42 American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) ISSN (Print) 2313-4410, ISSN (Online) 2313-4402 © Global Society of Scientific Research and Researchers http://asrjetsjournal.org/ Soft Pre-Open Sets In Soft Bitopological Spaces Ameer Mohammed Hussein Hasan* Department of mathematics Faculty of mathematics and computer science university of kufa, Najaf, Iraq Email: ameerm.hasan@uokufa.edu.iq Abstract In this work , I introduce the concept of soft bitopological space on a soft set and some definitions on soft pre- open set on soft bitopological space . Also introduce soft pre separation axioms , Spre- T , Spre- 1T and Spre- 2T , with study some properties in soft bitopological space. Keywords: Soft set; Soft pre-open; Soft topology; Soft bitopological spaces. 1. Introduction Many classical methods have been used to solve some complicated problems in engineering economics and environment. For instance, the interval mathematics, theory of fuzzy, theory of probability, and sets which can consider as mathematical tools for dealing with uncertainties since all these theories have their own problems and difficulties. In [2], the author in 1999 introduced the notion of soft set, which is free of difficulties in solving aforementioned problems, and it has been applied over many different fields. In 2011, Naim Cagman and his colleagues introduced a new concept of soft set called soft topology define by using the soft power set of soft set , and this first idea to soft mathematical concepts and structures that are based on the operations of theoretic soft set [6]. In 2011[7], the authors defined the concept of"soft topology on the collection of soft sets over"with some basic notations of soft topological spaces. In [6], the notion of soft topology was more general than that in [7]. Therefore, algebraists continue investigating the work of Cagman [6]"and follow their notations and mathematical formalism. In 2013 J. Subhashini and C. Sekar defined soft pre-open sets [4]"by following Cagman’s theory of soft topology. Therefore, this paper has introduced soft bitopological space relying on [6] and defined the soft pre-open set of soft bitopological space. Also, I have discussed soft pre separation axioms , Spre- T , Spre- 1T and Spre- 2T . ------------------------------------------------------------------------ * Corresponding author. http://asrjetsjournal.org/ American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 26, No 4 , pp 42-53 43 2. Preliminaries Through this section , I give and introduction some important definitions and facts about soft topology and recall primary definition soft set that need in this work . 2.1. Definition [2] The set of ordered pairs represents a soft set AF on the universe U , where is mapping and Af is called an approximate function of the soft set AF . However , the set of all soft sets over U is denoted by )(USS . 2.2. Definition [5] Let )(USF SA ∈ . If for all Ax∈ , then AF is called an empty set and denoted by φF . Moreover , If Uf A = for all Ax∈ , then AF is called an A -universal soft set and denoted by AF ˆ . But , If EA = , then A -universal soft set is called universal soft set and denoted by EF~ . 2.3. Definition [5] Let )(, USFF SBA ∈ . If )()( xfxf BA ⊆ for all x , then , AF is a soft subset of BF and denoted by . 2.4. Definition [5] Let )(, USFF SBA ∈ . Then, the soft union is denoted by BA FF ∪~ , however , the soft intersection is denoted by BA FF ∩~ Also , the soft difference of AF and BF is denoted by BA FF ∆~ , are defined by the approximate functions , , , respectively, on the onther hand , the soft complement c AF ~ of AF is defined by the approximate function )(=)( xfxf c AcA , where )(xf c A is the complement of the set )(xf A ; that is, )(=)( xfUxf A c A − for all Ax∈ . It is easy to see that A cc A FF =)( ~~ and E c FF ~ ~ =φ . 2.5. Proposition [5] Let )(,, USFFF SCBA ∈ . Then , 1- . American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 26, No 4 , pp 42-53 44 2- . 3- . 4- . 5- . 6- . 7- . 8- )~(~)~(=)~(~&)~(~)~(=)~(~ CABACBACABACBA FFFFFFFFFFFFFF ∩∪∩∪∩∪∩∪∩∪ . 2.6. Definition [6] Let )(USF SA ∈ ."The soft power set of AF is defined by". and its cardinality is defined by , where |)(| xf A is the cardinality of )(xf A . 2.7. Definition [6] Let )(USF SA ∈ . A soft topology on AF , denoted by τ~ , is a collection of soft subsets of AF having the following properties: • τφ ~, ∈AFF . • . • . 2.8. Definition [6] Let )~,( τAF be a soft space on AF . Every element of τ~ is called a soft open sets . 2.9. Definition [6] Let )~,( τAF be a soft space on AF and AB FF ⊆~ . Then , the collection American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 26, No 4 , pp 42-53 45 },~:~{=~ NIiFFF iABiABF ⊆∈∈∩ ττ is called a soft subspace topology on BF . Hence, )~,( BFBF τ is called a soft topological subspace of )~,( τAF . 2.10. Definition [6] Let )~,( τAF be a soft space on AF and AB FF ⊆~ . The soft interior of BF , denoted  BF , is defined as the union of all soft open subsets of BF . Note that  BF is the biggest soft open set that is contained by BF . 2.11. Theorem [6] Let )~,( τAF be a soft space on AF and ACB FFF ⊆~, . Then , 1-  BB FF =)( 2- CB FF ⊆~ . Then ,  CB FF ⊆~ 3-  )~(=~ CBCB FFFF ∩∩ 4-  )~(~~ CBCB FFFF ∪⊆∪ 2.12. Definition [6] Let )~,( τAF be a soft space on AF and AB FF ⊆~ . Then, the soft closure of BF , denoted BF is defined as the soft intersection of all soft closed superset of BF . Note that BF is the smallest soft closed set that containing BF . 2.13. Theorem [6] Let )~,( τAF be a soft space on AF and ACB FFF ⊆~, . Then , 1- BB FF =)( . 2- )(=)( ~~ c B c B FF . 3- CB FF ⊆~ . Then , . American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 26, No 4 , pp 42-53 46 4- . 5- . 2.14. Theorem [6] Let )~,( τAF be a soft space on AF and AB FF ⊆~ Then BBB FFF ⊆⊆ ~~ . 2.15. Definition Let )~,( 1τAF and )~,( 2τAF be the two different soft topologies on AF . Then )~,~,( 21 ττAF is called a soft bitopological space . 2.16. Example Let },,,{= 4321 uuuuU , },,{= 321 wwwE , },{= 21 wwA such that EA⊆~ and then })}{,{(= 111 uwFA })}{,{(= 212 uwFA })}{,{(= 324 uwFA })}{,{(= 425 uwFA })},{,{(= 4326 uuwFA })}{,(}),{,{(= 32117 uwuwFA })}{,(}),{,{(= 42118 uwuwFA })}{,((}),{,{(= 322110 uwuwFA American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 26, No 4 , pp 42-53 47 })}{,(}),{,{(= 422111 uwuwFA })},{,(}),{,{(= 4322112 uuwuwFA })}{,(}),,{,{(= 4221114 uwuuwFA AA FF = 15 φFFA = 16 Then },{=~ 1 AFFφτ and },,,{=~ 1122 AAA FFFFφτ are a soft topology of AF then )~,~,( 21 ττAF , is a soft bitopological space . 2.17. Definition Let )~,~,( 21 ττAF be a soft bitopological space over AF and AB FF ⊆~ . Then },~:~{=~ 11 NIiFFFF iABiAB ⊆∈∈∩ ττ and },~:~{=~ 22 NIiFFFF iABiAB ⊆∈∈∩ ττ are said to be the relative topologies on BF . Then )~,~,( 21 BBB FFF ττ is called a relative soft bitopological space of )~,~,( 21 ττAF 2.18. Theorem If )~,~,( 21 ττAF is a soft bitopological space then 21 ~~~ ττ ∩ is a soft topological space over AF . Proof :- • 21 ~~~, ττφ ∩∈AFF . • Let },{ IiF iA ∈ be a family of soft sets in 121 ~~~~ τττ ∈⇒∩ iAF and 2 ~τ∈ iAF for all Ii∈ . Therefore 1 ~τ∈ ∈ iAIi F and 2 ~τ∈ ∈ iAIi F . Thus 21 ~~~ ττ ∩∈ ∈ iAIi F . • Let . Then 1 ~τ∈ iAF and . Since 11= ~τ∈ ′ iA n i F1 and 21= ~τ∈ ′ iA n i F1 . Therefore NnniF iA n i ∈′′≤≤∩∈ ′ ,,1~~~ 211= ττ1 . American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 26, No 4 , pp 42-53 48 2.19. Remark If )~,~,( 21 ττAF is a soft bitopological space then 21 ~~~ ττ ∪ is not a soft topological space over AF . 2.20. Example Let us consider 2.16 and let },,{=~ 11 AA FFFφτ and },,{=~ 22 AA FFFφτ are soft topology of AF then )~,~,( 21 ττAF is soft bitopological space . Now },,,{=~~~ 2121 AAA FFFFφττ ∪ . If take 21 , AA FF , 21211121 ~~~})}{,(}),{,{(=~ ττ ∪∉∪ uxuxFF AA . Thus 21 ~~~ ττ ∪ is not soft topology on AF . 3. Some Definition of Soft Pre-open set in soft bitopological space In this section introduce some definitions of soft pre-open set , soft pre-closed , soft pre-neighborhood , soft pre- closure and soft pre-interior on soft bitopological space . 3.1. Definition Let )~,~,( 21 ττAF be soft bitopological space and let AB FF ⊆~ , BF is called soft pre-open set with respect to the two soft topological spaces 1 ~τ and 2 ~τ if )(~ BB FF ⊆ . 3.2. Notes 1- The set of all soft pre-open set with respect to the two soft topologies is denoted by Pre( AF ). 2- The relative soft bitopological space for BF with respect to Soft pre-open sets is the collection Pre BFAF )( given by Pre )}(:~{=)( ACBCBFA FPreFFFF ∈∩ 3- Any 2 ~τ -open soft set is not necessarily to be soft pre-open set . 4- Any soft pre-open set is not necessarily to be of 1 ~τ -open ( 2 ~τ -open ) soft set . 3.3. Example Let us consider example 2.20 , )~,~,( 21 ττAF is soft bitopological space . Take AA FF ⊆~ 1 then 111 )(~ AAA FFF ⇒⊆  is soft per-open set with respect to the two soft topological spaces 1 ~τ and 2 ~τ . 3.4. Remarks 1- The intersection of any soft pre-open sets is not necessary a soft pre-open set American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 26, No 4 , pp 42-53 49 2- The union of any soft pre-open sets is soft pre-open set with respect to soft bitopological space . The example to part (1) is simply . The following proof explain the part (2) of the remark 3.4 . Let BF and CF be a two soft pre-open sets with respect to soft bitopological space )~,~,( 21 ττAF . i.e. )(~ BB FF ⊆ and )(~ CC FF ⊆ with respect to the two soft topological spaces 1 ~τ and  )~(~)(~)(~~~ 2 CBCBCB FFFFFF ∪⊆∪⊆∪⇒τ . Since then )~(~~ CBCB FFFF ∪⊆∪ with respect to the two soft topological spaces 1 ~τ and 2 ~τ . 3.5. Definition Let )~,~,( 21 ττAF is soft bitopological space and let AB FF ⊆~ , BF is called soft pre-closed set of AF if and only if c BF ~ is soft pre-open set of AF . 3.6. Definition Let )~,~,( 21 ττAF is soft bitopological space and AF∈α , AB FF ⊆~ is said to be soft pre-neighborhood of a point α if there is a soft pre-open set CF such that BC FF ⊆∈ ~α . The set of all soft pre-neighborhoods of a point α is denoted by Spre- )(~ αυ . 3.7. Definition Let )~,~,( 21 ττAF is soft bitopological space , and AB FF ⊆~ . A point AF∈α is said to be soft pre-interior point of BF with respect to the two soft topological spaces 1 ~τ and 2 ~τ if there is a soft pre-open CF such that BC FF ⊆∈ ~α . The set of all soft pre-interior points of BF with respect to the two soft topological spaces 1 ~τ and 2 ~τ denoted by Spre-int( BF ). 3.8. Definition Let )~,~,( 21 ττAF is soft bitopological space . A point α is called soft pre-limit point of soft subset BF of AF with respect to the two soft topological spaces 1 ~τ and 2 ~τ if and only if for each a soft pre-open set CF containing another point different from α in BF , that is φα ≠∩ BC FF ~})/{( . The set of all soft pre-limit points of BF be denoted by Spre-lm( BF ). 3.9. Definition American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 26, No 4 , pp 42-53 50 Let )~,~,( 21 ττAF is soft bitopological space , and AB FF ⊆~ , the intersection of all soft pre-closed sets containing BF is called soft pre-closure of BF , and is denoted by Spre-cl( BF ). In the year 2014 J. Subhashinin and Dr. C.Sekar [3] by depending on the [6] and [4] introduces soft pre separation axioms , soft P T -space and some of its properties in the soft topological spaces . Now begin to important section to discuss soft pre separation axioms and some result . 4. The Separation Axioms in Soft Bitopological Space In section four , I introduce some soft pre separation axioms , Spre- T , Spre- 1T and Spre- 2T and illustrate transmission this Properties to The relative soft bitopological space with some result of soft pre separation axioms . 4.1. Definition Let )~,~,( 21 ττAF is soft bitopological space , then )~,~,( 21 ττAF is called Spre- T space if and only if for all pair of soft point AF∈21,αα such that 21 αα ≠ , there exists soft pre-open set BF containing 1α but not 2α or soft pre-open set CF containing 2α but not 1α . 4.2. Theorem A soft bitopological space )~,~,( 21 ττAF is Spre- T space if and only if for each distinct soft points 21,αα in AF , Spre-cl( }{ 1α )≠ Spre-cl( }{ 2α ) . Proof :- Let AF∈21,αα such that 21 αα ≠ and Spre-cl( }{ 1α )≠ Spre-cl( }{ 2α ) . Then there exists at least one soft point 3α in AF such that , ∈3α Spre-cl( }{ 1α ) but ∉3α Spre-cl( }{ 2α ) . Suppose ∈3α Spre-cl( }{ 1α ) , to show that ∉1α Spre-cl( }{ 2α ) . If ∈1α Spre-cl( }{ 2α ) , then ⊂~}{ 1α Spre- cl( }{ 2α ) . So Spre-cl( }{ 1α )⊂~ Spre-cl(Spre-cl( }{ 2α ))=Spre-cl( }{ 2α ) , hence ∈3α Spre-cl( }{ 1α ) , then ∈3α Spre-cl( }{ 2α ) which is contradiction . Hence ∉1α Spre-cl( }{ 2α ) , consequently AF∈1α -Spre-cl( }{ 2α ) but Spre-cl( }{ 2α ) is soft pre-closed , so AF -Spre-cl( }{ 2α ) is soft pre-open which contains 1α but not 2α . It follows that )~,~,( 21 ττAF is Spre- T space . Conversely , since )~,~,( 21 ττAF is Spre- T space, then for each tow distinct soft points AF∈21,αα there American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 26, No 4 , pp 42-53 51 exists soft pre-open set BF such that BF∈1α , BF∉2α . BA FF − is soft closed set which does not contain 1α but contains 2α , by definition (3.9) Spre-cl( }{ 2α ) is the"soft intersection of all soft pre-closed"which contain }{ 2α . Thus , Spre-cl( }{ 2α ) BA FF −⊂~ then BA FF −∉1α .This implies that ∉1α Spre-cl( }{ 2α ) . So we have ∈1α Spre-cl( }{ 1α ) , ∉1α Spre-cl( }{ 2α ) . Therefore Spre-cl( }{ 1α )≠ Spre-cl( }{ 2α ) 4.3. Theorem Every soft subspace of Spre- T space is Spre- T space . Proof :- Let )~,~,( 21 BBB FFF ττ be a soft sub space of Spre- T space )~,~,( 21 ττAF . To prove that the soft sub space is Spre- T space , let BF∈21,ββ such that 21 ββ ≠ . Since AB FF ⊆~ then AF∈≠ 21 ββ and )~,~,( 21 ττAF is Spre- T space , then there is a soft pre-open set CF in AF , such that CC FF ∉∈ 21 ,ββ . So BC FF ∩~ is soft pre-open set in BF and BC FF ∩∈ ~ 1β and BC FF ∩∉ ~ 2β . Hence )~,~,( 21 BBB FFF ττ is Spre- T space . 4.4. Definition Let )~,~,( 21 ττAF is soft bitopological space , then )~,~,( 21 ττAF is called Spre- 1T space if and only if for all pair of soft point AF∈21,αα , there are two soft pre-open sets CB FF , such that BF contains 1α but not 2α and CF contains 2α but not 1α . 4.5. Theorem Every soft subspace of Spre- 1T space is Spre- 1T space . Proof :- Let )~,~,( 21 BBB FFF ττ be a soft sub space of Spre- 1T space )~,~,( 21 ττAF . To prove that the soft sub space is Spre- 1T space , let BF∈21,ββ such that 21 ββ ≠ . Since AB FF ⊆~ then AF∈≠ 21 ββ and )~,~,( 21 ττAF is Spre- T space , then there exists two soft pre-open set CF , DF in AF , such that CF∈1β but CF∉2β and DF∈2β but DF∉1β . Then we obtain two soft set BDDBCC FFFFFF ∩∩ ~=,~= 11 are soft pre-open sets in BF , we have 11 CF∈β but 1212 ; DC FF ∈∉ ββ , but 11 DF∈β . Hence )~,~,( 21 BBB FFF ττ is Spre- 1T space . American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 26, No 4 , pp 42-53 52 4.6. Theorem If Every singleton soft subset of soft bitopological space )~,~,( 21 ττAF is soft pre-closed , then )~,~,( 21 ττAF is Spre- 1T space . Proof :-This is clearly seen . 4.7. Theorem A soft bitopological space )~,~,( 21 ττAF is a Spre- 1T space if and only if Spre-cl φα =})({ , for each AF∈α . Proof :-This is clearly by using prove contradiction . 4.8. Definition Let )~,~,( 21 ττAF is soft bitopological space , then )~,~,( 21 ττAF is called Spre- 2T space (Spre-Hausdorf) if and only if for each pair of distinct soft point AF∈21,αα , there exists two soft pre-open sets CB FF , in AF such that CB FF ∈∈ 21 ,αα and φ=~ CB FF ∩ . 4.9. Theorem Each soft subspace of Spre- 2T space is Spre- 2T space . Proof :- Let )~,~,( 21 BBB FFF ττ be a soft sub space of Spre- 2T space )~,~,( 21 ττAF and let φ≠BF be a soft subset of AF , and BF∈≠ 21 αα then AF∈21,αα , since )~,~,( 21 ττAF is Spre- 2T space ,there exists two soft pre- open sets CD FF , in AF such that CD FF ∈∈ 21 ,αα and φ=~ CD FF ∩ . So BCBD FFFF ∩∩ ~,~ are soft pre-open sets in BF and BCBD FFFF ∩∈∩∈ ~,~ 21 αα ; and φ=~)~(=)~(~)~( BCDBCBD FFFFFFF ∩∩∩∩∩ . Hence )~,~,( 21 BBB FFF ττ is a Spre- 2T space . 4.10. Theorem Each singleton soft subset of Spre- 2T space is a soft pre-closed . Proof :-This is clearly seen . American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 26, No 4 , pp 42-53 53 5. Conclusion In the conclusion of a work paper , many of the basic concepts on soft bitopological space , introduced soft bitopology . Furthermore , introduced relative soft bitopological space , soft pre-open set and some definitions on bitopology by"soft pre-open set as (soft pre-closed , soft pre-neighborhood , soft pre-interior ,soft pre-limit point and soft pre-closure) these"definitions using in other sections from the work and introduce some soft pre separation axioms and studied Properties on soft bitopological space with some important results, one could study the soft ideal bitopology and get some important results too . References [1] Basavaraj M. Ittanagi , Soft Bitopological Spaces , International Journal of Computer Applications , Volume 107 , No. 7, December 2014. [2] D.A. Molodtsov ,Soft set theory-first results , Computers and Mathematics with Applications 37 (1999) 19-31. [3] J.Subhashini and C.Sekar , Soft pre T1 Space in the Soft Topological Spaces , International Journal of Fuzzy Mathematics and Systems , Volume 4 , Number 2 (2014) , pp. 203-207. [4] J. Subhashinin and Dr. C. Sekar , Local properties of soft P-open and soft P-closed sets , Proceedings of National Conference on Discrete Mathematic and Optimization Techniques (2014) 89-100. [5] Naim Cagman and Serdar Enginoglu , Soft set theory and uni-int decision making , European Journal of Operational Research 207 (2010) 848-855 [6] Naim Cagman , Serkan Karatas and Serdar Enginoglu , Soft topology , Computers and Mathematics with Applications 62 (2011) 351-358 . [7] Sabir Hussain and Bashir Ahmad , Some properties of soft topological spaces , Computers and Mathematics with Applications 62 (2011) 4058-4067 .