( 21 EX Ξ·Ξ· is a π‘·π‘·π‘·π‘·π‘»π‘»πŸŽπŸŽβˆ— if and only if every soft point Ee Xx )(ξ∈ is a Ξ»p -closed soft set. Lemma 2.1 [11] Let ),,,( 21 EX Ξ·Ξ· be a sbts. Then ),,,( 21 EX Ξ·Ξ· is a π‘·π‘·π‘·π‘·π‘»π‘»πŸŽπŸŽβˆ— if and only if for all EXyx )(, ΞΎΞ²Ξ± ∈ , Ξ²Ξ± yx β‰  there exists cEG 1212),( Ξ·Ξ· βˆͺ∈ such that ),(~ EGx ∈α and ),(~ EGy ∈/Ξ² . Definition 2.19 [9] Let ),,,( 21 EX Ξ·Ξ· be a sbts and let EXSSEG )(),( ∈ . A soft set ),( EG is said to be a generalized pairwise closed soft set [briefly, gp -closed soft set] if ),(~),(12 EHEGscl βŠ† whenever ),(~),( EHEG βŠ† and ),( EH is a p -open soft set. Definition 2.20 [9] A soft bitopological space ),,,( 21 EX Ξ·Ξ· is called a pairwise soft 𝑇𝑇1 2οΏ½ βˆ— [briefly, 𝑃𝑃𝑃𝑃𝑇𝑇1 2οΏ½ βˆ— ] if every gp -closed soft set is a p -closed soft set. American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2017) Volume 28, No 1, pp 225-247 232 Theorem 2.9 [9] Let ),,,( 21 EX Ξ·Ξ· be a soft bitopological space. Then, ),,,( 21 EX Ξ·Ξ· is a 𝑃𝑃𝑃𝑃𝑇𝑇1 2οΏ½ βˆ— if and only if every soft point either p -open soft set or p -closed soft set. Theorem 2.10 [9] In any sbts ),,,( 21 EX Ξ·Ξ· , every soft point either p -closed soft set or its complement is a gp -closed soft set. 3. More on 𝒑𝒑𝝀𝝀 βˆ’closed soft sets In this section, we give some additional properties of Ξ»p -closed soft sets and introduce and study a related new pairwise soft separation axiom 𝑃𝑃𝑃𝑃𝑇𝑇1 4οΏ½ βˆ— . Theorem 3.1 Let ),,,( 21 EX Ξ·Ξ· be a sbts and EXSSEG )(),( ∈ . The arbitrary intersection of Ξ»p -closed soft sets is a Ξ»p -closed soft set. Proof. Let Ei XCSPiEG ),,(}:),{( 21 Ξ·Ξ·Ξ»βŠ†βˆ†βˆˆ . Then, βˆ†βˆˆβˆ€βˆˆ iXCSPEG Ei ),,(),( 21 Ξ·Ξ·Ξ» . Thus, for all βˆ†βˆˆi there exist c i EF 12),( η∈ and Ξ›βˆˆ pi EH Ξ·),( such that ),(~),(=),( EHEFEG iii ∩ β‡’ =),( EGii βˆ†βˆˆο‰ )],(~),[( EHEF iii ∩ βˆ†βˆˆο‰ )],([)],([= EHEF iiii βˆ†βˆˆβˆ†βˆˆ  . Since βˆ†βˆˆβˆ€βˆˆ iEF c i 12),( Ξ· , then c ii EF 12),( η∈ βˆ†βˆˆο‰ . Also, since βˆ†βˆˆβˆ€βˆˆ Ξ› iEH pi Ξ·),( , then Ξ›βˆ†βˆˆ ∈ pii EH Ξ·),( [for Ξ›pΞ· is an Alexandroff soft topology]. Therefore, ),( EGii βˆ†βˆˆο‰ is a Ξ»p -closed soft set. Remark 3.1 The union of any two Ξ»p -closed soft sets may not be a Ξ»p -closed soft set as shown in the following example. Example 3.1 Let },{= yxX , },{= 21 eeE and let )},(),,(),,~(),,~{(= 211 EGEGEXEφη , )},(),,~(),,~{(=2 EHEXEφη , where })}{,(}),{,{(=),( 211 yexeEG , American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2017) Volume 28, No 1, pp 225-247 233 })}{,(}),{,{(=),( 212 xeyeEG , })}{,(}),{,{(=),( 21 xexeEH . Then, ),,,( 21 EX Ξ·Ξ· is a sbts. Consequently, )},(),,(),,(),,(),,(),,~(),,~{(= 212112 EPEPEHEGEGEXEφη , where )},(}),{,{(=),( 211 XexeEP , })}{,(),,{(=),( 212 xeXeEP . It is easy to prove that )},(),,{(= 2112 EMEMp ηη Ξ› where )},(}),{,{(=),( 211 Ο†exeEM , })}{,(),,{(=),( 212 xeeEM Ο† . Let })}{,(),,{(=),( 21 yeeEK Ο† , then ),( EK is a p -closed soft set. It is clear that ),( EK and ),( 2 EG are Ξ»p -closed soft sets but )},(}),{,{(=),(~),( 212 XeyeEGEK βˆͺ is not Ξ»p -closed soft set. Moreover, )},(),,{(=),,( 21121221 EMEMXCSP c E ηηηηλ . Lemma 3.1 Let ),,,( 21 EX Ξ·Ξ· be a sbts and let ),( Exe be a soft point in ),~( EX . If c e c e ExExsker ),(=),(12 , then c e Ex ),( is a p -open soft set. Proof. Assume that c e Ex ),( is not a p -open soft set, then the only p -open soft superset of c e Ex ),( is ),~( EX . Therefore, ),~(=),(12 EXExsker c e , a contradiction. Theorem 3.2 Let ),,,( 21 EX Ξ·Ξ· be a sbts. Then ),,,( 21 EX Ξ·Ξ· is a 𝑃𝑃𝑃𝑃𝑇𝑇1 2οΏ½ βˆ— iff every soft set on X is a Ξ»p -closed soft set. American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2017) Volume 28, No 1, pp 225-247 234 Proof. For any soft set ),( EG over X , let }),(~:),{(=),( 12 c ee EGxExEM βˆˆβˆˆΞ·ο• . Then, ),( EM is a p -open soft set. Moreover, every soft point in ),( EM is a p -open soft set. We set )},(~:),{(=),( EMyEyEF c βˆˆΞ±Ξ±ο‰ , then ),( EF is a p -closed soft set. Now, we take cc EMEGEN ),(~),(=),( ∩ . Let ),(~ ENw ∈γ . Then cEGw ),(~∈γ and cEMw ),(~∈γ . Therefore, ),( EwΞ³ is a p -closed soft set because if ),( EwΞ³ is not p -closed soft set, then by 𝑃𝑃𝑃𝑃𝑇𝑇1 2οΏ½ βˆ— property and Theorem 2.9 we conclude that ),( EwΞ³ is p -open soft set. It follows that ),(~),( EMEw βŠ†Ξ³ which contradicts with cEMw ),(~∈γ . Consequently, every soft point in ),( EN is a p -closed soft set. Put )},(~:),{(=),( ENzEzEH c βˆˆΞ²Ξ²ο‰ implies )}],(~:),{([=),( 1212 ENzEzskerEHsker c βˆˆΞ²Ξ²ο‰ )},(~:),({= 12 ENzEzsker c βˆˆΞ²Ξ²ο‰ [ by Theorem 3.32 in [10]] )},(~:),{(= ENzEz c βˆˆΞ²Ξ²ο‰ [for cEz ),( Ξ² is p -open soft set] ),(= EH . Hence, ),( EH is a Ξ›p -soft set. Now, since cEGEM ),(~),( βŠ† , then cEGy ),(~∈α ),(~ EMy βˆˆβˆ€ Ξ± implies cEyEG ),(~),( Ξ±βŠ† ),(~ EMy βˆˆβˆ€ Ξ± . It follows that ),(~),( EFEG βŠ† . Similarly, since cEGEN ),(~),( βŠ† , then ),(~),( EHEG βŠ† . Consequently, ),(~),(~),( EFEHEG βˆ©βŠ† . On the other hand, let ),(~),(~ EFEHze ∩∈ . Assume that ),(~ EGze ∈/ , then c e EGz ),(~∈ . We have two cases: Case(1): If ),( Eze is p -open soft point, then ),(~ EMze ∈ it follows that c e EzEF ),(~),( βŠ† which contradicts with ),(~ EFze ∈ . Case(2): If ),( Eze is not p -open soft point, then ),(~ EMze ∈/ it follows that c e EMz ),(~∈ . Therefore, ),(~ ENze ∈ which implies that c e EzEH ),(~),( βŠ† which contradicts with ),(~ EHze ∈ . In both cases we have a contradiction. Hence, ),(~ EGze ∈ . Consequently, ),(~),(=),( EFEHEG ∩ . From the above we conclude that, ),( EG is a Ξ»p -closed soft set. Conversely, let ),( Exe be a soft point in X . Then, c e Ex ),( is a soft set over X . Therefore, by hypothesis, c e Ex ),( is a Ξ»p -closed soft set. Now, if ),( Exe is a p -open soft set, then we are done. If ),( Exe is not American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2017) Volume 28, No 1, pp 225-247 235 p -open soft set, then c e Ex ),( is not p -closed soft set. But c e Ex ),( is a Ξ»p -closed soft set, then there exist a Ξ›p -soft set ),( EH and a p -closed soft set ),( EF such that ),(~),(=),( EFEHEx c e ∩ . But the only p -closed soft superset of c e Ex ),( is ),~( EX . Therefore, ),(=),( EHEx c e . Consequently, c e c e ExExsker ),(=),(12 . Hence, by Lemma 3.1, c e Ex ),( is a p -open soft set. Therefore, ),( Exe is a p -closed soft set. Definition 3.1 A soft topological space ),,( EX Ξ· is said to be a soft 𝑇𝑇1 4οΏ½ [briefly, 𝑃𝑃𝑇𝑇1 4οΏ½ ] if for every finite soft set ),( EG over X and every ),(~ EGxe ∈/ , there exists a soft set cex EM Ξ·Ξ·ο•βˆˆ),( such that ),(~ EMx ex e ∈/ and ),(~),( EMEG exβŠ† . Example 3.2 Let ZX = , where Z denote the set of all integer numbers, E be a non-empty set of parameters such that }{= 1 dEE βˆͺ and 1Ed ∈/ and let }),(),(~0:)(),{()},~{(= setsoftfiniteisEGandEGZSSEGE c dEd βˆˆβˆˆο•Ο†Ξ· . It is easy to prove that dΞ· is a soft topology on Z . Now, let ),( EF be a finite soft set and ),(~ EFxe ∈/ , then we have four cases: Case(1): dex 0β‰  , 0β‰ x and ),(~0 EFd ∈/ . In this case, we have c d EF ),(~0 ∈ . But, ),( EF is finite soft set, then cEF ),( is an open soft set. It follows that ),( EF is a closed soft set. Take ),(=),( EFEM ex . Case(2): dex 0β‰  , 0β‰ x and ),(~0 EFd ∈ . In this case, we have c ed Ex ),(~0 ∈ . So, c e Ex ),( is an open soft set, c e ExEF ),(~),( βŠ† and c ee Exx ),(~∈/ . Take c e ex ExEM ),(=),( . Case(3): dex 0β‰  , 0=x . In this case, we have de β‰  . So, c ed Ex ),(~0 ∈ . Therefore, c e Ex ),( is an open soft set. Take c e ex ExEM ),(=),( . Case(4): dex 0= . In this case, we have de = and 0=x implies ),(~0 EFd ∈/ . Therefore, cEF ),( is an open soft set. It follows that ),( EF is a closed soft set. Take ),(=),( EFEM ex . Consequently, ),,( EZ dΞ· is a 𝑃𝑃𝑇𝑇1 4οΏ½ . Theorem 3.3 Let ),,( EX Ξ· be a soft topological space. Then, American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2017) Volume 28, No 1, pp 225-247 236 1) If ),,( EX Ξ· is a 𝑃𝑃𝑇𝑇1 4οΏ½ , then it is 0ST . 2) If ),,( EX Ξ· is a 𝑃𝑃𝑇𝑇1 2οΏ½ , then it is 𝑃𝑃𝑇𝑇1 4οΏ½ . Proof. Straightforward. Definition 3.2 A sbts ),,,( 21 EX Ξ·Ξ· is said to be a pairwise soft 𝑇𝑇1 4οΏ½ βˆ— [briefly, 𝑃𝑃𝑃𝑃𝑇𝑇1 4οΏ½ βˆ— ] if for every finite soft set ),( EG over X and every ),(~ EGxe ∈/ , there exists a soft set cex EM 1212),( Ξ·Ξ· ο•βˆˆ such that ),(~ EMx ex e ∈/ and ),(~),( EMEG exβŠ† . Theorem 3.4 Let ),,,( 21 EX Ξ·Ξ· be a sbts. Then, ),,,( 21 EX Ξ·Ξ· is a 𝑃𝑃𝑃𝑃𝑇𝑇1 4οΏ½ βˆ— iff every finite soft set is a Ξ»p -closed soft set. Proof. Let ),( EG be a finite soft set over X . Since, ),,,( 21 EX Ξ·Ξ· is a 𝑃𝑃𝑃𝑃𝑇𝑇1 4οΏ½ βˆ— , then for every ),(~ EGxe ∈/ there exist cex EM 1212),( Ξ·Ξ· ο•βˆˆ such that ),(~ EMx ex e ∈/ and ),(~),( EMEG exβŠ† . Set )},(~),(),,(~:),{(=),( 12),(~ EMEGEMxEMEH exex e ex EGex βŠ†βˆˆ/∈ ∈/ η and )},(~),(),,(~:),{(=),( 12),(~ EMEGEMxEMEF exex e cex EGex βŠ†βˆˆ/∈ ∈/ η . It is clear that ),( EH is a Ξ›p -soft set and ),( EF is a p -closed soft set. Moreover, ),(~),(=),( EFEHEG ∩ , Therefore, ),( EG is a Ξ»p -closed soft set. Conversely, let ),( EG be a finite soft set over X and let ),(~ EGxe ∈/ . Since ),( EG is a Ξ»p -closed soft set, then ),(~),(=),( EFEHEG ∩ , where ),( EH is a Ξ›p -soft set and ),( EF is a p -closed soft set. So, ),(~ EHxe ∈/ or ),(~ EFxe ∈/ . If ),(~ EHxe ∈/ , then ),(~ 12 EHskerxe ∈/ . It follows that there exists a p -open soft set ),( EN such that ),(~),( ENEH βŠ† and ),(~ ENxe ∈/ , in this case take ),(=), ENEM ex . If ),(~ EFxe ∈/ , take ),(=), EFEM ex . Consequently, ),,,( 21 EX Ξ·Ξ· is a 𝑃𝑃𝑃𝑃𝑇𝑇1 4οΏ½ βˆ— . Theorem 3.5 Let ),,,( 21 EX Ξ·Ξ· be a sbts. Then, 1) If ),,( 1 EX Ξ· or ),,( 1 EX Ξ· is a 𝑃𝑃𝑇𝑇1 4οΏ½ , then ),,,( 21 EX Ξ·Ξ· is a 𝑃𝑃𝑃𝑃𝑇𝑇1 4οΏ½ βˆ— . 2) If ),,,( 21 EX Ξ·Ξ· is a 𝑃𝑃𝑃𝑃𝑇𝑇1 4οΏ½ βˆ— , then ),,( EX pΛη is a 𝑃𝑃𝑇𝑇1 4οΏ½ . Proof. Straightforward. American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2017) Volume 28, No 1, pp 225-247 237 Theorem 3.6 Let ),,,( 21 EX Ξ·Ξ· be a sbts. Then, 1) If ),,,( 21 EX Ξ·Ξ· is a 𝑃𝑃𝑃𝑃𝑇𝑇1 4οΏ½ βˆ— , then it is 𝑃𝑃𝑃𝑃𝑇𝑇0βˆ—. 2) If ),,,( 21 EX Ξ·Ξ· is a 𝑃𝑃𝑃𝑃𝑇𝑇1 2οΏ½ βˆ— , then it is 𝑃𝑃𝑃𝑃𝑇𝑇1 4οΏ½ βˆ— . Proof. (1) : Let EXyx )(, ΞΎΞ²Ξ± ∈ such that Ξ²Ξ± yx β‰  . Since ),( ExΞ± is a finite soft set and ),(~ Exy Ξ±Ξ² ∈/ , then by 𝑃𝑃𝑃𝑃𝑇𝑇1 4οΏ½ βˆ— property there exists cEG 1212),( Ξ·Ξ· βˆͺ∈ such that ),(~),( EGEx βŠ†Ξ± and ),(~ EGy ∈/Ξ² . It follows that ),(~ EGx ∈α and ),(~ EGy ∈/Ξ² . Therefore, by Lemma 2.1, ),,,( 21 EX Ξ·Ξ· is a 𝑃𝑃𝑃𝑃𝑇𝑇1 4οΏ½ βˆ— . (2) : It follows from Theorems 3.2 and 3.4. Remark 3.2 The following examples shows that the converse of items (1) and (2) in Theorem 3.6 are not true in general. Example 3.3 In Example 3.1, it is easy to prove that ),,,( 21 EX Ξ·Ξ· is a 𝑃𝑃𝑃𝑃𝑇𝑇0βˆ— . Let )},(}),{,{(=),( 21 XeyeEF . We note that ),(~ 1 EFxe ∈/ and ),( EF is finite soft set. But, there is no p - open or p -closed soft set ),( EM such that ),(~),( EMEF βŠ† and ),(~ 1 EMxe ∈/ . Hence, ),,,( 21 EX Ξ·Ξ· is not 𝑃𝑃𝑃𝑃𝑇𝑇1 4οΏ½ βˆ— . Example 3.4 From Example 3.2 and Theorem 3.3, we have ),,,( EZ dd Ξ·Ξ· is a 𝑃𝑃𝑃𝑃𝑇𝑇1 4οΏ½ βˆ— , but it is not 𝑃𝑃𝑃𝑃𝑇𝑇1 2οΏ½ βˆ— because ),(0 Ed is a soft point in Z but it is neither p -open nor p -closed soft set. Corollary 3.1 If X is a finite set, then ),,,( 21 EX Ξ·Ξ· is a 𝑃𝑃𝑃𝑃𝑇𝑇1 2οΏ½ βˆ— iff it is a 𝑃𝑃𝑃𝑃𝑇𝑇1 4οΏ½ βˆ— . 4. Pairwise locally closed soft sets In this section, we introduce the notion of pairwise locally closed soft sets. Some basic properties of them and their relationships with different types of soft sets are studied. Definition 4.1 A soft set ),( EG is said to be a pairwise locally closed soft set in a soft bitopological space ),,,( 21 EX Ξ·Ξ· [briefly, p - locally -closed soft set] if ),(~),(=),( EHEFEG ∩ , where ),( EF is a p - closed soft set and ),( EH is a p -open soft set. The family of all p - locally closed soft sets we denoted by EXPLCS ),,( 21 Ξ·Ξ· . Theorem 4.1 Let ),,,( 21 EX Ξ·Ξ· be a sbts. Then, American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2017) Volume 28, No 1, pp 225-247 238 every p -open ( p -closed) soft set is p -locally closed soft set. Proof. Immediate from definition. Remark 4.1 The union (intersection) of two p locally closed soft sets need not be a p -locally closed soft set as shown in the following example Example 4.1 In Example 3.1, it is clear that ),( 2 EG and ),( EK are p -locally closed soft sets [by Proposition 4.1] but ),(~),( 2 EKEG βˆͺ is not p -locally closed soft set. Proposition 4.1 Let ),,,( 21 EX Ξ·Ξ· be a sbts. Then, every p -locally closed soft set is Ξ»p -closed soft set. Proof. Straightforward Theorem 4.2 A soft set ),( EG in a sbts ),,,( 21 EX Ξ·Ξ· is a p -locally closed soft set iff ),(~),(=),( 12 EHEGsclEG ∩ for some 12),( η∈EH . Proof. Let ),( EL be a p -locally closed soft set. Then, there exist p -closed soft set ),( EF and p -open soft set ),( EH such that ),(~),(=),( EGEFEL ∩ . It follows that ),(~),(~),( 1212 EGsclEFELscl βˆ©βŠ† which implies that ),(~),(12 EFELscl βŠ† and so ),(~),(~),(~),(12 EGEFEGELscl βˆ©βŠ†βˆ© . Therefore, ),(~),(~),(12 ELEGELscl βŠ†βˆ© . On the other hand, since ),(~),( 12 ELsclEL βŠ† , then ),(~),(~),(~),( 12 EGELsclEGEL βˆ©βŠ†βˆ© . It follows that ),(~),(~),( 12 EGELsclEL βˆ©βŠ† . Consequently, ),(~),(=),( 12 EGELsclEL ∩ . The sufficiency of the theorem is clear. Definition 4.2 A sbts ),,,( 21 EX Ξ·Ξ· is said to be a pairwise soft π‘‡π‘‡πΏπΏβˆ— [briefly, π‘ƒπ‘ƒπ‘ƒπ‘ƒπ‘‡π‘‡πΏπΏβˆ—] if every soft point in X is a p -locally closed soft set. Theorem 4.3 Let ),,,( 21 EX Ξ·Ξ· be a sbts. Then 1) If ),,,( 21 EX Ξ·Ξ· is a 𝑃𝑃𝑃𝑃𝑇𝑇1 2οΏ½ βˆ— , then it is π‘ƒπ‘ƒπ‘ƒπ‘ƒπ‘‡π‘‡πΏπΏβˆ—. 2) If ),,,( 21 EX Ξ·Ξ· is a π‘ƒπ‘ƒπ‘ƒπ‘ƒπ‘‡π‘‡πΏπΏβˆ—, then it is 𝑃𝑃𝑃𝑃𝑇𝑇0βˆ—. American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2017) Volume 28, No 1, pp 225-247 239 Proof. (1) : Since ),,,( 21 EX Ξ·Ξ· is a 𝑃𝑃𝑃𝑃𝑇𝑇1 2οΏ½ βˆ— , then every soft point is a p -open soft set or a p -closed soft set. It follows that, by Proposition 4.1, every soft point is a p -locally closed soft set. Consequently, ),,,( 21 EX Ξ·Ξ· is a π‘ƒπ‘ƒπ‘ƒπ‘ƒπ‘‡π‘‡πΏπΏβˆ—. (2) : Since ),,,( 21 EX Ξ·Ξ· is a π‘ƒπ‘ƒπ‘ƒπ‘ƒπ‘‡π‘‡πΏπΏβˆ—, then every soft point is a p -locally closed soft set. It follows that, by Proposition 4.1, every soft point is a Ξ»p -closed soft set. Therefore, ),,,( 21 EX Ξ·Ξ· is a 𝑃𝑃𝑃𝑃𝑇𝑇0βˆ— [by Theorem 2.8]. Remark 4.2 The converse of items (1) and (2) in the above theorem is not true in general which is shown in the following example. Example 4.2 Let ),,,( 21 EX Ξ·Ξ· be the same in Example 3.1. Since ),(=),( 11 EyEP e c , then ),( 1 Eye is a p -closed soft set. It follows that ),( 1 Eye is a p -locally closed soft set. Also, ),(=),( 22 EyEP e c is a p - locally closed soft set. Now, since c e EGEHEx ),(~),(=),( 21 ∩ , then ),( 1 Exe is a p -locally closed soft set. Also, c e EGEHEx ),(~),(=),( 12 ∩ . It follows that ),( 2 Exe is a a p -locally closed soft set. Therefore, every soft point in X is a p -locally closed soft set. Hence, ),,,( 21 EX Ξ·Ξ· is a π‘ƒπ‘ƒπ‘ƒπ‘ƒπ‘‡π‘‡πΏπΏβˆ—. On the other hand, ),,,( 21 EX Ξ·Ξ· is not 𝑃𝑃𝑃𝑃𝑇𝑇1 2οΏ½ βˆ— because ),( 1 Exe neither p -open nor p -closed soft set. Remark 4.3 The concepts of π‘ƒπ‘ƒπ‘ƒπ‘ƒπ‘‡π‘‡πΏπΏβˆ— and 𝑃𝑃𝑃𝑃𝑇𝑇1 4οΏ½ βˆ— are independent as shown in the following example. Example 4.3 In Example 4.2, we have ),,,( 21 EX Ξ·Ξ· is a π‘ƒπ‘ƒπ‘ƒπ‘ƒπ‘‡π‘‡πΏπΏβˆ— but it is not 𝑃𝑃𝑃𝑃𝑇𝑇1 2οΏ½ βˆ— which implies that ),,,( 21 EX Ξ·Ξ· is not 𝑃𝑃𝑃𝑃𝑇𝑇1 4οΏ½ βˆ— [by Corollary 3.1]. From Examples 3.2 and 3.4, we have ),,,( EX dd Ξ·Ξ· is a 𝑃𝑃𝑃𝑃𝑇𝑇1 4οΏ½ βˆ— but it is not π‘ƒπ‘ƒπ‘ƒπ‘ƒπ‘‡π‘‡πΏπΏβˆ— for ),(0 Ed is not p -locally closed soft set. 5. On Ξ»p - open soft sets In this section, we introduce the notion of pairwise Ξ» -open soft sets. Some basic properties of them and their relationships with different types of soft sets are studied. Definition 5.1 A soft set ),( EG in a sbts ),,,( 21 EX Ξ·Ξ· is said to be a pairwise Ξ» -open soft set[briefly, Ξ»p -open soft set] if its complement is a Ξ»p -closed soft set. Clearly, ),( EG is a Ξ»p -open soft set iff ),(~),(=),( EMEHEG βˆͺ , where 12),( η∈EH and βˆ¨Ξ›βˆˆ p c pEM Ξ·Ξ· =),( . We denoted the family of all Ξ»p -open soft sets by EXOSP ),,( 21 Ξ·Ξ·Ξ» . American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2017) Volume 28, No 1, pp 225-247 240 Theorem 5.1 Let ),,,( 21 EX Ξ·Ξ· be a sbts and EXSSEG )(),( ∈ . Then: 1) Every p -open(closed) soft set is a Ξ»p -open soft set. 2) Every ∨p -soft set is a Ξ»p -open soft set. 3) An arbitrary union of Ξ»p -open soft sets is a Ξ»p -open soft set. Proof. Straightforward. Corollary 5.1 The family of all Ξ»p -open soft sets is a supra soft topology. Proof. The proof is direct from Theorem 5.1. Remark 5.1 The family of all Ξ»p -open soft sets may not be a soft topology as shown by the following example. Example 5.1 In Example 3.1, we have 1,...,6}=:),{(=)( 1212 iEMXSS i c E ηη where )},(}),{,{(=),( 211 Ο†exeEM , })}{,(),,{(=),( 212 xeeEM Ο† , )},(),,{(=),( 213 XeeEM Ο† , )},(),,{(=),( 214 Ο†eXeEM , )},(}),{,{(=),( 215 XeyeEM , })}{,(),,{(=),( 216 yeXeEM . It is easy to show that )},(),,{(=),,( 65121221 EMEMXOSP c E ηηηηλ . It is clear that EXOSPEGEP ),,(),(),,( 2112 ηηλ∈ but ),(=),(~),( 112 EMEGEP ∩ which is not Ξ»p - open soft set. Theorem 5.2 Let ),,,( 21 EX Ξ·Ξ· be a sbts and EXSSEG )(),( ∈ . The following statements are equivalent. 1) ),( EG is a Ξ»p -open soft set. 2) ),(~),(=),( EMEHEG βˆͺ , where 12),( η∈EH , ∨∈ pEM Ξ·),( . 3) ),(~),(=),( 12 EMEGsintEG βˆͺ , where ∨∈ pEM Ξ·),( . 4) ),(~),(=),( * 1212 EGskerEGsintEG βˆͺ . American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2017) Volume 28, No 1, pp 225-247 241 Proof. (2)(1)β‡’ : Let ),( EG be a Ξ»p -open soft set. Then, ),(~),(=),( EMEHEG βˆͺ , where 12),( η∈EH , c pEM Ξ›βˆˆΞ·),( , but βˆ¨Ξ› p c p Ξ·Ξ· = [by Theorem 2.7], then (2) holds. (3)(2)β‡’ : Let ),(~),(=),( EMEHEG βˆͺ , where 12),( η∈EH , ∨∈ pEM Ξ·),( . Since ),(~),( EGEH βŠ† , then ),(~),( 1212 EGsintEHsint βŠ† . Therefore, ),(~),( 12 EGsintEH βŠ† . It follows that ),(~),(~),(~),( 12 EMEGsintEMEH βˆͺβŠ†βˆͺ . So, ),(~),(~),( 12 EMEGsintEG βˆͺβŠ† , but ),(~),(12 EGEGsint βŠ† and ),(~),( EGEM βŠ† then ),(~),(~),(12 EGEMEGsint βŠ†βˆͺ . Therefore, ),(~),(=),( 12 EMEGsintEG βˆͺ . Hence, (3) holds. (3)β‡’ (4): Let ),(~),(=),( 12 EMEGsintEG βˆͺ , where ∨∈ pEM Ξ·),( . Then βˆͺ~),(=),( 12 EGsintEG ),(* 12 EMsker . Since ),(~),( EGEM βŠ† , then ),(~),( * 12 * 12 EGskerEMsker βŠ† which implies that ),(~),(~),(~),( * 1212 * 1212 EGskerEGsintEMskerEGsint βˆͺβŠ†βˆͺ . It follows that ),(~),(~),( * 1212 EGskerEGsintEG βˆͺβŠ† , but ),(~),(12 EGEGsint βŠ† and ),(~),(* 12 EGEGsker βŠ† , then ),(~),(~),( * 1212 EGEGskerEGsint βŠ†βˆͺ . Therefore, ),(~),(=),( * 1212 EGskerEGsintEG βˆͺ . Hence, (4) holds. (1)(4)β‡’ : Since 1212 ),( η∈EGsint , ∨∈ pEGsker Ξ·),(* 12 , then ),( EG is a Ξ»p -open soft set. Hence, (1) holds. Definition 5.2 Let ),,,( 21 EX Ξ·Ξ· be a sbts, and let EXSSEG )(),( ∈ . The Ξ»p -soft closure of ),( EG , denoted by ),( EGsclpΞ» , is defined by )},(~),(:),,(),{(=),( 21 EFEGXCSPEFEGscl Ep βŠ†βˆˆ ηηλλ  . Lemma 5.1 Let ),,,( 21 EX Ξ·Ξ· be a sbts and EXSSEG )(),( ∈ . The Ξ»p -soft closure of ),( EG is the smallest Ξ»p -closed soft superset of ),( EG . Proof. Immediate from Theorem 3.1. Theorem 5.3 Let ),,,( 21 EX Ξ·Ξ· be a sbts and let EXSSEHEG )(),(),,( ∈ . Then 1) ),~(=),~( EXEXsclpΞ» ; ),~(=),~( EEsclp φφλ . American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2017) Volume 28, No 1, pp 225-247 242 2) ),(~),( EGsclEG pΞ»βŠ† . 3) ),(~),(),(~),( EHsclEGsclEHEG pp λλ βŠ†β‡’βŠ† . 4) ),( EG is a Ξ»p -closed soft set ),(=),( EGEGsclpλ⇔ . 5) )],(~),[(~),(~),( EHEGsclEHsclEGscl ppp βˆͺβŠ†βˆͺ λλλ . 6) ),(=)],([ EGsclEGsclscl ppp λλλ . Proof. Straightforward. Corollary 5.2 Ξ»pscl is a supra soft closure operator and it is induced a supra soft topology given by }),(=),(:)(),{(= cc pEp EGEGsclXSSEG λλη ∈ which is precisely EXOSP ),,( 21 Ξ·Ξ·Ξ» . Remark 5.2 The equality in Theorem 5.3 (5) may not be satisfied as shown in the following example. Example 5.2 Consider the Example 3.1, since cEH ),( })}){,(}),{,{((= 21 yeye is p -closed soft set, then cEH ),( is a Ξ»p -closed soft set. Therefore, cc p EHEHscl ),(=),(Ξ» [ by Theorem 5.3 (4) ]. Similarly, cc p EGEGscl ),(=),( 11Ξ» . It follows that })}{,(}),{,{(=),(~),( 211 XeyeEGsclEHscl c p c p λλ βˆͺ . On the other hand, ),~(=})}{,(}),{,{(=]),(~),[( 211 EXXeyesclEGEHscl p cc p λλ βˆͺ . Hence, c p c p cc p EGsclEHsclEGEHscl ),(~),(]),(~),[( 11 λλλ βˆͺβ‰ βˆͺ . Theorem 5.4 Let ),,,( 21 EX Ξ·Ξ· be a sbts and EXSSEG )(),( ∈ . Then, ),,~(),(~),(),(~ EEGEOEGsclx expe φλ β‰ βˆ©β‡”βˆˆ ),( EO exβˆ€ , where ),( EO ex is a Ξ»p -open soft set containing ex . Proof. Let ),(~ EGsclx pe λ∈ and assume that there exists 12),( η∈EO ex such that ),~(=),(~),( EEGEO ex Ο†βˆ© . Then c ex EOEG ),(~),( βŠ† which implies that c ex c expp EOEOsclEGscl ),(=),(~),( λλ βŠ† , therefor ),~(=),(~),( EEOEGscl exp φλ ∩ , a contradiction. Conversely, assume that ),(~ EGsclx pe λ∈/ . Then c pe EGsclx )],([~ λ∈ it follows tha cEGscl )],([ 12 is a Ξ»p -open soft set containing ex . Thus, by hypothesis ),~(),(~)],([ 12 EEGEGscl c Ο†β‰ βˆ© which contradicts American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2017) Volume 28, No 1, pp 225-247 243 with ),(~),( EGsclEG pΞ»βŠ† . Corollary 5.3 Let ),,,( 21 EX Ξ·Ξ· be a sbts. Then, Epp c XSS )(12 βŠ†βŠ†βŠ† ∨ ληηη . Proof. Straightforward. Remark 5.3 The equality in Corollary 5.3 may not be satisfied as shown in the following example. Example 5.3 In Example 5.1, clear that Ep XSS )(≠λη . Also, we have λη pEP ∈),( 1 but ∨∈/ pEP Ξ·),( 1 because ),(})}{,(}),{,{(=),( 1111 * 12 EPyexeEPsker β‰  . Hence, ληη pp β‰ βˆ¨ . Lemma 5.2 Let ),,,( 21 EX Ξ·Ξ· be a sbts. Then, the supra soft topology λη p is a soft topology on X iff the finite intersection of Ξ»p -open soft sets is a Ξ»p -open soft set or equivalently, the finite union of Ξ»p -closed soft sets is a Ξ»p -closed soft set. Proof. Suppose that λη p is a soft topology on X . Let ),( EG and ),( EH be two Ξ»p -closed soft sets. Then, cEG ),( and cEH ),( are Ξ»p -open soft sets. Therefore, cc EHEG ),(~),( ∩ is a Ξ»p -open soft set[for λη p is a soft topology]. Consequently, ),(~),( EHEG βˆͺ is a Ξ»p -closed soft set. Conversely, it is obvious. Definition 5.3 A sbts ),,,( 21 EX Ξ·Ξ· is called a pairwise soft π‘‡π‘‡πœ†πœ†βˆ— [briefly, π‘ƒπ‘ƒπ‘ƒπ‘ƒπ‘‡π‘‡πœ†πœ†βˆ— ] if λη p is a soft topology on X . Theorem 5.5 Every 𝑃𝑃𝑃𝑃𝑇𝑇1 2οΏ½ βˆ— is a π‘ƒπ‘ƒπ‘ƒπ‘ƒπ‘‡π‘‡πœ†πœ†βˆ—. Proof. Let ),,,( 21 EX Ξ·Ξ· be a 𝑃𝑃𝑃𝑃𝑇𝑇1 2οΏ½ βˆ— space. Then, by Theorem 3.2, every soft set is a Ξ»p -closed soft set. Consequently, the union of any two Ξ»p -closed soft sets is a Ξ»p -closed soft set. Hence, by Lemma 5.2, λη p is a soft topology on X . Hence, ),,,( 21 EX Ξ·Ξ· is a π‘ƒπ‘ƒπ‘ƒπ‘ƒπ‘‡π‘‡πœ†πœ†βˆ—. Remark 5.4 A π‘ƒπ‘ƒπ‘ƒπ‘ƒπ‘‡π‘‡πœ†πœ†βˆ— need not be a 𝑃𝑃𝑃𝑃𝑇𝑇1 2οΏ½ βˆ— or 𝑃𝑃𝑃𝑃𝑇𝑇0βˆ— as shown by the following example. Example 5.4 Let },,{= zyxX , },{= 21 eeE and let American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2017) Volume 28, No 1, pp 225-247 244 )},(),,(),,(),,~(),,~{(= 3211 EGEGEGEXEφη , )},(),,(),,(),,~(),,~{(= 3212 EHEHEHEXEφη , such that })},{,(}),,{,{(=),( 211 yxezxeEG , })},{,(}),,{,{(=),( 212 zyezyeEG , })}{,(}),{,{(=),( 213 yezeEG , })},{,(}),,{,{(=),( 211 zxeyxeEH , })},{,(}),{,{(=),( 212 yxezeEH , })}{,(),,{(=),( 213 xeeEH Ο† . It is easily seen that ),,,( 21 EX Ξ·Ξ· is a sbts and )},(),,(),,(),,(),,(),,(),,(),,~(),,~{(= 32132112 EPEHEHEHEGEGEGEXEφη , where )},(}),,{,{(=),( 21 XezyeEP . Consequently, }),(,),{(= 2112 cc p EGEGηη Ξ› . By studying all soft sets such that 50}1,2,3,...,=:),{(=)( 1212 iEMXSS i c E ηη we found that: c EXCSP 121221 =),,( Ξ·Ξ·Ξ·Ξ·Ξ»  . Moreover, ληηηλ pEXCSP =),,( 21 . Now, let λη pEKEN ∈),(),,( , then EXCSPEKEN ),,(),(),,( 21 ηηλ∈ . Therefore, EXCSPEKEN ),,(),(~),( 21 ηηλ∈∩ [by Theorem 3.1]. Hence, λη pEKEN ∈∩ ),(~),( . Therefore, λη p is a soft topology. Consequently, ),,,( 21 EX Ξ·Ξ· is a π‘ƒπ‘ƒπ‘ƒπ‘ƒπ‘‡π‘‡πœ†πœ†βˆ— but it is not 𝑃𝑃𝑃𝑃𝑇𝑇1 2οΏ½ βˆ— because ),( 1 Eye neither p -open soft set nor p -closed soft set. Also, ),,,( 21 EX Ξ·Ξ· is not 𝑃𝑃𝑃𝑃𝑇𝑇0βˆ— for 21 ee zy β‰  but there is no p -open soft set contains one of them but not American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2017) Volume 28, No 1, pp 225-247 245 contains the other. Theorem 5.6 A sbts ),,,( 21 EX Ξ·Ξ· is a 𝑃𝑃𝑃𝑃𝑇𝑇1 4οΏ½ βˆ— if it is a 𝑃𝑃𝑃𝑃𝑇𝑇0βˆ— and π‘ƒπ‘ƒπ‘ƒπ‘ƒπ‘‡π‘‡πœ†πœ†βˆ—. Proof. Suppose that ),,,( 21 EX Ξ·Ξ· is a 𝑃𝑃𝑃𝑃𝑇𝑇0βˆ— and π‘ƒπ‘ƒπ‘ƒπ‘ƒπ‘‡π‘‡πœ†πœ†βˆ— . Let ),( EF be a finite soft set. Then )},(~:),{(=),( EFxExEF ee βˆˆο• [by Proposition 2.1]. Since ),,,( 21 EX Ξ·Ξ· is a 𝑃𝑃𝑃𝑃𝑇𝑇0βˆ—, then ),( Exe is a 𝑝𝑝𝑝𝑝-closed soft set for all ),(~ EFxe ∈ . But, ),,,( 21 EX Ξ·Ξ· is a π‘ƒπ‘ƒπ‘ƒπ‘ƒπ‘‡π‘‡πœ†πœ†βˆ—, then πœ‚πœ‚π‘π‘πœ†πœ† is a soft topology on X . It follows that, by Lemma 5.2,the finite union of Ξ»p -closed soft sets is a Ξ»p -closed soft set. Consequently, )},(~:),{( EFxEx ee βˆˆο• is a Ξ»p -closed soft set. Therefore, ),( EF is a Ξ»p -closed soft set. Hence, by Theorem 3.4 we conclude that ),,,( 21 EX Ξ·Ξ· is a 𝑃𝑃𝑃𝑃𝑇𝑇1 4οΏ½ βˆ— . Remark 5.5 A πππππ“π“π›Œπ›Œβˆ— space need not be π‘·π‘·π‘·π‘·π‘»π‘»πŸπŸ πŸ’πŸ’οΏ½ βˆ— . In Example 5.4, we proved that ),,,( 21 EX Ξ·Ξ· is a π‘·π‘·π‘·π‘·π‘»π‘»π€π€βˆ— . Since X is a finite set and ),,,( 21 EX Ξ·Ξ· is not π‘·π‘·π‘·π‘·π‘»π‘»πŸπŸ 𝟐𝟐� βˆ— , then by Corollary 3.1 we conclude that ),,,( 21 EX Ξ·Ξ· is not π‘·π‘·π‘·π‘·π‘»π‘»πŸπŸ πŸ’πŸ’οΏ½ βˆ— . 6. Conclusion The notions of pairwise Ξ» -closed soft sets, pairwise Ξ» -open soft sets and pairwise locally closed soft sets are turn out to be useful in the study of some soft bitopologies which are not * 1PST . In this paper, we introduce new classes of soft sets called Ξ»p -closed soft sets, Ξ»p -open soft sets and p -locally closed soft sets in soft bitopological spaces. It turn out that Ξ»p -closed soft sets, Ξ»p -open soft sets and p -locally closed soft sets are weaker forms of p -open(closed) soft sets. We also conclude several important properties of such soft sets. Moreover, we introduce and study a related pairwise soft separation axioms, namely, π‘·π‘·π‘·π‘·π‘»π‘»πŸπŸ πŸ’πŸ’οΏ½ βˆ— , π‘·π‘·π‘·π‘·π‘»π‘»π‘³π‘³βˆ— and π‘·π‘·π‘·π‘·π‘»π‘»π€π€βˆ— . We studied the relationships between these types of separation axioms. Acknowledgements The authors express their sincere thanks to the reviewers for their valuable suggestions. The authors are also thankful to the editors-in-chief and managing editors for their important comments which helped to improve the presentation of the paper. References [1] A. Aygunoglu and H. Aygun, Some notes on soft topological spaces, Neural Comput. Applic., vol. 21 (Suppl 1), pp. S113βˆ’S119, 2012. [2] Basavaraj Ittanagi, Soft bitopological spaces, International journal of Computer Applications, vol. 107 American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2017) Volume 28, No 1, pp 225-247 246 no. 7, pp. 1βˆ’ 4, 2011. [3] N. Cagman, S. Karatas and S. Enginoglu, Soft topology, Comput. Math. Appl. vol. 62, pp. 351βˆ’ 358, 2011. [4] S. A. El-Sheikh and A. M. Abd El-latif, Decompositions of some types of supra soft sets and soft continuity, International Journal of Mathematics Trends and Technology, vol. 9, no. 1, pp. 37βˆ’56, 2014. [5] D. N. Georgiou a and A. C. Megaritis, Soft set theory and topology, Appl. Gen. Topol., vol. 15, no. 1, pp. 93βˆ’109, 2014. [6] H. Hazra, P. Majumdar and S. K.Samanta, Soft topology, Fuzzy Inform. Eng., vol. 4, no. 1, pp.105βˆ’ 115, 2012. [7] S. Hussain and B. Ahmad, Some properties of soft topological spaces, Comput. Math. Appl., vol. 62, pp. 4058βˆ’ 4067, 2011. [8] M. Irfan Ali, M. Shabir and M. Naz, Algebraic structures of soft sets associated with new operations, Comput. Math. Appl., vol. 61, pp. 2647βˆ’ 2654, 2011. [9] A. Kandil, O. A. E. Tantawy, S. A. El-Sheikh and Shawqi A. Hazza, Generalized pairwise closed soft sets and the associate pairwise soft separation axioms, South Asian J. Math., vol. 6, no. 2, pp., 43βˆ’57, 2016. [10] A. Kandil, O. A. E. Tantawy, S. A. El-Sheikh and Shawqi A. Hazza, Pairwise open (closed) soft sets in a soft bitopological spaces, Ann. Fuzzy Math. Inform., vol. 11, no. 4, pp. 571βˆ’588, 2016. [11] A. Kandil, O. A. E. Tantawy, S. A. El-Sheikh and Shawqi A. Hazza, Pairwise soft separation axioms in soft bitopological spaces, Ann. Fuzzy Math. Inform., vol. xx, no. xx, pp. 1-20, 2017. [12] A. Kandil, O. A. E. Tantawy, S. A. El-Sheikh and Shawqi A. Hazza, Some types of pairwise soft sets and the associated soft topologies, Journal of Intelligent and Fuzzy Syst., vol. 32, no. 2, pp. 1007βˆ’ 1018, 2017. [13] K. Kannan, Soft generalized closed sets in soft topological spaces, Journal of Theoretical and Appl. Inform. Technology, vol. 37, no. 1, pp. 17βˆ’21, 2012. [14] D. Molodtsov, Soft set theory-First results, Comput. Math. Appl., vol. 37, pp. 19βˆ’ 31, 1999. [15] Sk. Nazmul and S. K. Samanta, Neighbourhood properties of soft topological spaces, Ann. Fuzzy Math. Inform. , vol. 6, no. 1, pp. 1βˆ’15, 2013. American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2017) Volume 28, No 1, pp 225-247 247 [16] Sk. Nazmul and S. K. Samanta, Some properties of soft topologies and group soft topologies, Ann. Fuzzy Math. Inform.m vol. 8, no. 4, pp. 645βˆ’ 661, 2014. [17] Ningxin Xie, Soft points and the structure of soft topological spaces, Ann. Fuzzy. Math. inform., vol. 10, no. 2, pp. 309βˆ’322, 2015. [18] E. Peyghan, B. Samadi and A. Tayebi, About soft topological spaces, Journal of New Results in Science, vol. 2, pp. 60βˆ’75, 2013. [19] D. Pie and D. Miao, From soft sets to information systems, Granular computing, IEEE Inter. Conf., vol. 2, pp. 617βˆ’ 621, 2005. [20] M. Shabir and M. Naz, On soft toplogical spaces, Comput. Math. Appl., vol. 61, no. 7, pp. 1786βˆ’ 1799, 2011. [21] Sujoy Das and S. K. Samanta, Soft metric, Ann. Fuzzy Math. Inform., vol.6, pp. 77βˆ’94, 2013. [22] I. Zorlutuna, M. Akdag, W. K. Min and S. Atmaca, Remarks on soft topological spaces, Ann. Fuzzy Math. Inform., vol. 3, no. 2, pp. 171βˆ’ 185, 2012.