Gazi University Journal of Science GU J Sci 27(4):1077-1083 (2014) On Soft Preopen Sets and Soft Pre Separation Axioms Metin AKDAG1,♠, Alkan OZKAN1 1 Cumhuriyet University Science Faculty Department of Mathematics 58140 Sivas Received:03/06/2014 Accepted:07/08/2014 ABSTRACT Arockiarani and Lancy [7], defined soft pre-open (closed) sets on soft topology. In this paper, we are continue investigating the properties of soft pre-open (closed) sets and define soft preclosure and soft preinterior in soft topological spaces. We are also introduce and research basic properties of the concepts of soft pre-regular spaces, soft P₃-spaces, soft pre-normal spaces and soft P₄-spaces in soft topological spaces, which are basic for further research on soft topology and will fortify the footing of the theory of soft topological space. Keywords: 1. INTRODUCTION Molodtsov [2] introduced the concept of a soft set in order to solve complicated problems in the economics, engineering, and environmental areas, because no mathematical tools can successfully deal with the various kinds of uncertainties in these problems. In the recent years in development in the fields of soft set theory and its application has been taking place in a rapid pace. Of late many authors [6,8,9,21,22,23] have studied various properties of soft topological spaces. This is because of the general nature of parameterization expressed by a soft set. Shabir and Naz [4] introduced the notion of soft topological spaces which are defined over an initial universe with a fixed set of parameters. Later, Akdag et al.[8], Aygunoglu et al [6] and Hussain et al are continued to study the properties of soft topological space. They got many important results in soft topological spaces. Weak forms of soft open sets were first studied by Chen [5]. He ♠Corresponding author, e-mail: [email protected] investigated soft semi-open sets in soft topological spaces and studied some properties of it. Arockiarani and Lancy are defined soft β-open sets and continued to study weak forms of soft open sets in soft topological space. Akdag and Ozkan [20], defined soft α-open and soft α-closed sets in soft topological spaces and studied many important results and some properties of it. Pre-open sets were introduced by Mashhour et al. [10] and since then different topological properties have been defined in terms of pre-open sets and investigated by many researchers [11,12,13,14,15,16,17,18,19]. In this paper we extend the notion of preopenness to soft sets [9]. We define soft preinteriors and preclosures and investigate their properties. Most attention is paid to the extension of separation notions to soft topological spaces with the help of soft pre-open sets. We obtain interesting characterization of soft pre separation axioms. 1078 GU J Sci, 27(4):1077-1083 (2014)/Metin AKDAĞ, Alkan ÖZKAN 2. PRELIMINARIES Definition 1. [2] Let be an initial universe and be a set of parameters. Let () denote the power set of and be a non-empty subset of . A pair (, ) is called a soft set over , where is a mapping given by : → (). In other words, a soft set over is a parameterized family of subsets of the universe . For ∈ , () may be considered as the set of -approximate elements of the soft set (, ). (, ), if ⊂ , and Definition 2. [2] Let (, ) and (, ) be two soft sets over a common universe . (, ) ⊂ () ⊂ () for all ∈ . Definition 3. [3] Two soft sets (, ) and (, ) over a common universe are said to be soft equal if (, ) is a soft subset of (, ) and (, ) is a soft subset of (, ). Definition 4. [3] A soft set (, ) over is called a null soft set, denoted by ∅, if ∈ , () = ∅. Definition 5. [3] A soft set (, ) over is called an absolute soft set, denoted by , if ∈ , () = . Definition 6. [2] The union of two soft sets of (, ) and (, ) over the common universe is the soft set (, ), where = ∪ and for all ∈ , (, ) = (, ). We write (, ) ∪ F(e),ife ∈ A − B () = G(e),ife ∈ B − A F(e) ∪ G(e), ife ∈ (A ∩ B) Definition 7. [2] The intersection (, ) of two soft sets (, ) and (, ) over a common universe , denoted (, ), is defined as = ∩ , and () = () ∩ () for all ∈ . (, ) ∩ Definition 8. [1] For a soft set (, ) over the relative complement of (, ) is denoted by (, )& and is defined by (, )& = ( & , ), where & : → () is a mapping given by & (') = − (') for all α∈A. Definition 9. [4] Let (, ) be a soft set over and ( ∈ . We say that ( ∈ (, ) read as ( belongs to the soft set (, ), whenever ( ∈ (') for all ' ∈ . Note that for ( ∈ , ( ∉ (, ) if ( ∉ (')for some ' ∈ . Definition 10. [4] Let ( ∈ ; then ((, ) denotes the soft set over for which ((') = {(}, for all ' ∈ . Lemma 1. [22] Let (, ) be a soft set over and ( ∈ . Then: (, ); (1) ( ∈ (, )iff ((, ) ⊂ (, ) = - then ( ∉ (, ). (2) if((, ) ∩ Definition 11. [4] Let (, ) be a soft set over and Y be a non-empty subset of . Then the subsoft set of (, ) over Y denoted by ( . , ), is defined as follows . (') = / ∩ ('), for all ' ∈ . (, ). In other words ( . , ) = /0 ∩ Definition 12. [4] Let 1 be the collection of soft sets over , then τ is said to be a soft topology on if satisfies the following axioms. (1) ∅, belong to 1, (2) the union of any number of soft sets in 1 belongs to 1, (3) the intersection of any two soft sets in 1 belongsto 1. The triplet (, 1, ) is called a soft topological space over . Let (, 1, ) be a soft topological space over , then the members of τ are said to be soft open sets in . Let (, 1, ) be a soft topological space over . A soft set (, ) over is said to be a soft closed set in , if its relative complement (, )& belongs to 1. If (, 1, ) is a a soft topological space with 1 = {∅, }, then 1 is called the soft indiscrete topology on and (, 1, ) is said to be a soft indiscrete topological space. If (, 1, ) is a a soft topological space with 1 is the collection of all soft sets which can be defined over , then 1 is called the soft discrete topology on and (, 1, ) is said to be a soft discrete topological space. Throughout the paper, the space and / (or (, 1, ) and (/, 2, )) stand for soft topological spaces assumed unless otherwise stated. Let (, ) be a subset of a soft topological space . The closure and interior of (, ) are denoted by 34((, )) and 567((, )), respectively. Definition 13. [4] Let (, 1, ) be a soft topological space over , (, ) be a soft set over and ( ∈ . Then ( is said to (, ). be soft interior point of (, ) if there exists a soft open set (, ) such that ( ∈ (, ) ⊂ Definition 14. [4] Let (, 1, ) be a soft topological space over , (, ) be a soft set over and ( ∈ . Then (, ) is (, ). said to be a soft neighborhood of ( if there exists a soft open set (, ) such that ( ∈ (, ) ⊂ Definition 15. [5] Let (, 1, ) be a soft topological space over and (, ) be a soft set over . (1) The soft interior of (, ) is the soft set {(8, ): (8, )59:9;<7;=6:6>(8, ) ⊂ (, )} 567((, )) =∪ (2) The soft closure of (, ) is the soft set {(, ): (, )59:9;<734;9>:6>(, ) ⊂ (, )}. 34((, )) =∩ GU J Sci, 27(4):1077-1083 (2014)/Metin AKDAĞ, Alkan ÖZKAN 3. SOME PROPERTIES OF SOFT PRE-OPEN SETS AND PRE-CLOSED SETS 1079 Definition 16. [7] Let (, ) be any soft set of a soft topological space (, 1, ). (, ) is called 567(34((, ))), and (1) (, ) soft pre-open set of if (, ) ⊂ 34(567((, ))). (2) (, ) soft pre-closed set of if (, ) ⊃ Theorem 1. (1) Arbitrary union of soft pre-open sets is a soft pre-open sets, and (2) Arbitrary intersection of soft pre-closed sets is a soft pre-closed set. Proof (1) Let {(@ , ): ' ∈ A} be a collection of soft pre-open sets. Then, for each ' ∈ A, 567(34((@ , ))). Now (@ , ) ⊂ (@ , ) ⊂ 567(34((@ , ))) ⊂ (34((@ , ))) = 567(34(∪ ((@ , ))). ∪ 567 ∪ ∪ (@ , ) is a soft pre-open set. Hence ∪ (2) Follows easily by taking complements. Remark 1. It is obvious that every soft open (closed) set is a soft pre-open (pre-closed) set. That the converse is false, is shown by following Example. Example 1. Let = {(₁, (₂, (₃, (₄}, = {₁, ₂, ₃} and 1 = {-, , (₁, ), (₂, ), (₃, ), . . . (₁₅, )} where (₁, ), (₂, ), (₃, ), . . . (₁₅, ) are soft sets over , defined as follows (₁, ) = {(₁, {(₁}), (₂, {(₂, (₃}), (₃, {(₁, (₄})}, (₂, ) = {(₁, {(₂, (₄}), (₂, {(₁, (₃, (₄}), (₃, {(₁, (₂, (₃})}, (₃, ) = {(₂, {(₃}), (₃, {(₁})}, (₄, ) = {(₁, {(₁, (₂, (₄}), (₂, ), (₃, )}, (₅, ) = {(₁, {(₁, (₃}), (₂, {(₂, (₄}), (₃, {(₂})}, (₆, ) = {(₁, {(₁}), (₂, {(₂})}, (₇, ) = {(₁, {(₁, (₃}), (₂, {(₂, (₃, (₄}), (₃, {(₁, (₂, (₄})}, (₈, ) = {(₂, {(₄}), (₃, {(₂})}, (₉, ) = {(₁, ), (₂, ), (₃, {(₁, (₂, (₃})}, (₁₀, ) = {(₁, {(₁, (₃}), (₂, {(₂, (₃, (₄}), (₃, {(₁, (₂})}, (₁₁, ) = {(₁, {(₁, (₂, (₄}), (₂, ), (₃, {(₁, (₂, (₃})}, (₁₂, ) = {(₁, {(₁}), (₂, {(₂, (₃, (₄}), (₃, {(₁, (₂, (₄})}, (₁₃, ) = {(₁, {(₁}), (₂, {(₂, (₄}), (₃, {(₂})}, (₁₄, ) = {(₂, {(₃, (₄}), (₃, {(J , (₂})}, (₁₅, ) = {(₁, {(₁}), (₂, {(₂, (₃}), (₃, {(₁})}. Then 1 defines a soft topology on and thus (, 1, ) is a soft topological space over . Clearly the soft closed sets are 0, -, (J , )& , (K , )& , (L , )& , . . . , (JM , )& . Then, let us take (, ) = {(₃, {(₂})} then 34((, )) = (J , )& , 567(34((, ))), hence (, ) is soft pre-open but not soft open. 567(34((, ))) = (₈, ), and so (, ) ⊂ Let (, ) be a subset of a soft topological space . The soft preclosure and preinterior of (, ) are denoted by 9=34((, )) and 9=567((, )), respectively. Definition 17. Let (, 1, ) be a soft topological space over and (, ) be a soft set over . Then: {(8, ): (8, )59:9;<7=N − ;=6:6>(8, ) ⊂ (, )} is called preinterior. (1) 9=567((, )) =∪ {(, ): (, )59:9;<7=N − 34;9>:6>(, ) ⊂ (, )} is called preclosure. (2) 9=34((, )) =∩ Clearly 9=567((, )) is the largest soft pre-open set over which is contained in (, ) and 9=34((, )) is the smallest soft pre-closed set over which contains (, ). Theorem 2. Let (, ) be any soft set in a soft space (, 1, ). Then, (1) 9=34((, )& ) = − 9=567((, )) and (2) 9=567((, )& ) = − 9=34((, )). (, ) is precisely the complement of a pre-closed set Proof (1) We see that a soft pre-open set (, ) ⊂ (, )& . Thus (, ) ⊃ {(, )& : (, )599;<7=N − 34;9>:6>(, ) ⊃ (, )& } 9=567((, )) =∪ (, )& } = −∩ {(, ): (, )599;<7=N − 34;9>:6>(, ) ⊃ = − 9=34((, )& ), whence 9=34((, )& ) = − 9=567((, )). (, ), (8, ) = (O, )& ⊂ (, )& . (2) Next let (8, ) be any soft pre-open set. Then for a soft pre-closed set (O, ) ⊃ {(8, )& : (8, )599;<7=N − ;=6:6>(8, ) ⊂ (, )& } 9=34((, )) =∩ {(8, ): (8, )599;<7=N − ;=6:6>(8, ) ⊂ (, )& } = −∪ = − 9=567((, )& ). Thus 9=567((, )& ) = − 9=34((, )). Theorem 3. ln a soft topological space (, 1, ), a soft set (, ) is soft pre-closed (pre-open) iff (, ) = 9=34(, ) ((, ) = 9=567(, )). (, ) and (, ) is soft pre-closed. Proof Let (, ) be soft pre-closed in (, 1, ). Since (, ) ⊂ (, )},and (, ) ∈ {(, ): (, )599;<7=N − 34;9>:6>(, ) ⊂ (, ) implies that (, ) ⊂ {(, ): (, )599;<7=N − 34;9>:6>(, ) ⊂ (, )}, i.e. (, ) =∩ (, ) = 9=34((, )). Conversely suppose that (, ) = 9=34((, )), i.e. 1080 GU J Sci, 27(4):1077-1083 (2014)/Metin AKDAĞ, Alkan ÖZKAN {(, ): (, )599;<7=N − 34;9>:6>(, ) ⊂ (, )}. (, ) =∩ (, )}. This implies that (, ) ∈ {(, ): (, )599;<7=N − 34;9>:6>(, ) ⊂ Hence (, ) is soft pre-closed. For the soft preopenness case use the definition of soft preinteriors. Proposition 1. In a soft space (, 1, ), the folowing hold for soft preclosure: (1) 9=34(∅) = ∅, (2) 9=34((, )) is soft pre-closed in (, 1, ) for each soft subset (, ) of , 9=34((, )), if (, ) ⊂ (, ), (3) 9=34((, )) ⊂ (4) 9=34(9=34((, ))) = 9=34((, )). Proof Easy. Theorem 4. In a soft topological space the following relations hold: (, )) ⊃ 9=34((, )), 9=34((, )) ∪ (1) 9=34((, ) ∪ (, )) ⊂ 9=34((, )). 9=34((, )) ∩ (2) 9=34((, ) ∩ Proof Easy. Remark 2. Similar results hold for soft preinteriors. Definition 18. [4] Let (, 1, ) be a soft topological space over and / be a non-empty subset of . Then 1. = {( . , )|(, ) ∈ 1} is said to be the soft relative topology on / and (/, 1. , ) is called a soft subspace of (, 1, ). We can easily verify that 1. is, in fact, a soft topology on /. Theorem 5. Let (, ) ⊂ / ⊂ , where (, 1, ) is a soft topological space and (/, 1. , ) is a soft pre-open subspace of (, 1, ). Then (, ) is soft pre-open in iff it is soft pre-open in /. / = (, ), so 567(34((, ))). But (, ) ⊂ / implies that (, ) ∩ Proof Since (, ) is soft pre-open in , (, ) ⊂ /) ⊂ /)). Therefore (, ) is soft pre-open in /. 567(34((, ) ∩ that ((, ) ∩ 567. Q34. R(, )ST which is soft pre-open in /. There exists an Conversely let (, ) be soft pre-open in /. Then (, ) ⊂ /0 = 567. Q34. R(, )ST. Thus (, ) ⊂ /0 . But / is soft pre-open in (8, ) ∩ open soft set (8, ) in such that (8, ) ∩ 567(34(/)). . Thus, / ⊂ 567(34(/)) ⊂ /0))) (8, ) ∩ 567(34((8, ) ∩ Hence we have (, ) ⊂ 567 V34 Q34. R(, )STW ⊂ 567 V34 Q34R(, )STW = 567 Q34R(, )ST. = 567 U34 V567. Q34. R(, )STWX ⊂ Hence (, ) is soft pre-open in . 4. SOME PROPERTIES OF SOFT PRE-SEPARATIONS AXIOMS Arokia and Arockiarani [24], defined some notions of soft pre-separation axioms. The this section, we introduce and study the new concepts of soft pre-separation axioms and investigated basic properties of these concepts in soft topolgical space. Definition 19. [24] Let (, 1, ) be a soft topological space over and (, Y ∈ such that ( ≠ Y. Then a soft topological space (, 1, ) is said to be soft ₀-space if there exist soft pre-open sets (, ) and (, ) such that ( ∈ (, ) and Y ∉ (, ) or Y ∈ (, ) and ( ∉ (, ). Example 2. A discrete soft topological space is a soft ₀-space since every (₁ ∈ is a soft pre-open set in the discrete space. A characterization for soft ₀-space is the following. Theorem 6. A soft topological space (, 1, ) is soft ₀-space iff soft preclosures of any two different soft singletons are distinct Proof Let (, 1, ) be soft ₀-space, and ((₁, ), ((₂, ) be two soft singletons, where x₁≠x₂. (, 1, ) being soft ₀ (O, ) ⊂ ((₂, )& . space, there exists a soft pre-open set (O, ) such that ((₁, ) ⊂ \ (O, )& , \ 9=34(((₂, )). 9=34(((₂, )) ⊂ (O, )& . This implies ((₂, ) ⊂ Since ((₁, ) ⊄ ((₁, ) ⊄ But \ ((₁, ) ⊄ 9=34(((₁, )). Hence 9=34(((₁, )) ≠ 9=34(((₂, )). Conversely, let ((₁, ) and ((₂, ) be any two soft singletons such that (₁ ≠ (₂. By hypothesis ((₁, ) ∉ 9=34(((₂, )) or ((₂, ) ∉ 9=34(((₁, )). Then since ((₁, ) ∉ 9=34(((₂, )), (=34(((₂, )))& is a soft pre-open set such that (₁ ∈ (9=34(((₂, )))& or since ((₂, ) ∉ 9=34(((₁, )), (=34(((₁, )))& is a soft pre-open set such that (₂ ∈ (=34(((₁, )))& . Thus (, 1, ) is soft ₀-space. Theorem 7. A soft subspace of a soft ₀-space is soft ₀. Proof Let / be a soft subspace of a soft ₀-space and let Y₁, Y₂ be two distinct soft points of /. Since these soft points /0 is a are also in , then at least one a soft pre-open set (, ) containing one soft point but not the other. Then (, ) ∩ soft pre-open set containing one soft point but not the other. Definition 20. [24] Let (, 1, ) be a soft topological space over and (, Y ∈ such that ( ≠ Y. Then a soft topological space (, 1, ) is said to be soft ₁-space if there exist soft pre-open sets (, ) and (, ) such that ( ∈ (, ) and Y ∉ (, ) and Y ∈ (, ) and ( ∉ (, ). Example 3. Let = {(₁, (₂}, = {₁, ₂} and 1 = {-, , (₁, ), (₂, )} where (₁, ) = {(₁, {(₁})}, (₂, ) = {(₁, {(₁, (₂}), (₂, {(₂})}. Then 1 defines a soft topology on and thus (, 1, ) is a soft topological space over . For (₁, (₂ ∈ , since (₁ ≠ (₂, we can take soft pre-open sets (₁, ) and (₂, ) satisfyig (₁ ∈ (₁, ) and (₂ ∉ (₁, ); and (₂ ∈ (₂, ) and (₁ ∉ (₂, ). Hence the soft topological (, 1, ) is a soft ₁-space. GU J Sci, 27(4):1077-1083 (2014)/Metin AKDAĞ, Alkan ÖZKAN 1081 Remark 3. Obviously every soft ₁-space is soft ₀-space but the converse is not always true, as shown in the next example. Example 4. Let = {(₁, (₂}, = {₁, ₂} and 1 = {-, 0 , (₁, ), (₂, )} where (₁, ) = {(₁, {(₁})}, (₂, ) = {(₁, {(₁}), (₂, {(₂})}. Then τ defines a soft topology on and thus (, 1, ) is a soft topological space over X. For (₁, (₂ ∈ , since (₁ ≠ (₂, we can take soft pre-open sets (₁, ) and (₂, ) satisfyig (₁ ∈ (₁, ) and (₂ ∉ (₁, ). Thus the soft topological (, 1, ) is a soft ₀-space but not soft ₁-space. Theorem 8. A soft subspace of a soft ₁-space is soft ₁. Proof Let / be a soft subspace of a soft ₀-space and let Y₁, Y₂ be two distinct soft points of /. Since these soft points are also in , then there exist soft pre-open sets (, ) and (, ) such that Y₁ ∈ (, ) and Y₂ ∉ (, ); and Y₂ ∈ (, ) /0 and (, ) ∩ /0 are soft pre-open set containing one soft point but not the other in /. and Y₁ ∉ (, ). Then (, ) ∩ Therefore / is a soft ₁-space. Theorem 9. Let (, 1, ) be a soft topological space over . If every soft point of a soft topological space (, 1, ) is a soft pre-closed set, then (, 1, ) is a soft P₁-space. Proof Let (₁ be a soft point of which is a soft pre-closed set then {(₁}& is a soft pre-open set. Then for distinct soft points (₁, (₂, we have {(₁}& , {(₂}& are soft pre-open sets such that (₂ ∈ {(₁}& and (₂ ∉ {(₁}; (₁ ∈ {(₂}& and (₁ ∉ {(₂}. Definition 21. [24] Let (, 1, ) be a soft topological space over and (, Y ∈ such that ( ≠ Y. Then a soft topological space (, 1, ) is said to be soft ₂-space if there exist soft pre-open sets (, ) and (, ) such that ( ∈ (, ), Y ∈ (, ) = -. (, ) and (, ) ∩ Example 5. Let = {(₁, (₂}, = {₁, ₂} and 1 = {-, , (₁, ), (₂, )} where (₁, ) = {(₁, {(₁}), (₂, {(₁})}, (₂, ) = {(₁, {(₂}), (₂, {(₂})}. Then 1 defines a soft topology on and thus (, 1, ) is a soft topological space over . For (₁, (₂ ∈ , since (₁ ≠ (₂, we can take soft pre-open sets (₁, ) and (₂, ) satisfyig (₁ ∈ (₁, ), (₂ ∈ (₂, ) and (₂, ) = -. Therefore the soft topological (, 1, ) is a soft ₂-space. (₁, ) ∩ Remark 4. Obviously every soft ₂-space is soft ₁-space but, the converse is not always true, as shown in the next example. (₂, ) ≠ -. Example 6. It is shown by Example 3 because of (₁, ) ∩ Theorem 10. A soft subspace of a soft ₂-space is soft ₂. Proof Let (, 1, ) be a soft ₂-space and / be a soft subspace of . Let Y₁ and Y₂ be two distinct soft points of /. Since is soft ₂-space, there exist two disjoint soft pre-open sets (, ) and (, ) such that Y₁ ∈ (, ), Y₂ ∈ (, ). Then (, ) ∩ / and (, ) ∩ / are soft pre-open sets satisfying the requirements for / to be a soft ₂-space. (, ) for Theorem 11. Let (, 1, ) be a soft topological space over and ( ∈ . If is a soft ₂-space, then ((, ) =∩ each soft pre-open set (, ) with ( ∈ (, ). Proof Suppose there exists ] ∈ such that ( ≠ ] and ] ∈∩ (') for some ' ∈ . Since is soft ₂-space, there exist (, ) = - and so (, ) ∩ soft pre-open sets (, ) and (, ) such that ( ∈ (, ) and ] ∈ (, ) and (, ) ∩ (], ) = - and (') ∩ ](') = ∅. This contradicts the fact that ] ∈∩ (') for some. This completes the proof. Corollary 1. Let (, 1, ) be a soft topological space over and ( ∈ . If and are finite, and if is a soft ₂-space, then ((, ) is a soft pre-open set for ( ∈ . Definition 22. Let (, 1, ) be a soft topological space over , and let (, ) be a soft pre-closed set in and ( ∈ such (₂, ) and that ( ∉ (, ). If there exist soft pre-open sets (₁, ) and (₂, ) such that ( ∈ (₁, ), (, ) ⊂ (₂, ) = -, then (, 1, ) is called a soft pre-regular space. (₁, ) ∩ Proposition 2. Let (, 1, ) be a soft topological space over . If every soft pre-open set of is closed, then is soft preregular space. Proof Let every soft pre-open set in is closed, and let (^, ) be a soft pre-closed set in and ( ∈ such that ( ∈ (^, )& . Then (^, ) and (^, )& are soft pre-open sets, which containing (^, ) and (, respectively. Viz ( ∈ (^, )& , (^, ) = -, then is a soft pre-regular space. (^, ) and (^, )& ∩ (^, ) ⊂ Example 7. Since every soft pre-open set in discrete topology on is closed, is soft pre-regular. Lemma 2. Let (, 1, ) be a soft topological space over , and let (, ) be a soft pre-closed set in and ( ∈ such that ( ∉ (, ). If (, 1, ) is a soft pre-regular space, then there exist soft pre-open sets (, ) such that ( ∈ (, ) and (, ) = -. (, ) ∩ Theorem 12. Let (, 1, ) be a soft topological space over and ( ∈ . If is a soft pre-regular space, then: (, ) = -. (1) For a soft pre-closed set (, ), ( ∉ (, ) iff ((, ) ∩ (, ) = -. (2) For a soft pre-open set (, ), ( ∉ (, ) iff ((, ) ∩ (, ) = Proof (1) Let ( ∉ (, ). By Lemma 2, there exists a soft pre-open set (, ) such that ( ∈ (, ) and (, ) ∩ (, ) = -. (, ), we have ((, ) ∩ -. Since ((, ) ⊂ The converse is obtained by Lemma 1(2). (2) Let ( ∉ (, ). Then there are two cases: (i) ( ∉ (') for all α∈E and (ii) ( ∉ (') and ( ∈ (_)for some ', _ ∈ . (, ) = -. In the other case, ( ∈ & (') and ( ∉ & (_) for some ', _ ∈ and so In case (i) it is obvious that ((, ) ∩ & (, )& = -. So ((, ) ⊂ (, ) but this contradicts (, ) is a soft pre-closed set such that ( ∉ (, )& , by (1), ((, ) ∩ (, ) = -. ( ∉ (') for some ' ∈ . Consequently, we have ((, ) ∩ The converse is obvious. Theorem 13. Let (, 1, ) be a soft topological space over and ( ∈ . Then the following are equivalent: (1) (, 1, ) is a soft pre-regular space. 1082 GU J Sci, 27(4):1077-1083 (2014)/Metin AKDAĞ, Alkan ÖZKAN (, ) = -, there exist soft pre-open sets (₁, ) and (₂, ) (2) For each soft pre-closed set (, ) such that ((, ) ∩ (₂, ) = -. (₁, ), (, ) ⊂ (₂, ) and (₁, ) ∩ such that ((, ) ⊂ Proof The proof is obvious, obtained by Theorem 12(1) and Lemma 1(1). Theorem 14. Let (, 1, ) be a soft topological space over and ( ∈ . If is a soft pre-regular space, then: (1) For a soft pre-open set (, ), ( ∈ (, ) iff ( ∈ (') for some ' ∈ . {((, ): ( ∈ (')for some' ∈ }. (2) For a soft pre-open set (, ), (, ) =∪ (3) For each ', _ ∈ , [email protected] = 1` . (, ) = -. By the Proof (1) Assume that for some ' ∈ , ( ∈ (') and ( ∉ (, ). Then by Theorem 12(2), ((, ) ∩ assumption, this is a contradiction and so ( ∈ (, ). The converse is obvious. (, ). (2) It follows from (1) and ( ∈ (, ) iff ((, ) ⊂ (3) It is obtained from (2). Theorem 15. Let (, 1, ) be a soft topological space over . If (, 1, ) is a soft pre-regular space, then the following are equivalent: (1) (, 1, ) is a soft ₁-space. (, ) and (2) For (, Y ∈ with ( ≠ Y, there exist soft pre-open sets (, ) and (, ) such that ((, ) ⊂ (, ) = -, and (Y, ) ⊂ (, ) = -. (, ) and ((, ) ∩ (Y, ) ∩ (, ) = -. Hence we (, ), and by Theorem 12, ( ∉ (, ) iff ((, ) ∩ Proof It is obvious that ( ∈ (, ) iff ((, ) ⊂ have that the above statements are equivalent. Definition 23. Let (, 1, ) be a soft topological space over . Then (, 1, ) is said to be a soft ₃-space if it is a soft pre-regular and soft ₁-space. Theorem 16. Let (, 1, ) be a soft topological space over . If (, 1, ) is a soft ₃-space, then for each ( ∈ , ((, ) is soft pre-closed. Proof We show that ((, )& is soft pre-open. For each Y ∈ − {(}, since (, 1, ) is a soft pre-regular and P₁-space, by (a , ) ve ((, ) ∩b(a , ) =Φ. Theorem 15, there exists a soft pre-open set (a , )such that (Y, ) ⊂ a∈cd{e} (a , ) ⊂ a∈cd{e} (a , ) = (, ) where (') =∪a∈cd{e} . a (') for ((, )& . For the other inclusion, let ∪ So ∪ all ' ∈ . Moreover, from the definition of the relative complement and Definition 10, we know that ((, )& = (( & , ), where ( & (') = − {(} for each ' ∈ . Now, for each Y ∈ − {(} and for each ' ∈ , ( & (') = − {(} = a∈cd{e} (a , ), ∪ ∪a∈cd{e} {Y} =∪a∈cd{e} Y(') ⊂∪a∈cd{e} (') = ('). By Definition 2, this implies that ((, )& ⊂ a∈cd{e} (a , ). Since (, ) is soft pre-open for each ∈ − {(} ∈ − {(} , consequently, ((, ) is and so ((, )& =∪ soft pre-closed. Theorem 17. A soft ₃-space is soft ₂. Proof Let (, 1, ) be any soft ₃-space. For (, Y ∈ with ( ≠ Y, by the above Theorem 16, (Y, ) is soft pre-closed and ( ∉ (Y, ). From the soft regularity, there exist soft pre-open sets (₁, ) and (₂, ) such that ( ∈ (₁, ), Y ∈ (₂, ) = -. Thus (, 1, ) is soft ₂. (₂, ) and (₁, ) ∩ (Y, ) ⊂ Definition 24. A soft topological space (, 1, ) is said to be a soft pre-normal space if for every pair of disjoint pre (₁, ) closed soft sets (, ) and (, ), there exist two disjoint soft pre-open sets (₁, ) and (₂, ) such that (, ) ⊂ (₂, ). and (, ) ⊂ Remark 5. We give an example to show that every soft pre-normal space does not have to be both soft pre-regular and soft ₁-space. Example 8. Let = {(₁, (₂, (₃}, = {₁, ₂} and 1 = {-, , (₁, ), (₂, ), (₃, )} where (₁, ) = {(₁, {(₁}), (₂, {(₁})}, (₂, ) = {(₁, {(₂}), (₂, {(₂})}, (₃, ) = {(₁, {(₁, (₂}), (₂, {(₁, (₂})}. Then 1 defines a soft topology on and therefore (, 1, ) is a soft topological space over . Also (, 1, ) soft prenormal space over , but neither soft pre-regular nor soft ₁-space. Theorem 18. A soft topological space (, 1, ) is soft pre-normal iff for any soft pre-closed set (, ) and soft pre-open (, ) and 9=34((, )) ⊂ (, ). set (, ) containing (, ), there exists a soft pre-openset (, ) such that (, ) ⊂ Proof Let (, 1, ) be soft pre-normal space and (, ) be a soft pre-closed set and (, ) be a soft pre-open set containing (, ) ⇒ (, ) and (, )& are disjoint soft pre-closed sets ⇒ ∃ two disjoint soft pre-open sets (, ), (, ) (, ) and (, )& ⊂ (, ). such that (, ) ⊂ (, )& ⇒ 9=34((, )) ⊂ 9=34((, )& ) = (, )& . Now (, ) ⊂ & & (, ). Also, (, ) ⊂ (, ) ⇒ (, ) ⊂ (, ) ⇒ 9=34((, )) ⊂ (^, )& , then by hypothesis there Conversely, let (h, ) and (^, ) be any disjoint pair soft pre-closed sets ⇒ (h, ) ⊂ (, ) and exists a soft pre-open set (, ) such that (h, ) ⊂ (9=34((, )))& ⇒ (, ) and (9=34((, )))& are disjoint soft pre-open sets such (^, )& ⇒ ((^, )) ⊂ 9=34((, )) ⊂ (, ) and (^, ) ⊂ (9=34((, )))& . that (h, ) ⊂ Definition 25. A soft pre-normal ₁-space is called a soft ₄-space. Example 9. Let = {(₁, (₂}, = {₁, ₂} and 1 = {-, , (₁, ), (₂, )} where (₁, ) = {(₁, {(₁}), (₂, {(₁})}, (₂, ) = {(₁, {(₂}), (₂, {(₂})}. GU J Sci, 27(4):1077-1083 (2014)/Metin AKDAĞ, Alkan ÖZKAN 1083 Then 1 defines a soft topology on and hence (, 1, ) is a soft topological space over . Moreover (, 1, ) is a soft ₄space over . Theorem 19. Every soft ₄-space is soft ₃. Proof Let be a soft ₄-space. Since is also the soft ₁-space, only soft pre-regular is enough to show that space. Let {(} = (^, ) be a soft pre-closed set in and ( ∉ (^, ). Since is soft ₁-space, {(} is soft pre-closed. Then (^, ) ∩ (₁, ) and -, and since is soft pre-regular, there exist soft pre-open sets (₁, ) and (₂, ) in such that {(} ⊂ (₂, ) = -. Thus X is soft pre-regular, and so soft ₃-space. (₂, ) and (₁, ) ∩ (^, ) ⊂ Remark 6. Every soft ₄-space is soft ₃-space, every soft ₃-space is soft ₂-space, every soft ₂-space is soft ₁-space and every soft ₁-space is soft₀-space. 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