aa r X i v : . [ m a t h . G M ] J u l NEUTROSOPHIC SOFT FILTERS
NAIME DEMIRTAS¸
Abstract.
In this paper, the concept of neutrosophic soft filter and itsbasic properties are introduced. Later, we set up a neutrosophic soft topol-ogy with the help of a neutrosophic soft filter. We also give the notionsof the greatest lower bound and the least upper bound of the family ofneutrosophic soft filters, neutrosophic soft filter subbase and neutrosophicsoft filter base and explore some basic properties of them. Introduction
We can not solve the problems by using mathematical tools generally in thesocial life since in mathematics, the concepts are precise and not subjective. Todeal with this problem, researchers proposed several methods such as fuzzy settheory [12], rough set theory [7] and soft set theory [6]. Theories of fuzzy setsand rough sets can be considered as tools for dealing with vagueness but bothof these theories have their own difficulties. The reason for these difficulties is,possibly, the inadequacy of the parametrization tool of the theory as mentionedby Molodtsov [6] in 1999. Molodtsov initiated a novel concept of soft settheory which is a completely new approach for modeling uncertainities andsuccesfully applied it into several directions such as smoothness of functions,game theory, Riemann Integration, theory of measurement , and so on. Thefundamental concepts of neutrosophic set were introduced by Smarandache[10]. This theory is a generalization of classical sets, fuzzy set theory [12],intuitionistic fuzzy set theory [1], etc. Later some researchers [8, 9] studiedbasic concepts and properties of neutrosophic sets.The notion of neutrosophicsoft sets was first defined by Maji [5] and later, Deli and Broumi [3] modifiedit. Bera [2] introduced the concept of neutrosophic soft topological spaces.Also, neutrosophic soft point concept and neutrosophic soft T i -spaces werepresented by G¨un¨uuz Aras et al. [4].The main purpose of this paper is to introduce neutrosophic soft filters.Later we study some basic properties of neutrosophic soft filters and set up aneutrosophic soft topology with the help of a neutrosophic soft filter. Somenew notions in neutrosophic soft filters such as the greatest lower bound and Key words and phrases.
Neutrosophic soft set, neutrosophic soft topological space, neu-trosophic soft filter. the least upper bound of the family of neutrosophic soft filters, neutrosophicsoft filter subbase and neutrosophic soft filter base were introduced. Also, wegive some basic porperties of these concepts.2.
Introduction
We can not solve the problems by using mathematical tools generally in thesocial life since in mathematics, the concepts are precise and not subjective. Todeal with this problem, researchers proposed several methods such as fuzzy settheory [12], rough set theory [7] and soft set theory [6]. Theories of fuzzy setsand rough sets can be considered as tools for dealing with vagueness but bothof these theories have their own difficulties. The reason for these difficulties is,possibly, the inadequacy of the parametrization tool of the theory as mentionedby Molodtsov [6] in 1999. Molodtsov initiated a novel concept of soft settheory which is a completely new approach for modeling uncertainities andsuccesfully applied it into several directions such as smoothness of functions,game theory, Riemann Integration, theory of measurement , and so on. Thefundamental concepts of neutrosophic set were introduced by Smarandache[10]. This theory is a generalization of classical sets, fuzzy set theory [12],intuitionistic fuzzy set theory [1], etc. Later some researchers [8, 9] studiedbasic concepts and properties of neutrosophic sets.The notion of neutrosophicsoft sets was first defined by Maji [5] and later, Deli and Broumi [3] modifiedit. Bera [2] introduced the concept of neutrosophic soft topological spaces.Also, neutrosophic soft point concept and neutrosophic soft T i -spaces werepresented by G¨un¨uuz Aras et al. [4].The main purpose of this paper is to introduce neutrosophic soft filters.Later we study some basic properties of neutrosophic soft filters and set up aneutrosophic soft topology with the help of a neutrosophic soft filter. Somenew notions in neutrosophic soft filters such as the greatest lower bound andthe least upper bound of the family of neutrosophic soft filters, neutrosophicsoft filter subbase and neutrosophic soft filter base were introduced. Also, wegive some basic porperties of these concepts.3. Preliminaries
In this section, we present the basic definitions and results of neutrosophicsoft sets and neutrosophic soft topological spaces that we require in the nextsections.
Definition 1. [3] Let X be an initial universe set and E be a set of parameters.Let P ( X ) denote the set of all neutrosophic sets of X . Then a neutrosophicsoft set ( ∼ F , E ) over X is a set defined by a set value function ∼ F representing EUTROSOPHIC SOFT FILTERS 3 a mapping ∼ F : E → P ( X ), where ∼ F is called the approximate function of theneutrosophic soft set ( ∼ F , E ). In other words, the neutrosophic soft set is aparameterized family of some elements of the set P ( X ) and therefore it canbe written as a set of ordered pairs.( ∼ F , E ) = (cid:26)(cid:18) e, (cid:28) x, T ∼ F ( e ) ( x ) , I ∼ F ( e ) ( x ) , F ∼ F ( e ) ( x ) (cid:29) : x ∈ X (cid:19) : e ∈ E (cid:27) , where T ∼ F ( e ) ( x ) , I ∼ F ( e ) ( x ) , F ∼ F ( e ) ( x ) ∈ [0 ,
1] are respectively called the truth-membership, indeterminacy-membership and falsity-membership function of ∼ F ( e ). Since the supremum of each T, I, F is 1, the inequality 0 ≤ T ∼ F ( e ) ( x ) + I ∼ F ( e ) ( x ) + F ∼ F ( e ) ( x ) ≤ Definition 2. [2] Let ( ∼ F , E ) be a neutrosophic soft set the universe set X .The complementof ( ∼ F , E ) is denoted by ( ∼ F , E ) c and is defined by:( ∼ F , E ) c = (cid:26)(cid:18) e, (cid:28) x, F ∼ F ( e ) ( x ) , − I ∼ F ( e ) ( x ) , T ∼ F ( e ) ( x ) (cid:29) : x ∈ X (cid:19) : e ∈ E (cid:27) . It is obvious that (cid:18) ( ∼ F , E ) c (cid:19) c = ( ∼ F , E ). Definition 3. [5] Let ( ∼ F , E ) and ( ∼ G, E ) be two neutrosophic soft sets overthe universe set X . ( ∼ F , E ) is said to be a neutrosophic soft subset of ( ∼ G, E )if T ∼ F ( e ) ( x ) ≤ T ∼ G ( e ) ( x ), I ∼ F ( e ) ( x ) ≤ I ∼ G ( e ) ( x ), F ∼ F ( e ) ( x ) ≥ F ∼ G ( e ) ( x ), ∀ e ∈ E , ∀ x ∈ X . It is denoted by ( ∼ F , E ) ⊆ ( ∼ G, E ). ( ∼ F , E ) is said to be neutrosophicsoft equal to ( ∼ G, E ) if ( ∼ F , E ) is a neutrosophic soft subset of ( ∼ G, E ) and ( ∼ G, E )is a neutrosophic soft subset of ( ∼ F , E ). It is denoted by ( ∼ F , E ) = ( ∼ G, E ). Definition 4. [4] Let ( ∼ F , E ) and ( ∼ F , E ) be two neutrosophic soft sets overthe universe set X . Then their union is denoted by ( ∼ F , E ) ∪ ( ∼ F , E ) = ( ∼ F , E )and is defined by:( ∼ F , E ) = (cid:26)(cid:18) e, (cid:28) x, T ∼ F ( e ) ( x ) , I ∼ F ( e ) ( x ) , F ∼ F ( e ) ( x ) (cid:29) : x ∈ X (cid:19) : e ∈ E (cid:27) , where NAIME DEMIRTAS¸ T ∼ F ( e ) ( x ) = max (cid:26) T ∼ F ( e ) ( x ) , T ∼ F ( e ) ( x ) (cid:27) ,I ∼ F ( e ) ( x ) = max (cid:26) I ∼ F ( e ) ( x ) , I ∼ F ( e ) ( x ) (cid:27) ,F ∼ F ( e ) ( x ) = min (cid:26) F ∼ F ( e ) ( x ) , F ∼ F ( e ) ( x ) (cid:27) . Definition 5. [4] Let ( ∼ F , E ) and ( ∼ F , E ) be two neutrosophic soft sets overthe universe set X . Then their intersection is denoted by ( ∼ F , E ) ∩ ( ∼ F , E ) =( ∼ F , E ) and is defined by:( ∼ F , E ) = (cid:26)(cid:18) e, (cid:28) x, T ∼ F ( e ) ( x ) , I ∼ F ( e ) ( x ) , F ∼ F ( e ) ( x ) (cid:29) : x ∈ X (cid:19) : e ∈ E (cid:27) , where T ∼ F ( e ) ( x ) = min (cid:26) T ∼ F ( e ) ( x ) , T ∼ F ( e ) ( x ) (cid:27) ,I ∼ F ( e ) ( x ) = min (cid:26) I ∼ F ( e ) ( x ) , I ∼ F ( e ) ( x ) (cid:27) ,F ∼ F ( e ) ( x ) = max (cid:26) F ∼ F ( e ) ( x ) , F ∼ F ( e ) ( x ) (cid:27) . Definition 6. [4] A neutrosophic soft set ( ∼ F , E ) over the universe set X is saidto be a null neutrosophic soft set if T ∼ F ( e ) ( x ) = 0, I ∼ F ( e ) ( x ) = 0, F ∼ F ( e ) ( x ) = 1; ∀ e ∈ E , ∀ x ∈ X . It is denoted by 0 ( X,E ) . Definition 7. [4] A neutrosophic soft set ( ∼ F , E ) over the universe set X issaid to be an absolute neutrosophic soft set if T ∼ F ( e ) ( x ) = 1, I ∼ F ( e ) ( x ) = 1, F ∼ F ( e ) ( x ) = 0; ∀ e ∈ E , ∀ x ∈ X . It is denoted by 1 ( X,E ) .Clearly, 0 c ( X,E ) = 1 ( X,E ) and 1 c ( X,E ) = 0 ( X,E ) . Definition 8. [4] Let
N SS ( X, E ) be the family of all neutrosophic soft setsover the universe set X and τ ⊆ N SS ( X, E ). Then τ is said to be a neutro-sophic soft topology on X if:1 . ( X,E ) and 1 ( X,E ) belong to τ ,2 . the union of any number of neutrosophic soft sets in τ belongs to τ ,3 . the intersection of a finite number of neutrosophic soft sets in τ belongsto τ . EUTROSOPHIC SOFT FILTERS 5
Then (
X, τ, E ) is said to be a neutrosophic soft topological space over X .Each member of τ is said to be a neutrosophic soft open set. A neutrosophicsoft set ( ∼ F , E ) is called a neutrosophic soft closed set iff its complement ( ∼ F , E ) c is a neutrosophic soft open set. Definition 9. [4] Let
N SS ( X, E ) be the family of all neutrosophic soft setsover the universe set X . Then neutrosophic soft set x e ( α,β,γ ) is called a neu-trosophic soft point, for every x ∈ X , 0 < α, β, γ ≤ e ∈ E and is defined asfollows: x e ( α,β,γ ) ( e ′ )( y ) = (cid:26) ( α, β, γ ) if e ′ = e and y = x, (0 , , if e ′ = e or y = x. Definition 10. [4] Let ( ∼ F , E ) be a neutrosophic soft set over the universe set X . We say that x e ( α,β,γ ) ∈ ( ∼ F , E ) read as belonging to the neutrosophic softset ( ∼ F , E ) whenever α ≤ T ∼ F ( e ) ( x ), β ≤ I ∼ F ( e ) ( x ) and F ∼ F ( e ) ( x ) ≤ γ . Definition 11. [4] Let (
X, τ, E ) be a neutrosophic soft topological space over X . A neutrosophic soft set ( ∼ F , E ) in (
X, τ, E ) is called a neutrosophic softneighborhood of the neutrosophic soft point x e ( α,β,γ ) ∈ ( ∼ F , E ), if there exists aneutrosophic soft open set ( ∼ G, E ) such that x e ( α,β,γ ) ∈ ( ∼ G, E ) ⊆ ( ∼ F , E ). Theorem 1. [4] Let ( X, τ, E ) be a neutrosophic soft topological space and ( ∼ F , E ) be a neutrosophic soft set over X . Then ( ∼ F , E ) is a neutrosophicsoft open set if and only if ( ∼ F , E ) is a neutrosophic soft neighborhood of itsneutrosophic soft points. The neighborhood system of a neutrosophic soft point x e ( α,β,γ ) , denoted by U ( x e ( α,β,γ ) , E ), is the family of all its neighborhoods. Theorem 2. [4] The neighborhood system U ( x e ( α,β,γ ) , E ) at x e ( α,β,γ ) in a neu-trosophic soft topological space ( X, τ, E ) has the following properties:
1) If ( ∼ F , E ) ∈ U ( x e ( α,β,γ ) , E ), then x e ( α,β,γ ) ∈ ( ∼ F , E ),2) If ( ∼ F , E ) ∈ U ( x e ( α,β,γ ) , E ) and ( ∼ F , E ) ⊆ ( ∼ H, E ) then ( ∼ H , E ) ∈ U ( x e ( α,β,γ ) , E ),3) If ( ∼ F , E ), ( ∼ G, E ) ∈ U ( x e ( α,β,γ ) , E ) then ( ∼ F , E ) ∩ ( ∼ G, E ) ∈ U ( x e ( α,β,γ ) , E ),4) If ( ∼ F , E ) ∈ U ( x e ( α,β,γ ) , E ) then there exists a ( ∼ G, E ) ∈ U ( x e ( α,β,γ ) , E ) suchthat ( ∼ G, E ) ∈ U ( y e ′ ( α ′ ,β ′ ,γ ′ ) , E ) for each y e ′ ( α ′ ,β ′ ,γ ′ ) ∈ ( ∼ G, E ). NAIME DEMIRTAS¸
Definition 12.
Let (
X, τ, E ) be a neutrosophic soft topological space andG¸ ( x e ( α,β,γ ) , E ) be a family of some neutrosophic soft neighborhoods of neutro-sophic soft point x e ( α,β,γ ) . If, for each neutrosophic soft neighborhood ( ∼ G, E )of x e ( α,β,γ ) , there exists a ( ∼ H, E ) ∈ G¸ ( x e ( α,β,γ ) , E ) such that x e ( α,β,γ ) ∈ ( ∼ H, E ) ⊆ ( ∼ G, E ), then we say that G¸ ( x e ( α,β,γ ) , E ) is a neutrosophic soft neighborhoodbase at x e ( α,β,γ ) . Theorem 3.
If for each neutrosophic soft point x e ( α,β,γ ) there correspondsa family U ( x e ( α,β,γ ) , E ) such that the properties - in Theorem 13 aresatisfied, then there is a unique τ neutrosophic soft topological structure over X such that for each x e ( α,β,γ ) , U ( x e ( α,β,γ ) , E ) is the family of τ -neutrosophicsoft neighborhoods of x e ( α,β,γ ) .Proof. Let τ = (cid:26) ( ∼ G, E ) ∈ N SS ( X, E ) : x e ( α,β,γ ) ∈ ( ∼ G, E ) = ⇒ ( ∼ G, E ) ∈ U( x e ( α,β,γ ) , E ) (cid:27) .It is clear that, τ is a neutrosophic soft topology over X . The family τ certainly satisfies axioms 2 . and 3 . in Definition 8: for 3 . , this follows im-mediately from 2) in Theorem 13 and for 2 . , from 3) in Theorem 13. Theaxiom 1 . in Definition 8 is a result of 2) and 3) in Theorem 13. It remainsto show that, in the neutrosophic soft topology defined by τ , U ( x e ( α,β,γ ) , E )is the set of τ -neutrosophic soft neighborhoods of x e ( α,β,γ ) for each x e ( α,β,γ ) . Itfollows from 2) in Theorem 13 that every τ -neutrosophic soft neighborhoodof x e ( α,β,γ ) belongs to U ( x e ( α,β,γ ) , E ). Conversely, let ( ∼ G , E ) be a neutrosophicsoft set belonging to U ( x e ( α,β,γ ) , E ) and let ( ∼ G , E ) be the neutrosophic soft setof neutrosophic soft points y e ′ ( α ′ ,β ′ ,γ ′ ) such that ( ∼ G , E ) ∈ U ( y e ′ ( α ′ ,β ′ ,γ ′ ) , E ). Ifwe can show that x e ( α,β,γ ) ∈ ( ∼ G , E ), ( ∼ G , E ) ⊆ ( ∼ G , E ) and ( ∼ G , E ) ∈ τ ,then the proof will be complete. Since for every neutrosophic soft point y e ′ ( α ′ ,β ′ ,γ ′ ) ∈ ( ∼ G , E ) belongs to ( ∼ G , E ) by reason of 1) in Theorem 13 andthe hypothesis ( ∼ G , E ) ∈ U ( y e ′ ( α ′ ,β ′ ,γ ′ ) , E ), we obtain ( ∼ G , E ) ⊆ ( ∼ G , E ). Since( ∼ G , E ) ∈ U ( x e ( α,β,γ ) , E ) and ( ∼ G , E ) ⊆ ( ∼ G , E ), we have x e ( α,β,γ ) ∈ ( ∼ G , E ). Itremains to show that ( ∼ G , E ) ∈ τ , i.e. that ( ∼ G , E ) ∈ U ( y e ′ ( α ′ ,β ′ ,γ ′ ) , E ) for each y e ′ ( α ′ ,β ′ ,γ ′ ) ∈ ( ∼ G , E ). If y e ′ ( α ′ ,β ′ ,γ ′ ) ∈ ( ∼ G , E ) then by 4) in Theorem 13 thereis a neutrosophic soft set ( ∼ G , E ) such that for each z e ′′ ( α ′′ ,β ′′ ,γ ′′ ) ∈ ( ∼ G , E ) wehave ( ∼ G , E ) ∈ U ( z e ′′ ( α ′′ ,β ′′ ,γ ′′ ) , E ). Since ( ∼ G , E ) ∈ U ( z e ′′ ( α ′′ ,β ′′ ,γ ′′ ) , E ) means that EUTROSOPHIC SOFT FILTERS 7 z e ′′ ( α ′′ ,β ′′ ,γ ′′ ) ∈ ( ∼ G , E ), it follows that ( ∼ G , E ) ⊆ ( ∼ G , E ) and therefore, by 2) inTheorem 13, that ( ∼ G , E ) ∈ U ( y e ′ ( α ′ ,β ′ ,γ ′ ) , E ). (cid:3) Neutrosophic soft filters
Definition 13.
Let ℵ ⊆
N SS ( X, E ), then ℵ is called a neutrosophic soft filteron X if ℵ satisfies the following properties:( ℵ ) 0 ( X,E ) / ∈ ℵ ,( ℵ ) ∀ ( ∼ F , E ) , ( ∼ G, E ) ∈ ℵ = ⇒ ( ∼ F , E ) ∩ ( ∼ G, E ) ∈ ℵ ,( ℵ ) ∀ ( ∼ F , E ) ∈ ℵ and ( ∼ F , E ) ⊆ ( ∼ G, E ) = ⇒ ( ∼ G, E ) ∈ ℵ . Remark . It follows from ( ℵ ) and ( ℵ ) that every finite intersections ofneutrosophic soft sets of ℵ are not 0 ( X,E ) . Proposition 1.
The condition ( ℵ ) is equivalent to the following two condi-tions: ( ℵ a ) The intersection of two members of ℵ belongs to ℵ .( ℵ b ) 1 ( X,E ) belongs to ℵ . Example . The family ℵ = { ( X,E ) } is a neutrosophic soft filter over X . Theorem 4.
Let ( X,E ) = ( ∼ F , E ) ∈ N SS ( X, E ) . Then the family ℵ ( ∼ F ,E ) = (cid:26) ( ∼ G, E ) : ( ∼ F , E ) ⊆ ( ∼ G, E ) ∈ N SS ( X, E ) (cid:27) is a neutrosophic soft filter over X .Proof. Since 1 ( X,E ) ∈ ℵ and 0 ( X,E ) / ∈ ℵ , ∅ 6 = ℵ 6 = N SS ( X, E ). Suppose( ∼ H , E ) , ( ∼ H , E ) ∈ ℵ , then ( ∼ F , E ) ⊆ ( ∼ H , E ), ( ∼ F , E ) ⊆ ( ∼ H , E ). Thus T ∼ F ( e ) ( x ) ≤ min (cid:26) T ∼ H ( e ) ( x ) , T ∼ H ( e ) ( x ) (cid:27) , I ∼ F ( e ) ( x ) ≤ min (cid:26) I ∼ H ( e ) ( x ) , I ∼ H ( e ) ( x ) (cid:27) and F ∼ F ( e ) ( x ) ≤ max (cid:26) F ∼ H ( e ) ( x ) , F ∼ H ( e ) ( x ) (cid:27) for all x ∈ X . So ( ∼ F , E ) ⊆ ( ∼ H , E ) ∩ ( ∼ H , E ) and hence ( ∼ H , E ) ∩ ( ∼ H , E ) ∈ ℵ . (cid:3) Theorem 5.
Let ( X, τ, E ) be a neutrosophic soft topological space over X .The neighborhood system U ( x e ( α,β,γ ) , E ) is a neutrosophic soft filter for everyneutrosophic soft point x e ( α,β,γ ) . Also, it is called neutrosophic soft neighbor-hoods filter of the neutrosophic soft point x e ( α,β,γ ) .Proof. ( ℵ ) By 1) in Theorem 13, since x e ( α,β,γ ) ∈ ( ∼ G, E ), we obtain 0 ( X,E ) / ∈ U ( x e ( α,β,γ ) , E ). NAIME DEMIRTAS¸ ( ℵ ) This is clearly seen by 3) in Theorem 13.( ℵ ) This is clearly seen by 2) in Theorem 13. (cid:3) Now, we set up a neutrosophic soft topology with the help of a neutrosophicsoft filter.
Theorem 6.
If, for every x e ( α,β,γ ) , there exists a neutrosophic soft filter ℵ ( x e ( α,β,γ ) ) = U ( x e ( α,β,γ ) , E ) which satisfies the following two properties, then there exists aunique neutrosophic soft topology τ such that ℵ ( x e ( α,β,γ ) ) consists of the τ -neutrosophic soft neighborhoods of the neutrosophic soft point x e ( α,β,γ ) . (1) Every neutrosophic soft set in the neutrosophic soft filter ℵ ( x e ( α,β,γ ) )contains the neutrosophic soft point x e ( α,β,γ ) ,(2) For every ( ∼ G, E ) ∈ ℵ ( x e ( α,β,γ ) ) there exists a ( ∼ H, E ) ∈ ℵ ( x e ( α,β,γ ) ) suchthat for every y e ′ ( α ′ ,β ′ ,γ ′ ) ∈ ( ∼ H , E ), ( ∼ G, E ) ∈ ℵ ( y e ′ ( α ′ ,β ′ ,γ ′ ) ). Proof.
Since the axioms ( ℵ ), ( ℵ ), ( ℵ ), (1) and (2) are equivalent to theneighborhood axioms 1) − τ such that ℵ ( x e ( α,β,γ ) ) consists of the τ -neutrosophic soft neigh-borhoods of the neutrosophic soft point x e ( α,β,γ ) . (cid:3) Example . Let (
X, τ, E ) be a neutrosophic soft topological space and x e ( α,β,γ ) be a neutrosophic soft point over X . Since ( ∼ G, E ) cannot be an element ofG¸ ( x e ( α,β,γ ) , E ) for every ( ∼ H , E ) ∈ G¸ ( x e ( α,β,γ ) , E ) and ( ∼ H , E ) ⊆ ( ∼ G, E ), then theneutrosophic soft neighborhood base G¸ ( x e ( α,β,γ ) , E ) is not a neutrosophic softfilter over X .5. Comparison of neutrosophic soft filters
Definition 14.
Let ℵ and ℵ be neutrosophic soft filters over X . If ℵ ⊆ ℵ ,then ℵ is said to be finer that ℵ or ℵ coarser than ℵ .If also ℵ = ℵ , then ℵ is strictly finer than ℵ or ℵ is strictly coarserthan ℵ . If either ℵ ⊆ ℵ or ℵ ⊆ ℵ , then ℵ is comparable with ℵ . Theorem 7.
Let ( ℵ i ) i ∈ I be a family of neutrosophic soft filters over X . Then ℵ = ∩ i ∈ I ℵ i is a neutrosophic soft filter over X . In fact ℵ is the greatest lower bound of the family ( ℵ i ) i ∈ I . Proof. ( ℵ ) Since 0 ( X,E ) / ∈ ℵ i for each i ∈ I , then 0 ( X,E ) does not belong to ℵ = ∩ i ∈ I ℵ i . EUTROSOPHIC SOFT FILTERS 9 ( ℵ ) Let ( ∼ F , E ) , ( ∼ G, E ) ∈ ℵ = ∩ i ∈ I ℵ i . Then ( ∼ F , E ) , ( ∼ G, E ) ∈ ℵ i for each i ∈ I . Since ( ∼ F , E ) ∩ ( ∼ G, E ) ∈ ℵ i for each i ∈ I , so we obtain ( ∼ F , E ) ∩ ( ∼ G, E ) ∈ℵ = ∩ i ∈ I ℵ i .( ℵ ) Let ( ∼ F , E ) ∈ ℵ = ∩ i ∈ I ℵ i and ( ∼ F , E ) ⊆ ( ∼ G, E ). Since ( ∼ F , E ) ∈ ℵ i foreach i ∈ I and ( ∼ F , E ) ⊆ ( ∼ G, E ), we get ( ∼ G, E ) ∈ ℵ i for each i ∈ I . Hence( ∼ G, E ) ∈ ℵ = ∩ i ∈ I ℵ i . (cid:3) Now, we investigate the least upper bound of the family of neutrosophicsoft filters over X . Theorem 8.
Let S ⊆ N SS ( X, E ) . Then there exists a neutrosophic soft filter ℵ which contains the family S , if S has the following property: ”The all finiteintersections of neutrosophic soft sets of S are not ( X,E ) ”.Proof. Let S = (cid:26) ( ∼ F i , E ) : ∀ i ∈ J ( J is f inite ) , ∩ i ∈ J ( ∼ F i , E ) = 0 ( X,E ) (cid:27) . Thenwe give the family which consists of finite intersections of elements of S ; β = (cid:26) ( ∼ G, E ) : ∀ i ∈ J ( J is f inite ) , ( ∼ F i , E ) ∈ S and ( ∼ G, E ) = ∩ i ∈ J ( ∼ F i , E ) (cid:27) .Then the family ℵ ( S ) = (cid:26) ( ∼ H, E ) : ( ∼ G, E ) ∈ β and ( ∼ G, E ) ⊆ ( ∼ H, E ) (cid:27) is aneutrosophic soft filter over X .( ℵ ) 0 ( X,E ) ∈ β , for every ( ∼ H, E ) ∈ ℵ ( S ), ( ∼ H , E ) = 0 ( X,E ) and so 0 ( X,E ) / ∈ℵ ( S ).( ℵ ) Let ( ∼ H , E ) , ( ∼ H , E ) ∈ ℵ ( S ). There exist neutrosophic soft sets( ∼ G , E ) , ( ∼ G , E ) ∈ β such that ( ∼ G , E ) ⊆ ( ∼ H , E ) and ( ∼ G , E ) ⊆ ( ∼ H , E ).From the definition of β , 0 ( X,E ) = ( ∼ G , E ) ∩ ( ∼ G , E ) ∈ β . Since ( ∼ G , E ) ∩ ( ∼ G , E ) ⊆ ( ∼ H , E ) ∩ ( ∼ H , E ), we obtain ( ∼ H , E ) ∩ ( ∼ H , E ) ∈ ℵ ( S ).( ℵ ) Let ( ∼ H , E ) ∈ ℵ ( S ) and ( ∼ H , E ) ⊆ ( ∼ H , E ). Then there exists a neu-trosophic soft set ( ∼ G, E ) ∈ β such that ( ∼ G, E ) ⊆ ( ∼ H , E ). Since ( ∼ H , E ) ⊆ ( ∼ H , E ), we obtain ( ∼ H , E ) ∈ ℵ ( S ). (cid:3) Remark . The neutrosophic soft filter ℵ ( S ) in Theorem 26 is said to begenerated by S and S is said to be neutrosophic soft filter subbase of ℵ ( S ). Itis clear that S ⊆ ℵ ( S ). Theorem 9.
The neutrosophic soft filter ℵ ( S ) which is generated by S is thecoarsest neutrosophic soft filter which contains S .Proof. Suppose that S ⊆ ℵ . By Theorem 26, S ⊆ β ⊆ ℵ . By Remark 27,for every ( ∼ H , E ) ∈ ℵ ( S ) there exists a ( ∼ G, E ) ∈ β such that ( ∼ G, E ) ⊆ ( ∼ H, E ).Since β ⊆ ℵ , then ( ∼ G, E ) ∈ ℵ . Since ℵ is a neutrosophic soft filter, ( ∼ H, E ) ∈ℵ by ( ℵ ) in Definition 16. Hence we obtain ℵ ( S ) ⊆ ℵ . (cid:3) Theorem 10.
The family ( ℵ i ) i ∈ I of neutrosophic soft filters over X has aleast upper bound if and only if for all finite subfamilies ( ℵ i ) ≤ i ≤ n of ( ℵ i ) i ∈ I and all ( ∼ G i , E ) ∈ ℵ i (1 ≤ i ≤ n ) , ( ∼ G , E ) ∩ ... ∩ ( ∼ G n , E ) = 0 ( X,E ) .Proof. = ⇒ : If there exists a least upper bound of the family ( ℵ i ) i ∈ I , by ( ℵ )and ( ℵ ) in Definition 16, for all finite subfamilies ( ℵ i ) ≤ i ≤ n of ( ℵ i ) i ∈ I and all( ∼ G i , E ) ∈ ℵ i (1 ≤ i ≤ n ), the intersection ( ∼ G , E ) ∩ ... ∩ ( ∼ G n , E ) = 0 ( X,E ) . ⇐ =: Let ( ∼ G , E ) ∩ ... ∩ ( ∼ G n , E ) = 0 ( X,E ) for all finite subfamilies ( ℵ i ) ≤ i ≤ n of ( ℵ i ) i ∈ I and all ( ∼ G i , E ) ∈ ℵ i (1 ≤ i ≤ n ). Then the neutrosophic soft filter ℵ ( S ) generated by S = ∪ i ∈ I ℵ i = (cid:26) ( ∼ F , E ) : ( ∃ i ∈ I ) ( ∼ F , E ) ∈ ℵ i (cid:27) is the least upper bound of the family ( ℵ i ) i ∈ I by Theorem 28. (cid:3) Definition 15.
Let β ⊆ N SS ( X, E ), then β is said to be a neutrosophic softfilter base on X if( β ) β = ∅ and 0 ( X,E ) / ∈ β .( β ) The intersection of two members of β contain a member of β . Remark . β which is in Theorem 26 is a neutrosophic soft filter base. Remark . It is clear that, every neutrosophic soft filter is a neutrosophic softfilter base.
Example . Let (
X, τ, E ) be a soft topological space and x e ( α,β,γ ) be a neutro-sophic soft point over X . The neutrosophic soft neighborhood base G¸ ( x e ( α,β,γ ) , E )is a neutrosophic soft filter base over X .( β ) Clearly, G¸ ( x e ( α,β,γ ) , E ) = ∅ . For every ( ∼ H, E ) ∈ G¸ ( x e ( α,β,γ ) , E ), x e ( α,β,γ ) ∈ ( ∼ H , E ). Then ( ∼ H, E ) = 0 ( X,E ) . Hence we obtain 0 ( X,E ) / ∈ G¸ ( x e ( α,β,γ ) , E ).( β ) Let ( ∼ G, E ) , ( ∼ H , E ) ∈ G¸ ( x e ( α,β,γ ) , E ). Since ( ∼ G, E ) , ( ∼ H , E ) ∈ U ( x e ( α,β,γ ) , E ),we get ( ∼ G, E ) ∩ ( ∼ H, E ) ∈ U ( x e ( α,β,γ ) , E ). By Definition 14, there exists a EUTROSOPHIC SOFT FILTERS 11 ( ∼ K, E ) ∈ G¸ ( x e ( α,β,γ ) , E ) such that ( ∼ K, E ) ⊆ ( ∼ G, E ) ∩ ( ∼ H, E ). Hence we getG¸ ( x e ( α,β,γ ) , E ) is a neutrosophic soft filter base of neutrosophic soft neighbor-hoods filter U ( x e ( α,β,γ ) , E ) by Definition 30. Theorem 11.
Let ℵ be a neutrosophic soft filter over X and β ⊆ ℵ . Then β is a base of ℵ if and only if every member of ℵ contains a member of β .Proof. It is obvious from Theorem 26. (cid:3)
Definition 16.
Two neutrosophic soft filter bases β and β over X are equiv-alent if and only if every member of β contains a member of β and everymember of β contains a member of β . Remark . Two equivalent neutrosophic soft filter bases generate the sameneutrosophic soft filter.
Theorem 12.
Let ( X, τ, E ) be a soft topological space and x e ( α,β,γ ) be a neu-trosophic soft point over X . If G¸ ( x e ( α,β,γ ) , E ) and G¸ ( x e ( α,β,γ ) , E ) are differ-ent neutrosophic soft neighborhood bases of x e ( α,β,γ ) , then G¸ ( x e ( α,β,γ ) , E ) andG¸ ( x e ( α,β,γ ) , E ) are two equivalent neutrosophic soft filter bases.Proof. For each ( ∼ F , E ) ∈ G¸ ( x e ( α,β,γ ) , E ), by Example 33, ( ∼ F , E ) ∈ U ( x e ( α,β,γ ) , E ).Also, since G¸ ( x e ( α,β,γ ) , E ) ⊆ U ( x e ( α,β,γ ) , E ) there exists a ( ∼ F , E ) ∈ G¸ ( x e ( α,β,γ ) , E )such that ( ∼ F , E ) ⊆ ( ∼ F , E ). Similarly, for each ( ∼ F , E ) ∈ G¸ ( x e ( α,β,γ ) , E ), byExample 33, ( ∼ F , E ) ∈ U ( x e ( α,β,γ ) , E ). Since G¸ ( x e ( α,β,γ ) , E ) ⊆ U ( x e ( α,β,γ ) , E ),there exists a ( ∼ F , E ) ∈ G¸ ( x e ( α,β,γ ) , E ) such that ( ∼ F , E ) ⊆ ( ∼ F , E ). Hence weobtain G¸ ( x e ( α,β,γ ) , E ) and G¸ ( x e ( α,β,γ ) , E ) are equivalent by Definition 35. (cid:3) Theorem 13.
Let β , β be neutrosophic soft filter bases and ℵ , ℵ be neu-trosophic soft filters over X such that β ⊆ ℵ and β ⊆ ℵ . Then ℵ ⊆ ℵ ifand only if every member of β contains a member of β .Proof. = ⇒ : Let ℵ ⊆ ℵ and ( ∼ G , E ) ∈ β . Since β ⊆ ℵ ⊆ ℵ , then ( ∼ G , E ) ∈ℵ . Since β ⊆ ℵ , there exists a ( ∼ G , E ) ∈ β such that ( ∼ G , E ) ⊆ ( ∼ G , E ) byTheorem 34. ⇐ =: Let ( ∼ F , E ) ∈ ℵ . From Theorem 34, there exists a ( ∼ G , E ) suchthat ( ∼ G , E ) ⊆ ( ∼ F , E ). By hypothesis, there exists a ( ∼ G , E ) ∈ β suchthat ( ∼ G , E ) ⊆ ( ∼ G , E ). Then we obtain ( ∼ G , E ) ⊆ ( ∼ F , E ). Since β ⊆ ℵ ,( ∼ F , E ) ∈ ℵ by Definition 30. Hence we obtain ℵ ⊆ ℵ . (cid:3) Conclusion
In the present study, we have introduced neutrosophic soft filters which aredefined over an initial universe with a fixed set of parameters. We set up aneutrosophic soft topology with the help of a neutrosophic soft filter. Wefurther investigate some essential features and basic concepts of neutrosophicsoft filters. We expect that results in this paper will be helpfull for futurestudies in neutrosophic soft sets.
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