1 https://doi.org/10.30526/31.2.1953 Mathmatics | 137 8201( عام 2العدد) 13المجلد مجلة إبن الهيثم للعلوم الصرفة و التطبيقية Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 31 (2) 2018 Soft (1,2)*-Omega Separation Axioms and Weak Soft (1,2)*-Omega Separation Axioms in Soft Bitopological Spaces Sabiha I. Mahmood Asaad Adel Abdul-Hady Dept. of Mathematics/ College of Science/ University of Mustansiriyah ssabihaa@uomustansiriyah.edu.iq Received in:24/April/2018, Accepted in:6/June/2018 Abstract In the present paper we introduce and study new classes of soft separation axioms in soft bitopological spaces, namely, soft (1,2)*-omega separation axioms and weak soft (1,2)*- omega separation axioms by using the concept of soft (1,2)*-omega open sets. The equivalent definitions and basic properties of these types of soft separation axioms also have been studied. Keywords: Soft (1,2)*-ω-open sets, soft (1,2)*-ω- iT ~ -spaces, soft (1,2)*-α-ω- iT ~ -spaces, soft (1,2)*-pre-ω- iT ~ -spaces, soft (1,2)*-b-ω- iT ~ -spaces, and soft (1,2)*-β-ω- iT ~ -spaces, for 2,1,,0 2 1i . https://doi.org/10.30526/31.2.19523 mailto:ssabihaa@uomustansiriyah.edu.iq https://doi.org/10.30526/31.2.1953 Mathmatics | 138 8201( عام 2العدد) 13لمجلد ا مجلة إبن الهيثم للعلوم الصرفة و التطبيقية Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 31 (2) 2018 Introduction Soft set theory was firstly introduced by Molodtsov [1] in 1999 as a new mathematical tool for dealing with uncertainty while modeling problems in computer science, economics, engineering physics, medical sciences, and social sciences. In 2011 Shabir and Naz [2] introduced and studied the concept of soft topological spaces. In 2014 Senel and Çagman [3] investigated the notion of soft bitopological spaces over an initial universe set with a fixed set of parameters. In 2018 Mahmood and Abdul-Hady [4] introduced and studied new types of soft sets in soft bitopological spaces called soft (1,2)*-omega open sets and weak forms of soft (1,2)*-omega open sets such as soft (1,2)*-α-ω-open sets, soft (1,2)*-pre-ω-open sets, soft (1,2)*-b-ω-open sets and soft (1,2)*-β-ω-open sets. The main purpose of this paper is to introduce and study new types of soft separation axioms in soft bitopological spaces called soft (1,2)*-omega separation axioms and weak soft (1,2)*-omega separation axioms by using the notion of soft (1,2)*-omega open sets such as soft (1,2)*-ω- iT ~ -spaces, soft (1,2)*-α-ω- iT ~ - spaces, soft (1,2)*-pre-ω- iT ~ -spaces, soft (1,2)*-b-ω- iT ~ -spaces, and soft (1,2)*-β-ω- iT ~ -spaces , for 2,1,,0 2 1i . Moreover we study the fundamental properties and equivalent definitions of these types of soft separation axioms. 1. Preliminaries: Throughout this paper U is an initial universe set, )(UP is the power set of U, P is the set of parameters and PC . Definition (1.1) [1]: A soft set over U is a pair ),( CH , where H is a function defined by )(: UPCH  and C is a non-empty subset of P. Definition (1.2)[5]: A soft set ),( CH over U is called a soft point if there is exactly one Ce such that }{)( ueH  for some Uu and φ)(eH , }{\ eCe  and is denoted by }){,(~ ueu  . Definition (1.3)[5]: A soft point }){,(~ ueu  is called belongs to a soft set ),( CH if Ce and )(eHu , and is denoted by ),(~~ CHu . Definition (1.4) [5]: A soft set ),( CH over U is called countable (finite) if the set )(eH is countable (finite) Ce . Definition (1.5)[6]: A soft set ),( CH over U is called a null soft set with respect to C if for each Ce , φ)(eH , and is denoted by Cφ ~ . If PC , then ),( CH is called a null soft set and is denoted by φ~ . Definition (1.6)[6]: A soft set ),( CH over U is called an absolute soft set with respect to C if for each Ce , UeH )( , and is denoted by CU ~ . If PC , then ),( CH is called an absolute soft set and is denoted by U ~ . Definition (1.7)[6]: Let ),( 11 CH and ),( 22 CH be soft sets over a common universe U. Then we say that: (1) ),( 11 CH is a soft subset of ),( 22 CH denoted by ),( ~),( 2211 CHCH  if 21 CC  and )()( 21 eHeH  for each 1Ce . (2) The soft union of two soft sets ),( 11 CH and ),( 22 CH over a common universe U is the soft set ),( CH , where 21 CCC  , and Ce ,          2121 122 211 )()( )( )( )( CCeifeHeH CCeifeH CCeifeH eH  And we write ),( ~ ),(),( 2211 CHCHCH  . https://doi.org/10.30526/31.2.19523 https://doi.org/10.30526/31.2.1953 Mathmatics | 139 8201( عام 2العدد) 13لمجلد ا مجلة إبن الهيثم للعلوم الصرفة و التطبيقية Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 31 (2) 2018 (3) The soft intersection of two soft sets ),( 11 CH and ),( 22 CH over a common universe U is the soft set ),( CH , where 21 CCC  , and Ce , )()()( 21 eHeHeH  , and we write ),( ~ ),(),( 2211 CHCHCH  . (4) The soft difference of two soft sets ),( 11 CH and ),( 22 CH over a common universe U is the soft set ),( CH , where 21 CCC  , and Ce , )()()( 21 eHeHeH  , and we write ),(),(),( 2211 CHCHCH  . Definition (1.8)[2]: A soft topology on U is a collection τ~ of soft subsets of U ~ having the following properties: (i) τφ ~~~  and τ~~ ~ U . (ii) If τ~~),(),,( 21 PHPH , then τ ~~),( ~ ),( 21 PHPH  . (iii) If τ~~),( PH j ,  j , then τ ~~),(    j j PH . The triple ),~,( PU τ is called a soft topological space over U. The members of τ~ are called soft open sets over U. The complement of a soft open set is called soft closed. Definition (1.9)[3]: Let U be a non-empty set and let 1 ~τ and 2 ~τ be two soft topologies over U. Then ),~,~,( 21 PU ττ is called a soft bitopological space over U. Definition (1.10)[3]: A soft subset ),( PH of a soft bitopological space ),~,~,( 21 PU ττ is called soft 21 ~~ ττ -open if ),( ~ ),(),( 21 PHPHPH  such that 11 ~~),( τPH and 22 ~~),( τPH . The complement of a soft 21 ~~ ττ -open set in U ~ is called soft 21 ~~ ττ -closed. Definitions (1.11)[7]: A soft bitopological space ),~,~,( 21 PU ττ is called a soft (1,2)*- 0 ~ T -space if for any two distinct soft points x~ and y~ of U ~ , there exists a soft 21 ~~ ττ -open set in U ~ containing one of the soft points but not the other. Definition (1.12)[7]: A soft bitopological space ),~,~,( 21 PU ττ is called a soft (1,2)*- 2 1 ~ T -space if every soft singleton set in U ~ is either soft 21 ~~ ττ -open or soft 21 ~~ ττ -closed. Definition (1.13)[7]: A soft bitopological space ),~,~,( 21 PU ττ is called a soft (1,2)*- 1 ~ T -space if for any two distinct soft points x~ and y~ of U ~ , there exists a soft 21 ~~ ττ -open set in U ~ containing x~ but not y~ and a soft 21 ~~ ττ -open set in U ~ containing y~ but not x~ . Definition (1.14)[7]: A soft bitopological space ),~,~,( 21 PU ττ is called a soft (1,2)*- 2 ~ T -space if for any two distinct soft points x~ and y~ of U ~ , there are two soft 21 ~~ ττ -open sets ),( PH and ),( PK in U ~ such that ),(~~ PHx , ),(~~ PKy and φ ~),( ~ ),( PKPH  . Definition (1.15) [4]: A soft subset ),( PH of a soft bitopological space ),~,~,( 21 PU ττ is called soft (1,2)*-omega open (briefly soft (1,2)*-ω-open) if for each ),(~~ PHx , there exists a soft 21 ~~ ττ -open set ),( PO in U ~ such that ),(~~ POx and ),(),( PHPO  is a countable soft set. The complement of a soft (1,2)*-ω-open set is called soft (1,2)*-omega closed (briefly soft (1,2)*-ω-closed). Clearly, every soft 21 ~~ ττ -open set is soft (1,2)*-ω-open, but the converse in general is not true we can see in the following example: Example (1.16): Let }3,2,1{U and },{ 21 ppP  , and let )},(, ~, ~ {~ 11 PHU φτ  and , ~ {~2 Uτ )},(,~ 2 PHφ be soft topologies over U, where })}2,1{,(}),{,{(),( 211 pUpPH  and ),( 2 PH https://doi.org/10.30526/31.2.19523 https://doi.org/10.30526/31.2.1953 Mathmatics | 140 8201( عام 2العدد) 13لمجلد ا مجلة إبن الهيثم للعلوم الصرفة و التطبيقية Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 31 (2) 2018 })}3,1{,(}),{,{( 21 pUp . The soft sets in )},(),,(, ~, ~ { 21 PHPHU φ are soft 21 ~~ ττ -open sets in U ~ . Thus ),~,~,( 21 PU ττ is a soft bitopological space and })}1{,(}),{,{(),( 21 pUpPH  is a soft (1,2)*-ω-open set in U ~ , but is not soft 21 ~~ ττ -open. Definition (1.17) [4]: Let ),~,~,( 21 PU ττ be a soft bitopological space and UPH ~~),(  . Then: (i) The soft (1,2)*-omega closure (briefly soft (1,2)*-ω-closure) of ),( PH , denoted by (1,2)*- ),( PHclω is the intersection of all soft (1,2)*-ω-closed sets in U ~ which contains ),( PH . (ii) The soft (1,2)*-omega interior (briefly soft (1,2)*-ω-interior) of ),( PH , denoted by (1,2)*- ),int( PHω is the union of all soft (1,2)*-ω-open sets in U ~ which are contained in ),( PH . Theorem (1.18) [4]: If ),~,~,( 21 PU ττ is a soft bitopological space, and UPKPH ~~),(),,(  . Then: (i) ),( PH ~ (1,2)*- ),( PHclω ~ ),(~~ 21 PHclττ . (ii) (1,2)*- ),( PHclω is soft (1,2)*-ω-closed set in U ~ . (iii) ),( PH is soft (1,2)*-ω-closed iff (1,2)*- ),( PHclω  ),( PH . (iv) If ),(~),( PKPH  , then (1,2)*- ),( PHclω ~ (1,2)*- ),( PKclω . Definitions (1.19) [4]: A soft subset ),( PH of a soft bitopological space ),~,~,( 21 PU ττ is called: (i) Soft (1,2)*-α-ω-open if ~),( PH (1,2)*- (~~int( 21 clττω (1,2)*- ))),int( PHω . (ii) Soft (1,2)*-pre-ω-open if ~),( PH (1,2)*- )),(~~int( 21 PHclττω . (iii) Soft (1,2)*-b-ω-open if ~),( PH (1,2)*- )),(~~int( 21 PHclττω  ~ (~~ 21 clττ (1,2)*- )),int( PHω . (iv) Soft (1,2)*-β-ω-open if ~),( PH (~~ 21 clττ (1,2)*- ))),( ~~int( 21 PHclττω . Proposition (1.20) [4]: If ),~,~,( 21 PU ττ is a soft bitopological space, then the following hold: (i) Every soft 21 ~~ ττ -open set is soft (1,2)*-ω-open. (ii) Every soft (1,2)*-ω-open set is soft (1,2)*-α-ω-open. (iii) Every soft (1,2)*-α-ω-open set is soft (1,2)*-pre-ω-open. (iv) Every soft (1,2)*-pre-ω-open set is soft (1,2)*-b-ω-open. (v) Every soft (1,2)*-b-ω-open set is soft (1,2)*-β-ω-open. Definition (1.21) [4]: Let ),~,~,( 21 PU ττ be a soft bitopological space and UPH ~~),(  . Then the soft (1,2)*-α-ω-closure (resp. soft (1,2)*-pre-ω-closure, soft (1,2)*-b-ω-closure,soft (1,2)*-β-ω-closure) of ),( PH , denoted by (1,2)*-α- ),( PHclω (resp. (1,2)*-pre- ),( PHclω , (1,2)*-b- ),( PHclω , (1,2)*-β- ),( PHclω ) is the intersection of all soft (1,2)*-α-ω-closed (resp. soft (1,2)*-pre-ω-closed, soft (1,2)*-b-ω-closed, soft (1,2)*-β-ω-closed) sets in U ~ which contains ),( PH . Definition (1.22) [4]: A soft subset ),( PN of a soft bitopological space ),~,~,( 21 PU ττ is called a soft (1,2)*-ω-neighborhood (resp. soft (1,2)*-α-ω-neighborhood, soft (1,2)*-pre-ω- neighborhood, soft (1,2)*-b-ω-neighborhood, soft (1,2)*-β-ω-neighborhood) of a soft point x~ in U ~ if there exists a soft (1,2)*-ω-open (resp. soft (1,2)*-α-ω-open, soft (1,2)*-pre-ω-open, soft (1,2)*-b-ω-open, soft (1,2)*-β-ω-open) set ),( PH in U ~ such that ),(~),(~~ PNPHx  . Definition (1.23)[8]: Let ),~,~,( 21 PU ττ be a soft bitopological space over U and UYφ . Then } ~~),(: ~~ ),{(~ 1~1 ττ  PMYPMY  and ),(: ~~ ),{(~ ~2 PNYPNY τ } ~~ 2τ are called the relative soft topologies on Y ~ and ), ~,~,( ~2~1 PY YY ττ is called the relative soft bitopological space of ),~,~,( 21 PU ττ . 2. Soft (1,2)*-Omega Separation Axioms and Weak Soft (1,2)*-Omega Separation Axioms Now, we introduce and study new types of soft separation axioms in soft bitopological spaces called soft (1,2)*-ω-separation axioms and weak soft (1,2)*-ω-separation axioms such as soft (1,2)*-ω- iT ~ -spaces, soft (1,2)*-α-ω- iT ~ -spaces, soft (1,2)*-pre-ω- iT ~ -spaces, soft https://doi.org/10.30526/31.2.19523 https://doi.org/10.30526/31.2.1953 Mathmatics | 141 8201( عام 2العدد) 13لمجلد ا مجلة إبن الهيثم للعلوم الصرفة و التطبيقية Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 31 (2) 2018 (1,2)*-b-ω- iT ~ -spaces, and soft (1,2)*-β-ω- iT ~ -spaces, for 2,1,,0 2 1i . The fundamental properties and equivalent definitions of these types of soft separation axioms also, have been studied. Definitions (2.1): A soft bitopological space ),~,~,( 21 PU ττ is called a soft (1,2)*-ω- 0 ~ T -space (resp. soft (1,2)*-α-ω- 0 ~ T -space, soft (1,2)*-pre-ω- 0 ~ T -space, soft (1,2)*-b-ω- 0 ~ T -space, soft (1,2)*-β-ω- 0 ~ T -space) if for any two distinct soft points x~ and y~ of U ~ , there exists a soft (1,2)*-ω-open (resp. soft (1,2)*-α-ω-open, soft (1,2)*-pre-ω-open, soft (1,2)*-b-ω-open, soft (1,2)*-β-ω-open) set in U ~ containing one of the soft points but not the other. Proposition (2.2): Every soft (1,2)*- 0 ~ T -space is a soft (1,2)*-ω- 0 ~ T -space (resp. soft (1,2)*- α-ω- 0 ~ T -space, soft (1,2)*-pre-ω- 0 ~ T -space, soft (1,2)*-b-ω- 0 ~ T -space, soft (1,2)*-β-ω- 0 ~ T - space). Proof: It is obvious. Remark (2.3): The converse of proposition (2.2) is not true in general we can see by the following example: Example (2.4): Let },,{ cbaU and },{ 21 ppP  and let )},(, ~, ~ {~ 11 PHU φτ  and , ~, ~ {~2 φτ U )},( 2 PH be soft topologies over U, where ),( 1 PH })}{,(}),{,{( 21 apap , ),( 2 PH })}{,(}),{,{( 21 bpbp and ),( 3 PH })},{,(}),,{,{( 21 bapbap . The soft sets in ),,(, ~, ~ { 1 PHU φ ),,( 2 PH )},( 3 PH are soft 21 ~~ ττ -open. Thus ),~,~,( 21 PU ττ is a soft (1,2)*-ω- 0 ~ T -space (resp. soft (1,2)*-α-ω- 0 ~ T -space, soft (1,2)*-pre-ω- 0 ~ T -space, soft (1,2)*-b-ω- 0 ~ T -space, soft (1,2)*- β-ω- 0 ~ T -space), but is not soft (1,2)*- 0 ~ T -space, since }){,( 1 ap }){,( ~~ 2 apyx  , but there exists no soft 21 ~~ ττ -open set containing one of the soft points but not the other. Theorem (2.5): A soft bitopological space ),~,~,( 21 PU ττ is a soft (1,2)*-ω- 0 ~ T -space (resp. soft (1,2)*-α-ω- 0 ~ T -space, soft (1,2)*-pre-ω- 0 ~ T -space, soft (1,2)*-b-ω- 0 ~ T -space, soft (1,2)*- β-ω- 0 ~ T -space) if and only if (1,2)*- })~({xclω  (1,2)*- })~({yclω (resp. (1,2)*-α- })~({xclω  (1,2)*-α- })~({yclω , (1,2)*-pre- })~({xclω  (1,2)*-pre- })~({yclω , (1,2)*-b- })~({xclω  (1,2)*- b- })~({yclω , (1,2)*-β- })~({xclω  (1,2)*-β- })~({yclω ) for any two distinct soft points x~ and y~ of U ~ . Proof: Let Uyx ~~~,~  such that yx ~~  . Since ),~,~,( 21 PU ττ is a soft (1,2)*-ω- 0 ~ T -space, then there exists a soft (1,2)*-ω-open set ),( PH containing x~ , but not y~ . Therefore ),( ~ PHU is a soft (1,2)*-ω-closed set containing y~ , but not x~ . Hence (1,2)*- ),( ~~})~({ PHUycl ω . Since ),( ~~~ PHUx  , this implies that  ~~x (1,2)*- })~({yclω . So we get, ~~x (1,2)*- })~({xclω , but  ~~x (1,2)*- })~({yclω . Thus (1,2)*- })~({xclω  (1,2)*- })~({yclω . Conversely, to prove that ),~,~,( 21 PU ττ is a soft (1,2)*-ω- 0 ~ T -space. Let Uyx ~~~,~  such that yx ~~  . Since (1,2)*- })~({xclω  (1,2)*- })~({yclω , then there exists Uz ~~~  such that ~~z (1,2)*- })~({xclω , but  ~~z (1,2)*- })~({yclω . Suppose ~~z (1,2)*- })~({xclω , to show that  ~~x (1,2)*- })~({yclω . If ~~x (1,2)*- })~({yclω  }~{x ~ (1,2)*- })~({yclω  (1,2)*- })~({xclω ~ (1,2)*- clω ((1,2)*- })~({yclω ) = (1,2)*- })~({yclω . Since ~~z (1,2)*- })~({xclω  ~~z (1,2)*- })~({yclω which is a contradiction. Thus  ~~x (1,2)*- })~({yclω  ~~x U ~ - (1,2)*- })~({yclω , but (1,2)*- })~({yclω is soft (1,2)*-ω-closed, so U ~ - (1,2)*- })~({yclω is soft (1,2)*- ω-open which contains x~ , but not y~ . Therefore ),~,~,( 21 PU ττ is a soft (1,2)*-ω- 0 ~ T -space. https://doi.org/10.30526/31.2.19523 https://doi.org/10.30526/31.2.1953 Mathmatics | 142 8201( عام 2العدد) 13لمجلد ا مجلة إبن الهيثم للعلوم الصرفة و التطبيقية Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 31 (2) 2018 Similarly, we can prove other cases. Theorem (2.6): Every soft bitopological space ),~,~,( 21 PU ττ is a soft (1,2)*-ω- 0 ~ T -space (resp. soft (1,2)*-α-ω- 0 ~ T -space, soft (1,2)*-pre-ω- 0 ~ T -space, soft (1,2)*-b-ω- 0 ~ T -space, soft (1,2)*-β-ω- 0 ~ T -space). Proof: Let ),~,~,( 21 PU ττ be any soft bitopological space and Uyx ~~~,~  such that yx ~~  . Since }~{ ~ yU  is a soft (1,2)*-ω-open (resp. soft (1,2)*-α-ω-open, soft (1,2)*-pre-ω-open, soft (1,2)*-b-ω-open, soft (1,2)*-β-ω-open) subset of U ~ containing x~ , but not y~ . Therefore ),~,~,( 21 PU ττ is a soft (1,2)*-ω- 0 ~ T -space (resp. soft (1,2)*-α-ω- 0 ~ T -space, soft (1,2)*-pre-ω- 0 ~ T -space, soft (1,2)*-b-ω- 0 ~ T -space, soft (1,2)*-β-ω- 0 ~ T -space). Corollary (2.7): Every soft subspace of a soft (1,2)*-ω- 0 ~ T -space (resp. soft (1,2)*-α-ω- 0 ~ T - space, soft (1,2)*-pre-ω- 0 ~ T -space, soft (1,2)*-b-ω- 0 ~ T -space, soft (1,2)*-β-ω- 0 ~ T -space) is a soft (1,2)*-ω- 0 ~ T -space (resp. soft (1,2)*-α-ω- 0 ~ T -space, soft (1,2)*-pre-ω- 0 ~ T -space, soft (1,2)*-b-ω- 0 ~ T -space, soft (1,2)*-β-ω- 0 ~ T -space). Proof: It is obvious. Proposition (2.8): If ),~,( 1 PU τ or ), ~,( 2 PU τ is a soft 0 ~ T -space, then ),~,~,( 21 PU ττ is a soft (1,2)*-ω- 0 ~ T -space (resp. soft (1,2)*-α-ω- 0 ~ T -space, soft (1,2)*-pre-ω- 0 ~ T -space, soft (1,2)*-b- ω- 0 ~ T -space, soft (1,2)*-β-ω- 0 ~ T -space). Proof: It follows from the fact ~~iτ soft 21 ~~ ττ -open sets in U ~ , 2,1i and proposition (2.2). Remark (2.9): The converse of proposition (2.8) is not true in general in example (2.4), ),~,~,( 21 PU ττ is a soft (1,2)*-ω- 0 ~ T -space (resp. soft (1,2)*-α-ω- 0 ~ T -space, soft (1,2)*-pre-ω- 0 ~ T -space, soft (1,2)*-b-ω- 0 ~ T -space, soft (1,2)*-β-ω- 0 ~ T -space), but both ),~,( 1 PU τ and ),~,( 2 PU τ are not soft 0 ~ T -space. Definition (2.10): A soft bitopological space ),~,~,( 21 PU ττ is called a soft (1,2)*-ω- 2 1 ~ T -space (resp. soft (1,2)*-α-ω- 2 1 ~ T -space, soft (1,2)*-pre-ω- 2 1 ~ T -space, soft (1,2)*-b-ω- 2 1 ~ T -space, soft (1,2)*-β-ω- 2 1 ~ T -space) if every soft singleton set in U ~ is either soft (1,2)*-ω-open (resp. soft (1,2)*-α-ω-open, soft (1,2)*-pre-ω-open, soft (1,2)*-b-ω-open, soft (1,2)*-β-ω-open) or soft (1,2)*-ω-closed (resp. soft (1,2)*-α-ω-closed, soft (1,2)*-pre-ω-closed, soft (1,2)*-b-ω-closed, soft (1,2)*-β-ω-closed). Proposition (2.11): Every soft (1,2)*- 2 1 ~ T -space is a soft (1,2)*-ω- 2 1 ~ T -space (resp. soft (1,2)*-α-ω- 2 1 ~ T -space, soft (1,2)*-pre-ω- 2 1 ~ T -space, soft (1,2)*-b-ω- 2 1 ~ T -space, soft (1,2)*-β- ω- 2 1 ~ T -space). Proof: It is obvious. Remark (2.12): The converse of proposition (2.11) is not true in general. In example (2.4) ),~,~,( 21 PU ττ is a soft (1,2)*-ω- 2 1 ~ T -space (resp. soft (1,2)*-α-ω- 2 1 ~ T -space, soft (1,2)*-pre-ω- 2 1 ~ T -space, soft (1,2)*-b-ω- 2 1 ~ T -space, soft (1,2)*-β-ω- 2 1 ~ T -space), but is not soft (1,2)*- 2 1 ~ T - space. https://doi.org/10.30526/31.2.19523 https://doi.org/10.30526/31.2.1953 Mathmatics | 143 8201( عام 2العدد) 13لمجلد ا مجلة إبن الهيثم للعلوم الصرفة و التطبيقية Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 31 (2) 2018 Proposition (2.13): Every soft (1,2)*- 2 1 ~ T -space (resp. soft (1,2)*-ω- 2 1 ~ T -space, soft (1,2)*-α- ω- 2 1 ~ T -space, soft (1,2)*-pre-ω- 2 1 ~ T -space, soft (1,2)*-b-ω- 2 1 ~ T -space, soft (1,2)*-β-ω- 2 1 ~ T - space) is a soft (1,2)*- 0 ~ T -space (resp. soft (1,2)*-ω- 0 ~ T -space, soft (1,2)*-α-ω- 0 ~ T -space, soft (1,2)*-pre-ω- 0 ~ T -space, soft (1,2)*-b-ω- 0 ~ T -space, soft (1,2)*-β-ω- 0 ~ T -space). Proof: Let ),~,~,( 21 PU ττ be a soft (1,2)*-ω- 2 1 ~ T -space and let Uyx ~~~,~  such that yx ~~  . If }~{x is soft (1,2)*-ω-open, then }~{x is a soft (1,2)*-ω-open set containing x~ , but not y~ and if }~{x is soft (1,2)*-ω-closed, then }~{ ~ xU  is a soft (1,2)*-ω-open set containing y~ , but not x~ . Therefore ),~,~,( 21 PU ττ is a soft (1,2)*-ω- 0 ~ T -space. Similarly, we can prove that other cases. Remark (2.14): The soft (1,2)*- 0 ~ T -space may not be soft (1,2)*- 2 1 ~ T -space in general we can see in the following example: Example (2.15): Let },{ baU and },{ 21 ppP  and let )},(, ~, ~ {~ 11 PHU φτ  and , ~, ~ {~2 φτ U )},( 2 PH be soft topologies over U, where ),( 1 PH })}{,(}),{,{( 21 bpap , ),( 2 PH })}{,(}),{,{( 21 bpbp and ),( 3 PH })}{,(}),,{,{( 21 bpbap . The soft sets in ),,(, ~, ~ { 1 PHU φ )},(),,( 32 PHPH are soft 21 ~~ ττ -open sets. Thus ),~,~,( 21 PU ττ is a soft (1,2)*- 0 ~ T -space, but is not soft (1,2)*- 2 1 ~ T -space, since }~{})}{,{( 1 xap  is not soft 21 ~~ ττ -open and is not soft 21 ~~ ττ - closed. Theorem (2.16): Every soft bitopological space is a soft (1,2)*-ω- 2 1 ~ T -space (resp. soft (1,2)*-α-ω- 2 1 ~ T -space, soft (1,2)*-pre-ω- 2 1 ~ T -space, soft (1,2)*-b-ω- 2 1 ~ T -space, soft (1,2)*-β- ω- 2 1 ~ T -space). Proof: Let ),~,~,( 21 PU ττ be any soft bitopological space and Ux ~~~  . Since }~{ ~ xU  is a soft (1,2)*-ω-open (resp. soft (1,2)*-α-ω-open, soft (1,2)*-pre-ω-open, soft (1,2)*-b-ω-open, soft (1,2)*-β-ω-open) subset of U ~ , then }~{x is a soft (1,2)*-ω-closed (resp. soft (1,2)*-α-ω- closed, soft (1,2)*-pre-ω-closed, soft (1,2)*-b-ω-closed, soft (1,2)*-β-ω-closed) subset of U ~ . Therefore ),~,~,( 21 PU ττ is a soft (1,2)*-ω- 2 1 ~ T -space (resp. soft (1,2)*-α-ω- 2 1 ~ T -space, soft (1,2)*-pre-ω- 2 1 ~ T -space, soft (1,2)*-b-ω- 2 1 ~ T -space, soft (1,2)*-β-ω- 2 1 ~ T -space). Corollary (2.17): A soft bitopological space ),~,~,( 21 PU ττ is a soft (1,2)*-ω- 2 1 ~ T -space (resp. soft (1,2)*-α-ω- 2 1 ~ T -space, soft (1,2)*-pre-ω- 2 1 ~ T -space, soft (1,2)*-b-ω- 2 1 ~ T -space, soft (1,2)*-β-ω- 2 1 ~ T -space) iff it is a soft (1,2)*-ω- 0 ~ T -space (resp. soft (1,2)*-α-ω- 0 ~ T -space, soft (1,2)*-pre-ω- 0 ~ T -space, soft (1,2)*-b-ω- 0 ~ T -space, soft (1,2)*-β-ω- 0 ~ T -space). Proof: It follows that from the proposition (2.13) and theorem (2.16). Corollary (2.18): Every soft subspace of a soft (1,2)*-ω- 2 1 ~ T -space (resp. soft (1,2)*-α-ω- 2 1 ~ T -space, soft (1,2)*-pre-ω- 2 1 ~ T -space, soft (1,2)*-b-ω- 2 1 ~ T -space, soft (1,2)*-β-ω- 2 1 ~ T - space) is a soft (1,2)*-ω- 2 1 ~ T -space (resp. soft (1,2)*-α-ω- 2 1 ~ T -space, soft (1,2)*-pre-ω- 2 1 ~ T - space, soft (1,2)*-b-ω- 2 1 ~ T -space, soft (1,2)*-β-ω- 2 1 ~ T -space). Proof: It follows that from the theorem (2.16). https://doi.org/10.30526/31.2.19523 https://doi.org/10.30526/31.2.1953 Mathmatics | 144 8201( عام 2العدد) 13لمجلد ا مجلة إبن الهيثم للعلوم الصرفة و التطبيقية Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 31 (2) 2018 Proposition (2.19): If ),~,( 1 PU τ or ), ~,( 2 PU τ is a soft 2 1 ~ T -space, then ),~,~,( 21 PU ττ is a soft (1,2)*-ω- 2 1 ~ T -space (resp. soft (1,2)*-α-ω- 2 1 ~ T -space, soft (1,2)*-pre-ω- 2 1 ~ T -space, soft (1,2)*-b-ω- 2 1 ~ T -space, soft (1,2)*-β-ω- 2 1 ~ T -space). Proof: It follows from the fact ~~iτ soft 21 ~~ ττ -open sets in U ~ , 2,1i and proposition (2.11). Remark (2.20): The converse of proposition (2.19) is not true in general in example (2.4) ),~,~,( 21 PU ττ is a soft (1,2)*-ω- 2 1 ~ T -space (resp. soft (1,2)*-α-ω- 2 1 ~ T -space, soft (1,2)*-pre-ω- 2 1 ~ T -space, soft (1,2)*-b-ω- 2 1 ~ T -space, soft (1,2)*-β-ω- 2 1 ~ T -space), but both ),~,( 1 PU τ and ),~,( 2 PU τ are not soft (1,2)*- 2 1 ~ T -space. Definition (2.21): A soft bitopological space ),~,~,( 21 PU ττ is called a soft (1,2)*-ω- 1 ~ T -space (resp. soft (1,2)*-α-ω- 1 ~ T -space, soft (1,2)*-pre-ω- 1 ~ T -space, soft (1,2)*-b-ω- 1 ~ T -space, soft (1,2)*-β-ω- 1 ~ T -space) if for any two distinct soft points x~ and y~ of U ~ , there exists a soft (1,2)*-ω-open (resp. soft (1,2)*-α-ω-open, soft (1,2)*-pre-ω-open, soft (1,2)*-b-ω-open, soft (1,2)*-β-ω-open) set in U ~ containing x~ but not y~ and a soft (1,2)*-ω-open (resp. soft (1,2)*- α-ω-open, soft (1,2)*-pre-ω-open, soft (1,2)*-b-ω-open, soft (1,2)*-β-ω-open) set in U ~ containing y~ but not x~ . Proposition (2.22): Every soft (1,2)*-ω- 1 ~ T -space (resp. soft (1,2)*- 1 ~ T -space, soft (1,2)*-α-ω- 1 ~ T -space, soft (1,2)*-pre-ω- 1 ~ T -space, soft (1,2)*-b-ω- 1 ~ T -space, soft (1,2)*-β-ω- 1 ~ T -space) is a soft (1,2)*-ω- 2 1 ~ T -space (resp. soft (1,2)*- 2 1 ~ T -space, soft (1,2)*-α-ω- 2 1 ~ T -space, soft (1,2)*- pre-ω- 2 1 ~ T -space, soft (1,2)*-b-ω- 2 1 ~ T -space, soft (1,2)*-β-ω- 2 1 ~ T -space). Remark (2.23): The soft (1,2)*- 2 1 ~ T -space may not be soft (1,2)*- 1 ~ T -space in general we can see in the following example: Example (2.24): Let },{ baU and }{pP  and let )},(, ~, ~ {~1 PHU φτ  and } ~, ~ {~2 φτ U be soft topologies over U, where ),( PH })}{,{( ap . The soft sets in )},(, ~, ~ { PHU φ are soft 21 ~~ ττ - open sets. Thus ),~,~,( 21 PU ττ is a soft (1,2)*- 2 1 ~ T -space, but is not soft (1,2)*- 1 ~ T -space, since }){,( ap }){,(~~ bpyx  , but there exists no soft 21 ~~ ττ -open set containing y~ , but not containing x~ . Proposition (2.25): Every soft (1,2)*- 1 ~ T -space is a soft (1,2)*-ω- 1 ~ T -space (resp. soft (1,2)*- α-ω- 1 ~ T -space, soft (1,2)*-pre-ω- 1 ~ T -space, soft (1,2)*-b-ω- 1 ~ T -space, soft (1,2)*-β-ω- 1 ~ T - space). Proof: It is obvious. Remark (2.26): The converse of proposition (2.25) is not true in general. We see that in the following example: Example (2.27): Let },{ baU and },{ 21 ppP  and let )},(, ~, ~ {~ 11 PHU φτ  and , ~, ~ {~2 φτ U )},( 2 PH be soft topologies over U, where ),( 1 PH })}{,(}),{,{( 21 bpap and ),( 2 PH })}{,(}),{,{( 21 apbp . The soft sets in ),,(, ~, ~ { 1 PHU φ )},( 2 PH are soft 21 ~~ ττ -open. Thus ),~,~,( 21 PU ττ is a soft (1,2)*-ω- 1 ~ T -space (resp. soft (1,2)*-α-ω- 1 ~ T -space, soft (1,2)*-pre-ω- 1 ~ T -space, soft (1,2)*-b-ω- 1 ~ T -space, soft (1,2)*-β-ω- 1 ~ T -space), but is not soft (1,2)*- 1 ~ T -space. https://doi.org/10.30526/31.2.19523 https://doi.org/10.30526/31.2.1953 Mathmatics | 145 8201( عام 2العدد) 13لمجلد ا مجلة إبن الهيثم للعلوم الصرفة و التطبيقية Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 31 (2) 2018 Theorem (2.28): In a soft bitopological space ),~,~,( 21 PU ττ the following statements are equivalent. (i) ),~,~,( 21 PU ττ is a soft (1,2)*-ω- 1 ~ T -space (resp. soft (1,2)*-α-ω- 1 ~ T -space, soft (1,2)*-pre-ω- 1 ~ T -space, soft (1,2)*-b-ω- 1 ~ T -space, soft (1,2)*-β-ω- 1 ~ T -space) (ii) For each Ux ~~~  , }~{x is a soft (1,2)*-ω-closed (resp. soft (1,2)*-α-ω-closed, soft (1,2)*- pre-ω-closed, soft (1,2)*-b-ω-closed, soft (1,2)*-β-ω-closed) set in U ~ . (iii) Every soft subset of U ~ is the intersection of all soft (1,2)*-ω-open (resp. soft (1,2)*-α-ω- open, soft (1,2)*-pre-ω-open, soft (1,2)*-b-ω-open, soft (1,2)*-β-ω-open) sets containing it. (iv) The intersection of all soft (1,2)*-ω-open (resp. soft (1,2)*-α-ω-open, soft (1,2)*-pre-ω- open, soft (1,2)*-b-ω-open, soft (1,2)*-β-ω-open) sets containing the soft point Ux ~~~  is }~{x . Proof: )()( iii  . Let Ux ~~~  . To prove that }~{x is soft (1,2)*-ω-closed in U ~ . Let }~{ ~~ xy  yx ~~  . Since ),~,~,( 21 PU ττ is a soft (1,2)*-ω- 1 ~ T -space, then there is a soft (1,2)*-ω-open set ),( PH in U ~ such that ),(~~ PHy , but ),( ~~ PHx  φ ~),( ~ }~{ PHx   }~{x c PH ),(~  (1,2)*- })~({xclω ~ (1,2)*- )),(( c PHclω c PH ),( . Since c PHy ),( ~~    ~~y (1,2)*- })~({xclω  (1,2)*- })~({xclω }~{x . Therefore }~{x is a soft (1,2)*-ω-closed set in U ~ . )()( iiiii  . Let UPH ~~),(  and ),( ~~ PHy . Then c yPH }~{~),(  and c y}~{ is soft (1,2)*-ω- open in U ~ . Hence ),( PH }),(~~:}~{{ ~ cc PHyy  is the intersection of all soft (1,2)*-ω-open sets containing ),( PH . )()( iviii  . Obvious. )()( iiv  . Let Xyx ~~~,~  , yx ~~  . By our assumption there exist at least a soft (1,2)*-ω-open set containing x~ , but not y~ and also a soft (1,2)*-ω-open set containing y~ , but not x~ . Therefore ),~,~,( 21 PU ττ is a soft (1,2)*-ω- 1 ~ T -space. Similarly, we can prove that other cases. Theorem (2.29): Every soft bitopological space is a soft (1,2)*-ω- 1 ~ T -space (resp. soft (1,2)*- α-ω- 1 ~ T -space, soft (1,2)*-pre-ω- 1 ~ T -space, soft (1,2)*-b-ω- 1 ~ T -space, soft (1,2)*-β-ω- 1 ~ T - space). Proof: Let ),~,~,( 21 PU ττ be any soft bitopological space and Uyx ~~~,~  such that yx ~~  . Since }~{ ~ xU  and }~{ ~ yU  are soft (1,2)*-ω-open (resp. soft (1,2)*-α-ω-open, soft (1,2)*-pre-ω- open, soft (1,2)*-b-ω-open, soft (1,2)*-β-ω-open) sets in U ~ such that }~{ ~ yU  containing x~ , but not y~ and }~{ ~ xU  containing y~ , but not x~ . Therefore ),~,~,( 21 PU ττ is a soft (1,2)*-ω- 1 ~ T -space (resp. soft (1,2)*-α-ω- 1 ~ T -space, soft (1,2)*-pre-ω- 1 ~ T -space, soft (1,2)*-b-ω- 1 ~ T - space, soft (1,2)*-β-ω- 1 ~ T -space). Corollary (2.30): A soft bitopological space ),~,~,( 21 PU ττ is a soft (1,2)*-ω- 1 ~ T -space (resp. soft (1,2)*-α-ω- 1 ~ T -space, soft (1,2)*-pre-ω- 1 ~ T -space, soft (1,2)*-b-ω- 1 ~ T -space, soft (1,2)*-β- ω- 1 ~ T -space) iff it is a soft (1,2)*-ω- 2 1 ~ T -space (resp. soft (1,2)*-α-ω- 2 1 ~ T -space, soft (1,2)*- pre-ω- 2 1 ~ T -space, soft (1,2)*-b-ω- 2 1 ~ T -space, soft (1,2)*-β-ω- 2 1 ~ T -space). Proof: It follows that from the proposition (2.22) and theorem (2.29). Corollary (2.31): Every soft subspace of a soft (1,2)*-ω- 1 ~ T -space (resp. soft (1,2)*-α-ω- 1 ~ T - space, soft (1,2)*-pre-ω- 1 ~ T -space, soft (1,2)*-b-ω- 1 ~ T -space, soft (1,2)*-β-ω- 1 ~ T -space) is a https://doi.org/10.30526/31.2.19523 https://doi.org/10.30526/31.2.1953 Mathmatics | 146 8201( عام 2العدد) 13لمجلد ا مجلة إبن الهيثم للعلوم الصرفة و التطبيقية Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 31 (2) 2018 soft (1,2)*-ω- 1 ~ T -space (resp. soft (1,2)*-α-ω- 1 ~ T -space, soft (1,2)*-pre-ω- 1 ~ T -space, soft (1,2)*-b-ω- 1 ~ T -space, soft (1,2)*-β-ω- 1 ~ T -space). Proof: It follows that from the theorem (2.29). Proposition (2.32): If ),~,( 1 PU τ or ), ~,( 2 PU τ is a soft 1 ~ T -space, then ),~,~,( 21 PU ττ is a soft (1,2)*-ω- 1 ~ T -space (resp. soft (1,2)*-α-ω- 1 ~ T -space, soft (1,2)*-pre-ω- 1 ~ T -space, soft (1,2)*-b- ω- 1 ~ T -space, soft (1,2)*-β-ω- 1 ~ T -space). Proof: It follows from the fact ~~iτ soft 21 ~~ ττ -open sets in U ~ , 2,1i and proposition (2.25). Remark (2.33): The converse of proposition (2.32) is not true in general in example (2.27) ),~,~,( 21 PU ττ is a soft (1,2)*-ω- 1 ~ T -space (resp. soft (1,2)*-α-ω- 1 ~ T -space, soft (1,2)*-pre-ω- 1 ~ T -space, soft (1,2)*-b-ω- 1 ~ T -space, soft (1,2)*-β-ω- 1 ~ T -space), but both ),~,( 1 PU τ and ),~,( 2 PU τ are not soft 1 ~ T -space. Definition (2.34): A soft bitopological space ),~,~,( 21 PU ττ is called a soft (1,2)*-ω- 2 ~ T -space (resp. soft (1,2)*-α-ω- 2 ~ T -space, soft (1,2)*-pre-ω- 2 ~ T -space, soft (1,2)*-b-ω- 2 ~ T -space, soft (1,2)*-β-ω- 2 ~ T -space) if for any two distinct soft points x~ and y~ of U ~ , there are two soft (1,2)*-ω-open (resp. soft (1,2)*-α-ω-open, soft (1,2)*-pre-ω-open, soft (1,2)*-b-ω-open, soft (1,2)*-β-ω-open) sets ),( PH and ),( PK in U ~ such that ),(~~ PHx , ),(~~ PKy and φ~),( ~ ),( PKPH  . Proposition (2.35): Every soft (1,2)*-ω- 2 ~ T -space (resp. soft (1,2)*-α-ω- 2 ~ T -space, soft (1,2)* -pre-ω- 2 ~ T -space, soft (1,2)*-b-ω- 2 ~ T -space, soft (1,2)*-β-ω- 2 ~ T -space) is a soft (1,2)*-ω- 1 ~ T - space (resp. soft (1,2)*-α-ω- 1 ~ T -space, soft (1,2)*-pre-ω- 1 ~ T -space, soft (1,2)*-b-ω- 1 ~ T -space, soft (1,2)*-β-ω- 1 ~ T -space). Proof: Let Uyx ~~~,~  , yx ~~  . By our assumption there are two soft (1,2)*-ω-open sets ),( PH and ),( PK in U ~ such that ),(~~ PHx , ),(~~ PKy and ),( ~ ),( PKPH  φ~ . Thus ),( PH and ),( PK are soft (1,2)*-ω-open sets in U ~ such that ),( PH containing x~ , but not y~ and ),( PK containing y~ , but not x~ . Therefore ),~,~,( 21 PU ττ is a soft (1,2)*-ω- 1 ~ T -space. Similarly, we can prove that other cases. Remark (2.36): The converse of proposition (2.35) is not true in general. We see that in the following example: Example (2.37): Let X and },{ 21 ppP  and let 1 ~τ = cPHUPH ),(: ~~),{(  is finite}  ~ }~{φ and } ~, ~ {~2 φτ U be soft topologies over U. Thus ), ~,~,( 21 PU ττ is a soft (1,2)*-ω- 1 ~ T -space (resp. soft (1,2)*-α-ω- 1 ~ T -space, soft (1,2)*-pre-ω- 1 ~ T -space, soft (1,2)*-b-ω- 1 ~ T -space, soft (1,2)*-β-ω- 1 ~ T -space), clear that is not soft (1,2)*-β-ω- 2 ~ T -space. Proposition (2.38):(i): Every soft (1,2)*- 2 ~ T -space is a soft (1,2)*-ω- 2 ~ T -space. (ii) Every soft (1,2)*-ω- 2 ~ T -space is a soft (1,2)*-α-ω- 2 ~ T -space. (iii) Every soft (1,2)*-α-ω- 2 ~ T -space is a soft (1,2)*-pre-ω- 2 ~ T -space. (iv) Every soft (1,2)*-pre-ω- 2 ~ T -space is a soft (1,2)*-b-ω- 2 ~ T -space. (v) Every soft (1,2)*-b-ω- 2 ~ T -space is a soft (1,2)*-β-ω- 2 ~ T -space. Remark (2.39): The converse of proposition (2.38) is not true in general as shown by the following examples: https://doi.org/10.30526/31.2.19523 https://doi.org/10.30526/31.2.1953 Mathmatics | 147 8201( عام 2العدد) 13لمجلد ا مجلة إبن الهيثم للعلوم الصرفة و التطبيقية Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 31 (2) 2018 Example (2.40): Let },,{ cbaU  and },{ 21 ppP  and let )},(, ~, ~ {~ 11 PHU φτ  and , ~, ~ {~2 φτ U )},( 2 PH be soft topologies over U, where ),( 1 PH })}{,(}),,{,{( 21 apcap and ),( 2 PH })},{,(}),{,{( 21 cbpbp . The soft sets in ),,(, ~, ~ { 1 PHU φ )},( 2 PH are soft 21 ~~ ττ -open. Thus ),~,~,( 21 PU ττ is a soft (1,2)*-ω- 2 ~ T -space, but is not soft (1,2)*- 2 ~ T -space. Since }){,( 1 ap x ~ }){,(~ 2 apy  , but there exists no soft 21 ~~ ττ -open set ),( 1 PK containing x ~ and soft 21 ~~ ττ - open set ),( 2 PK containing y ~ such that φ~),( ~ ),( 21 PKPK  . Example (2.41): Let U and }{pP  and let )},(, ~, ~ {~ 11 PHU φτ  and 2 ~τ )},(,~, ~ { 2 PHU φ be soft topologies over U, where ),( 1 PH })}1{,{( p , ),( 2 PH })}1{,{(p and ),( 3 PH })}1,1{,{( p . The soft sets in ),,(, ~, ~ { 1 PHU φ ),( 2 PH , )},( 3 PH are soft 21 ~~ ττ -open. Thus ),~,~,( 21 PU ττ is a soft (1,2)*-α-ω- 2 ~ T -space, clear that is not soft (1,2)*-ω- 2 ~ T -space. Example (2.42): Let U and }{pP  and let )},(, ~, ~ {~1 PHU φτ  and } ~, ~ {~2 φτ U be soft topologies over U, where ),( PH })}1{,{(p . The soft sets in )},(, ~, ~ { PHU φ are soft 21 ~~ ττ - open. Thus ),~,~,( 21 PU ττ is a soft (1,2)*-pre-ω- 2 ~ T -space, but is not soft (1,2)*-α-ω- 2 ~ T -space. Since })2{,(p  yx ~~ })3{,(p , but there exists no a soft (1,2)*-α-ω-open set ),( 1 PK containing x~ and a soft (1,2)*-α-ω-open set ),( 2 PK containing y ~ such that  ~ ),( 1 PK φ~),( 2 PK . Theorem (2.43): For a soft bitopological space ),~,~,( 21 PU ττ the following statements are equivalent. (i) ),~,~,( 21 PU ττ is a soft (1,2)*-ω- 2 ~ T -space (resp. soft (1,2)*-α-ω- 2 ~ T -space, soft (1,2)*-pre- ω- 2 ~ T -space, soft (1,2)*-b-ω- 2 ~ T -space, soft (1,2)*-β-ω- 2 ~ T -space) (ii) If Ux ~~~  , then for each xy ~~  , there is a soft (1,2)*-ω-neighborhood (resp. soft (1,2)*-α- ω-neighborhood, soft (1,2)*-pre-ω-neighborhood, soft (1,2)*-b-ω-neighborhood, soft (1,2)*-β-ω-neighborhood) ),( PN of x~ such that  ~~y (1,2)*- ),( PNclω (resp.  ~~y (1,2)*-α- ),( PNclω ,  ~~y (1,2)*-pre- ),( PNclω ,  ~~y (1,2)*-b- ),( PNclω ,  ~~y (1,2)*-β- ),( PNclω ). (iii) For each Ux ~~~  , { ~  (1,2)*- :),( PNclω ),( PN is a soft (1,2)*-ω-neighborhood of x~ } (resp. { ~  (1,2)*-α- :),( PNclω ),( PN is a soft (1,2)*-α-ω-neighborhood of x~ }, { ~  (1,2)*- pre- :),( PNclω ),( PN is a soft (1,2)*-pre-ω-neighborhood of x~ }, { ~  (1,2)*-b- :),( PNclω ),( PN is a soft (1,2)*-b-ω-neighborhood of x~ }, { ~  (1,2)*-β- :),( PNclω ),( PN is a soft (1,2)*-β-ω-neighborhood of x~ }) }~{x . Proof: )()( iii  . Let Ux ~~~  . If Uy ~~~  such that xy ~~  , then there exists disjoint soft (1,2)*- ω-open sets ),( PH and ),( PK such that ),(~~ PHx and ),(~~ PKy . Hence  ~),(~~ PHx c PK ),( which implies that c PK ),( is a soft (1,2)*-ω-neighborhood of x~ . Also c PK ),( is soft (1,2)*-ω-closed and c PKy ),( ~~  . Let ),( PN c PK ),( . Then  ~~y (1,2)*- ),( PNclω . )()( iiiii  . Obvious. )()( iiii  . Let Uyx ~~~,~  , yx ~~  . By hypothesis, there is at least a soft (1,2)*-ω-neighborhood ),( PN of x~ such that  ~~y (1,2)*- ),( PNclω . We have ( ~~ x (1,2)*- c PNcl )),(ω which is soft (1,2)*-ω-open. Since ),( PN is a soft (1,2)*-ω-neighborhood of x~ , then there exists a soft (1,2)*-ω-open set ),( PH in U ~ such that  ~),(~~ PHx ),( PN and  ~ ),( PH ( (1,2)*- c PNcl )),(ω φ~ . Hence ),~,~,( 21 PU ττ is a soft (1,2)*-ω- 2 ~ T -space. Similarly, we can prove that other cases. https://doi.org/10.30526/31.2.19523 https://doi.org/10.30526/31.2.1953 Mathmatics | 148 8201( عام 2العدد) 13لمجلد ا مجلة إبن الهيثم للعلوم الصرفة و التطبيقية Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 31 (2) 2018 Now, we need the following lemma. Lemma (2.44): Let ),~,~,( 21 PU ττ be a soft bitopological space and UY  . If ),( PH is a soft (1,2)*-ω-open set in U ~ , then YPH ~~ ),(  is a soft (1,2)*-ω-open set in Y ~ . Proof: Let ),( PH be a soft (1,2)*-ω-open set in U ~ . To prove that YPH ~~ ),(  is a soft (1,2)*- ω-open set in Y ~ . Let YPHy ~~ ),(~~   ),(~~ PHy . Since ),( PH is soft (1,2)*-ω-open in U ~   a soft 21 ~~ ττ -open set ),( PV in U ~ such that ),(~~ PVy and ),(),( PHPV  is a soft countable set. Hence ),( ~~ PVY  is a soft YY 21 ~~ ττ -open set in Y ~ . Since  ~~ ()),( ~~ ( YPVY  )),( PH )),(),(( ~~ PHPVY   )),(),((~ PHPV  , then )),( ~~ ()),( ~~ ( PHYPVY   is soft countable. Thus YPH ~~ ),(  is a soft (1,2)*-ω-open set inY ~ . Proposition (2.45): Every soft subspace of a soft (1,2)*-ω- 2 ~ T -space is a soft (1,2)*-ω- 2 ~ T - space. Proof: Let ),~,~,( 21 PU ττ be a soft (1,2)*-ω- 2 ~ T -space and ),~,~,( ~ 2 ~ 1 PY YY ττ be a soft subspace of ),~,~,( 21 PU ττ . To prove that ), ~,~,( ~ 2 ~ 1 PY YY ττ is a soft (1,2)*-ω- 2 ~ T -space. Let Yyx ~~~,~  such that yx ~~  . Since UY ~~~  , then Uyx ~~~,~  . But ),~,~,( 21 PU ττ is a soft (1,2)*-ω- 2 ~ T -space, then there are two soft (1,2)*-ω-open sets ),( PH and ),( PK in U ~ such that ),(~~ PHx , ),(~~ PKy and φ ~),( ~ ),( PKPH  . By lemma (2.44), YPHPH ~~ ),(),(  and YPKPK ~~ ),(),(  are soft (1,2)*-ω-open sets in Y ~ such that ),(~~ PHx  and ),(~~ PKy  . Since  ~ ),( PH  ),( PK  ~ ) ~~ ),(( YPH  ~ ),(( PK ) ~ Y YPKPH ~~ )),( ~ ),((  φφ ~ ~~~  Y . Thus ),~,~,( ~ 2 ~ 1 PY YY ττ is a soft (1,2)*-ω- 2 ~ T -space. Remark (2.46): Soft subspace of a soft (1,2)*-α-ω- 2 ~ T -space is not a soft (1,2)*-α-ω- 2 ~ T - space as shown in the following example: Example (2.47): Let U and }{pP  and let )},(, ~, ~ {~ 11 PHU φτ  and 2 ~τ )},(,~, ~ { 2 PHU φ be soft topologies over U, where ),( 1 PH })}1{,{( p , ),( 2 PH })}1{,{(p and ),( 3 PH })}1,1{,{( p . Then ),~,~,( 21 PU ττ is a soft (1,2)*-α-ω- 2 ~ T -space. If  UY }1{ , then )},(,~, ~ {~ 1~1 PHYY φτ  and } ~, ~ {~ ~ 2 φτ Y Y  are soft topologies over Y. The soft sets in )},(,~, ~ { 1 PHY φ are soft 2 ~ 1 ~ ~~ YY ττ -open. Therefore ),~,~,( ~ 2 ~ 1 PY YY ττ is not a soft (1,2)*-α-ω- 2 ~ T - space, since })3{,(p  x~ })4{,(~ py  , but there exists no soft (1,2)*-α-ω-open sets ),( 1 PK and ),( 2 PK in Y ~ such that ),(~~),,(~~ 21 PKyPKx  and ),( ~ ),( 21 PKPK  φ ~ . Proposition (2.48): If ),~,( 1 PU τ or ), ~,( 2 PU τ is a soft 2 ~ T -space, then ),~,~,( 21 PU ττ is a soft (1,2)*-ω- 2 ~ T -space (resp. soft (1,2)*-α-ω- 2 ~ T -space, soft (1,2)*-pre-ω- 2 ~ T -space, soft (1,2)*-b- ω- 2 ~ T -space, soft (1,2)*-β-ω- 2 ~ T -space). Proof: It follows from the fact ~~iτ soft 21 ~~ ττ -open sets in U ~ , 2,1i and proposition (2.38). Remark (2.49): The converse of proposition (2.48) is not true in general. We see that by the following example: Example (2.50): Let },,,{ dcbaU and },{ 21 ppP  and let )},(, ~, ~ {~ 11 PHU φτ  and )},(,~, ~ {~ 22 PHU φτ  be soft topologies over U, where ),( 1 PH }),,{,{( 1 bap })},{,( 2 bap and ),( 2 PH })}{,(}),{,{( 21 apap . The soft sets in ),,(, ~, ~ { 1 PHU φ )},( 2 PH are soft 21 ~~ ττ -open. Thus ),~,~,( 21 PU ττ is a soft (1,2)*-ω- 2 ~ T -space (resp. soft (1,2)*-α-ω- 2 ~ T -space, soft (1,2)*- https://doi.org/10.30526/31.2.19523 https://doi.org/10.30526/31.2.1953 Mathmatics | 149 8201( عام 2العدد) 13لمجلد ا مجلة إبن الهيثم للعلوم الصرفة و التطبيقية Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 31 (2) 2018 pre-ω- 2 ~ T -space, soft (1,2)*-b-ω- 2 ~ T -space, soft (1,2)*-β-ω- 2 ~ T -space), but both ),~,( 1 PU τ and ),~,( 2 PU τ are not soft 2 ~ T -space. Definition (2.51): A soft function ),~,~,(),~,~,(: 2121 PVPUf σσττ  is called strongly soft (1,2)*-ω-continuous (resp. strongly soft (1,2)*-α-ω-continuous, strongly soft (1,2)*-pre-ω- continuous, strongly soft (1,2)*-b-ω-continuous, strongly soft (1,2)*-β-ω-continuous) if )),(( 1 PHf  is a soft 21 ~~ ττ -open set in U ~ for each soft (1,2)*-ω-open (resp. soft (1,2)*-α-ω- open, soft (1,2)*-pre-ω-open, soft (1,2)*-b-ω-open, soft (1,2)*-β-ω-open) set ),( PH in V ~ . Theorem (2.52): Let ),~,~,(),~,~,(: 2121 PVPUf σσττ  be a strongly soft (1,2)*-ω-continuous (resp. strongly soft (1,2)*-α-ω-continuous, strongly soft (1,2)*-pre-ω-continuous, strongly soft (1,2)*-b-ω-continuous, strongly soft (1,2)*-β-ω-continuous) injective function. If ),~,~,( 21 PV σσ is a soft (1,2)*-ω- iT ~ -space (resp. soft (1,2)*-α-ω- iT ~ -space, soft (1,2)*-pre-ω- iT ~ -space, soft (1,2)*-b-ω- iT ~ -space, soft (1,2)*-β-ω- iT ~ -space, then ),~,~,( 21 PU ττ is a soft (1,2)*- iT ~ -space, for 2,1,,0 2 1i . Proof: Suppose that ),~,~,( 21 PV σσ is a soft (1,2)*-ω- 2 ~ T -space. Let Uyx ~~~,~  such that yx ~~  . Since f is injective and ),~,~,( 21 PV σσ is a soft (1,2)*-ω- 2 ~ T -space, then there exists disjoint soft (1,2)*-ω-open sets ),( 1 PH and ),( 2 PH in V ~ such that ),(~)~( 1 PHxf  and ),( ~)~( 2 PHyf  . By definition (2.51), )),(( 1 1 PHf  and )),(( 2 1 PHf  are soft 21 ~~ ττ -open sets in U ~ such that )),((~~ 1 1 PHfx   , )),((~~ 2 1 PHfy   and φ ~)),(( ~ )),(( 2 1 1 1   PHfPHf  . Hence ),~,~,( 21 PU ττ is a soft (1,2)*- 2 ~ T -space. Similarly, we can prove that other cases. Definition (2.53): A soft function ),~,~,(),~,~,(: 2121 PVPUf σσττ  is called strongly soft (1,2)*-ω-open (resp. strongly soft (1,2)*-α-ω-open, strongly soft (1,2)*-pre-ω-open, strongly soft (1,2)*-b-ω-open, strongly soft (1,2)*-β-ω-open) if )),(( PHf is a soft 21 ~~ σσ -open set in V ~ for each soft (1,2)*-ω-open (resp. soft (1,2)*-α-ω-open, soft (1,2)*-pre-ω-open, soft (1,2)*-b- ω-open, soft (1,2)*-β-ω-open) set ),( PH in U ~ . Theorem (2.54): Let ),~,~,(),~,~,(: 2121 PVPUf σσττ  be a strongly soft (1,2)*-ω-open (resp. strongly soft (1,2)*-α-ω-open, strongly soft (1,2)*-pre-ω-open, strongly soft (1,2)*-b-ω-open, strongly soft (1,2)*-β-ω-open) bijective function. If ),~,~,( 21 PU ττ is a soft (1,2)*-ω- iT ~ -space (resp. soft (1,2)*-α-ω- iT ~ -space, soft (1,2)*-pre-ω- iT ~ -space, soft (1,2)*-b-ω- iT ~ -space, soft (1,2)*-β-ω- iT ~ -space), then ),~,~,( 21 PV σσ is a soft (1,2)*- iT ~ -space, for 2,1,,0 2 1i . Proof: Suppose that ),~,~,( 21 PU ττ is a soft (1,2)*-ω- 2 ~ T -space. Let Vyy ~~~,~ 21  such that 21 ~~ yy  . Since f is surjective, then there exists Uxx ~~~,~ 21  such that 11 ~)~( yxf  and 22 ~)~( yxf  . Since f is a function, then 21 ~~ xx  . But ),~,~,( 21 PU ττ is a soft (1,2)*-ω- 2 ~ T - space, then there exists disjoint soft (1,2)*-ω-open sets ),( 1 PH and ),( 2 PH in U ~ such that ),(~~ 11 PHx  and ),( ~~ 22 PHx  . By definition (2.53), )),(( 1 PHf and )),(( 2 PHf are soft 21 ~~ σσ -open sets in V ~ such that )),((~)~( 11 PHfxf  and )),(( ~)~( 22 PHfxf  . Since f is injective, then  ~ )),(( 1 PHf φ ~)),(( 2 PHf . Hence ), ~,~,( 21 PV σσ is a soft (1,2)*- 2 ~ T -space. By the same way we can prove that other cases. The following diagram shows the relation among soft (1,2)*- iT ~ -spaces, soft (1,2)*-ω- iT ~ -spaces, soft (1,2)*-α-ω- iT ~ -spaces, soft (1,2)*-pre-ω- iT ~ -spaces, soft (1,2)*-b-ω- iT ~ - spaces, and soft (1,2)*-β-ω- iT ~ -spaces, for 2,1,,0 2 1i . https://doi.org/10.30526/31.2.19523 https://doi.org/10.30526/31.2.1953 Mathmatics | 150 8201( عام 2العدد) 13لمجلد ا مجلة إبن الهيثم للعلوم الصرفة و التطبيقية Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 31 (2) 2018 References 1. Molodtsov, D. (1999), “Soft set theory-First results”, Computers and Mathematics with Applications,37 (4-5):19-31. 2. Shabir, M. and Naz, M. (2011), “On soft topological spaces”, Computers and Mathematics with Applications, 61(7):1786-1799. 3. Senel, G. and Çagman, N. (2014), “Soft closed sets on soft bitopological space”, Journal of new results in science, 5:57-66. 4. Mahmood, S. I. and Abdul-Hady, A. 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