Neutrosophic Soft Filters

2020·0.12 MB·English

by  Naime Demirtas

NEUTROSOPHIC SOFT FILTERS
NAIMEDEMIRTAS¸
9
1
Abstract. In this paper, the concept of neutrosophic soft filter and its
0
basicpropertiesareintroduced. Later,wesetupaneutrosophicsofttopol-
2
ogy with the help of a neutrosophic soft filter. We also give the notions
l
u of the greatest lower bound and the least upper bound of the family of
J neutrosophicsoft filters, neutrosophicsoft filtersubbaseandneutrosophic
soft filter base and explore some basic properties of them.
5
2

]
M
1. Introduction
G
We can not solve the problems by using mathematical tools generally in the
.
h sociallifesinceinmathematics,theconceptsarepreciseandnotsubjective. To
t
a dealwiththisproblem, researchersproposedseveral methodssuchas fuzzyset
m
theory [12], rough set theory [7] and soft set theory [6]. Theories of fuzzy sets
[ and rough sets can be considered as tools for dealing with vagueness but both

of these theories have their own difficulties. Thereason for these difficulties is,
1
v possibly,theinadequacyoftheparametrizationtoolofthetheoryasmentioned
7 by Molodtsov [6] in 1999. Molodtsov initiated a novel concept of soft set
1
theory which is a completely new approach for modeling uncertainities and
8
2 succesfully applied it into several directions such as smoothness of functions,
0 game theory, Riemann Integration, theory of measurement , and so on. The
.
8 fundamental concepts of neutrosophic set were introduced by Smarandache
0 [10]. This theory is a generalization of classical sets, fuzzy set theory [12],
9
intuitionistic fuzzy set theory [1], etc. Later some researchers [8,9] studied
1
: basic concepts and properties of neutrosophic sets.The notion of neutrosophic
v
soft sets was first defined by Maji [5] and later, Deli and Broumi [3] modified
i
X
it. Bera [2] introduced the concept of neutrosophic soft topological spaces.
r Also, neutrosophic soft point concept and neutrosophic soft T -spaces were
a i
presented by Gu¨nu¨uz 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 a
neutrosophic soft topology with the help of a neutrosophic soft filter. Some
new 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.
1
2 NAIMEDEMIRTAS¸
the least upper bound of the family of neutrosophic soft filters, neutrosophic
soft filter subbase and neutrosophic soft filter base were introduced. Also, we
give some basic porperties of these concepts.
2. Introduction
We can not solve the problems by usingmathematical tools generally in the
sociallifesinceinmathematics,theconceptsarepreciseandnotsubjective. To
dealwiththisproblem,researchers proposedseveral methodssuchas fuzzyset
theory [12], rough set theory [7] and soft set theory [6]. Theories of fuzzy sets
and rough sets can be considered as tools for dealing with vagueness but both
of these theories have their own difficulties. Thereason for these difficulties is,
possibly,theinadequacyoftheparametrizationtoolofthetheoryasmentioned
by Molodtsov [6] in 1999. Molodtsov initiated a novel concept of soft set
theory which is a completely new approach for modeling uncertainities and
succesfully applied it into several directions such as smoothness of functions,
game theory, Riemann Integration, theory of measurement , and so on. The
fundamental 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] studied
basic concepts and properties of neutrosophic sets.The notion of neutrosophic
soft sets was first defined by Maji [5] and later, Deli and Broumi [3] modified
it. Bera [2] introduced the concept of neutrosophic soft topological spaces.
Also, neutrosophic soft point concept and neutrosophic soft T -spaces were
i
presented by Gu¨nu¨uz 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 a
neutrosophic soft topology with the help of a neutrosophic soft filter. Some
new notions in neutrosophic soft filters such as the greatest lower bound and
the least upper bound of the family of neutrosophic soft filters, neutrosophic
soft filter subbase and neutrosophic soft filter base were introduced. Also, we
give some basic porperties of these concepts.
3. Preliminaries
In this section, we present the basic definitions and results of neutrosophic
soft sets and neutrosophic soft topological spaces that we require in the next
sections.
Definition 1. [3]LetX beaninitialuniversesetandE beasetofparameters.
Let P (X) denote the set of all neutrosophic sets of X. Then a neutrosophic
∼ ∼
soft set (F,E) over X is a set defined by a set value function F representing
NEUTROSOPHIC SOFT FILTERS 3
∼ ∼
a mapping F :E → P (X), where F is called the approximate function of the
∼
neutrosophic soft set (F,E). In other words, the neutrosophic soft set is a
parameterized family of some elements of the set P(X) and therefore it can
be written as a set of ordered pairs.
∼
(F,E) = e, x,T∼ (x),I∼ (x),F∼ (x) : x ∈ X : e∈ E ,
(cid:26)(cid:18) (cid:28) F(e) F(e) F(e) (cid:29) (cid:19) (cid:27)
where T∼ (x), I∼ (x), F∼ (x) ∈ [0,1] are respectively called the truth-
F(e) F(e) F(e)
membership, indeterminacy-membership and falsity-membership function of
∼
F(e). Since the supremum of each T, I, F is 1, the inequality 0 ≤ T∼ (x)+
F(e)
I∼ (x)+F∼ (x) ≤ 3 is obvious.
F(e) F(e)
∼
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 = e, x,F∼ (x),1−I∼ (x),T∼ (x) :x ∈ X :e ∈ E .
(cid:26)(cid:18) (cid:28) F(e) F(e) F(e) (cid:29) (cid:19) (cid:27)
∼ c ∼
It is obvious that (F,E)c = (F,E).
(cid:18) (cid:19)
∼ ∼
Definition 3. [5] Let (F,E) and (G,E) be two neutrosophic soft sets over
∼ ∼
the universe set X. (F,E) is said to be a neutrosophic soft subset of (G,E)
if T∼ (x) ≤ T∼ (x), I∼ (x) ≤ I∼ (x), F∼ (x) ≥ F∼ (x), ∀e ∈ E,
F(e) G(e) F(e) G(e) F(e) G(e)
∼ ∼ ∼
∀x ∈ X. It is denoted by (F,E) ⊆ (G,E). (F,E) is said to be neutrosophic
∼ ∼ ∼ ∼
softequalto(G,E)if(F,E)isaneutrosophicsoftsubsetof (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 over
1 2
∼ ∼ ∼
theuniverse set X. Then their union is denoted by (F ,E)∪(F ,E) = (F ,E)
1 2 3
and is defined by:
∼
(F3,E) = e, x,T∼ (x),I∼ (x),F∼ (x) :x ∈ X :e ∈ E ,
(cid:26)(cid:18) (cid:28) F3(e) F3(e) F3(e) (cid:29) (cid:19) (cid:27)
where
4 NAIMEDEMIRTAS¸
T∼ (x) = max T∼ (x),T∼ (x) ,
F3(e) (cid:26) F1(e) F2(e) (cid:27)
I∼ (x) = max I∼ (x),I∼ (x) ,
F3(e) (cid:26) F1(e) F2(e) (cid:27)
F∼ (x) = min F∼ (x),F∼ (x) .
F3(e) (cid:26) F1(e) F2(e) (cid:27)
∼ ∼
Definition 5. [4] Let (F ,E) and (F ,E) be two neutrosophic soft sets over
1 2
∼ ∼
the universe set X. Then their intersection is denoted by (F ,E)∩(F ,E) =
1 2
∼
(F ,E) and is defined by:
3
∼
(F3,E) = e, x,T∼ (x),I∼ (x),F∼ (x) : x ∈ X :e ∈ E ,
(cid:26)(cid:18) (cid:28) F3(e) F3(e) F3(e) (cid:29) (cid:19) (cid:27)
where
T∼ (x) = min T∼ (x),T∼ (x) ,
F3(e) (cid:26) F1(e) F2(e) (cid:27)
I∼ (x) = min I∼ (x),I∼ (x) ,
F3(e) (cid:26) F1(e) F2(e) (cid:27)
F∼ (x) = max F∼ (x),F∼ (x) .
F3(e) (cid:26) F1(e) F2(e) (cid:27)
∼
Definition 6. [4]Aneutrosophicsoftset(F,E)overtheuniversesetX issaid
to be a null neutrosophic soft set if T∼ (x) = 0, I∼ (x) = 0, F∼ (x) = 1;
F(e) F(e) F(e)
∀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 is
said to be an absolute neutrosophic soft set if T∼ (x) = 1, I∼ (x) = 1,
F(e) F(e)
F∼ (x) = 0; ∀e∈ E, ∀x ∈X. It is denoted by 1(X,E).
F(e)
Clearly, 0c = 1 and 1c = 0 .
(X,E) (X,E) (X,E) (X,E)
Definition 8. [4] Let NSS(X,E) be the family of all neutrosophic soft sets
over the universe set X and τ ⊆ NSS(X,E). Then τ is said to be a neutro-
sophic soft topology on X if:
1. 0 and 1 belong to τ,
(X,E) (X,E)
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 belongs
to τ.
NEUTROSOPHIC 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 neutrosophic
∼ ∼
softset(F,E)iscalledaneutrosophicsoftclosedsetiffitscomplement(F,E)c
is a neutrosophic soft open set.
Definition 9. [4] Let NSS(X,E) be the family of all neutrosophic soft sets
over the universe set X. Then neutrosophic soft set xe is called a neu-
(α,β,γ)
trosophic soft point, for every x ∈ X, 0 < α,β,γ ≤ 1, e ∈ E and is defined as
follows:
(α,β,γ) if e′ = e and y = x,
xe (e′)(y) =
(α,β,γ) (cid:26) (0,0,1) if e′ 6= e or y 6= x.
∼
Definition 10. [4] Let (F,E) be a neutrosophic soft set over the universe set
∼
X. We say that xe ∈ (F,E) read as belonging to the neutrosophic soft
(α,β,γ)
∼
set (F,E) whenever α≤ T∼ (x), β ≤ I∼ (x) and F∼ (x) ≤ γ.
F(e) F(e) F(e)
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 soft
∼
neighborhood of the neutrosophic soft point xe ∈ (F,E), if there exists a
(α,β,γ)
∼ ∼ ∼
neutrosophic soft open set (G,E) such that xe ∈ (G,E) ⊆ (F,E).
(α,β,γ)
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 neutrosophic
∼
soft open set if and only if (F,E) is a neutrosophic soft neighborhood of its
neutrosophic soft points.
The neighborhood system of a neutrosophic soft point xe , denoted by
(α,β,γ)
U(xe ,E), is the family of all its neighborhoods.
(α,β,γ)
Theorem 2. [4] The neighborhood system U(xe ,E) at xe in a neu-
(α,β,γ) (α,β,γ)
trosophic soft topological space (X,τ,E) has the following properties:
∼ ∼
1) If (F,E) ∈ U(xe ,E), then xe ∈ (F,E),
(α,β,γ) (α,β,γ)
∼ ∼ ∼ ∼
2)If(F,E) ∈ U(xe ,E)and(F,E) ⊆ (H,E)then(H,E) ∈ U(xe ,E),
(α,β,γ) (α,β,γ)
∼ ∼ ∼ ∼
3) If (F,E), (G,E) ∈U(xe ,E) then (F,E)∩(G,E) ∈ U(xe ,E),
(α,β,γ) (α,β,γ)
∼ ∼
4) If (F,E) ∈ U(xe ,E) then there exists a (G,E) ∈ U(xe ,E) such
(α,β,γ) (α,β,γ)
∼ ∼
that (G,E) ∈ U(ye′ ,E) for each ye′ ∈ (G,E).
(α′,β′,γ′) (α′,β′,γ′)
6 NAIMEDEMIRTAS¸
Definition 12. Let (X,τ,E) be a neutrosophic soft topological space and
G¸(xe ,E) be a family of some neutrosophic soft neighborhoods of neutro-
(α,β,γ)
∼
sophic soft point xe . If, for each neutrosophic soft neighborhood (G,E)
(α,β,γ)
∼ ∼
of xe , there exists a (H,E) ∈G¸(xe ,E) such that xe ∈ (H,E) ⊆
(α,β,γ) (α,β,γ) (α,β,γ)
∼
(G,E), then we say that G¸(xe ,E) is a neutrosophic soft neighborhood
(α,β,γ)
base at xe .
(α,β,γ)
Theorem 3. If for each neutrosophic soft point xe there corresponds
(α,β,γ)
a family U(xe ,E) such that the properties 1) - 4) in Theorem 13 are
(α,β,γ)
satisfied, then there is a unique τ neutrosophic soft topological structure over
X such that for each xe , U(xe ,E) is the family of τ-neutrosophic
(α,β,γ) (α,β,γ)
soft neighborhoods of xe .
(α,β,γ)
∼ ∼ ∼
Proof. Letτ = (G,E) ∈ NSS(X,E) :xe ∈ (G,E) =⇒ (G,E) ∈ U(xe ,E) .
(cid:26) (α,β,γ) (α,β,γ) (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. The
axiom 1. in Definition 8 is a result of 2) and 3) in Theorem 13. It remains
to show that, in the neutrosophic soft topology defined by τ, U(xe ,E)
(α,β,γ)
is the set of τ-neutrosophic soft neighborhoods of xe for each xe . It
(α,β,γ) (α,β,γ)
follows from 2) in Theorem 13 that every τ-neutrosophic soft neighborhood
∼
of xe belongs to U(xe ,E). Conversely, let (G ,E) be a neutrosophic
(α,β,γ) (α,β,γ) 1
∼
soft setbelongingtoU(xe ,E) andlet(G ,E)betheneutrosophicsoftset
(α,β,γ) 2
∼
of neutrosophic soft points ye′ such that (G ,E) ∈ U(ye′ ,E). If
(α′,β′,γ′) 1 (α′,β′,γ′)
∼ ∼ ∼ ∼
we can show that xe ∈ (G ,E), (G ,E) ⊆ (G ,E) and (G ,E) ∈ τ,
(α,β,γ) 2 2 1 2
then the proof will be complete. Since for every neutrosophic soft point
∼ ∼
ye′ ∈ (G ,E) belongs to (G ,E) by reason of 1) in Theorem 13 and
(α′,β′,γ′) 2 1
∼ ∼ ∼
the hypothesis (G ,E) ∈ U(ye′ ,E), we obtain (G ,E) ⊆ (G ,E). Since
1 (α′,β′,γ′) 2 1
∼ ∼ ∼ ∼
(G ,E) ∈ U(xe ,E) and (G ,E) ⊆ (G ,E), we have xe ∈ (G ,E). It
1 (α,β,γ) 2 1 (α,β,γ) 2
∼ ∼
remains to show that (G ,E) ∈ τ, i.e. that (G ,E) ∈ U(ye′ ,E) for each
2 2 (α′,β′,γ′)
∼ ∼
ye′ ∈ (G ,E). If ye′ ∈ (G ,E) then by 4) in Theorem 13 there
(α′,β′,γ′) 2 (α′,β′,γ′) 2
∼ ∼
is a neutrosophic soft set (G ,E) such that for each ze′′ ∈ (G ,E) we
3 (α′′,β′′,γ′′) 3
∼ ∼
have (G ,E) ∈ U(ze′′ ,E). Since (G ,E) ∈ U(ze′′ ,E) means that
1 (α′′,β′′,γ′′) 1 (α′′,β′′,γ′′)
NEUTROSOPHIC SOFT FILTERS 7
∼ ∼ ∼
ze′′ ∈ (G ,E), it follows that (G ,E) ⊆(G ,E) and therefore, by 2) in
(α′′,β′′,γ′′) 2 3 2
∼
Theorem 13, that (G ,E) ∈ U(ye′ ,E). (cid:3)
2 (α′,β′,γ′)
4. Neutrosophic soft filters
Definition 13. Letℵ ⊆ NSS(X,E), thenℵiscalled aneutrosophicsoftfilter
on X if ℵ satisfies the following properties:
(ℵ ) 0 ∈/ ℵ,
1 (X,E)
∼ ∼ ∼ ∼
(ℵ ) ∀(F,E), (G,E) ∈ ℵ =⇒ (F,E)∩(G,E) ∈ ℵ,
2
∼ ∼ ∼ ∼
(ℵ ) ∀(F,E) ∈ℵ and (F,E) ⊆ (G,E) =⇒ (G,E) ∈ ℵ.
3
Remark 1. It follows from (ℵ ) and (ℵ ) that every finite intersections of
1 2
neutrosophic soft sets of ℵ are not 0 .
(X,E)
Proposition 1. The condition (ℵ ) is equivalent to the following two condi-
2
tions:
(ℵ ) The intersection of two members of ℵ belongs to ℵ.
2a
(ℵ ) 1 belongs to ℵ.
2b (X,E)
Example 1. The family ℵ= {1 } is a neutrosophic soft filter over X.
(X,E)
∼
Theorem 4. Let 0(X,E) 6= (F,E) ∈ NSS(X,E). Then the family ℵ ∼ =
(F,E)
∼ ∼ ∼
(G,E) : (F,E) ⊆ (G,E) ∈ NSS(X,E) is a neutrosophic soft filter over X.
(cid:26) (cid:27)
Proof. Since 1 ∈ ℵ and 0 ∈/ ℵ, ∅ 6= ℵ 6= NSS(X,E). Suppose
(X,E) (X,E)
∼ ∼ ∼ ∼ ∼ ∼
(H ,E), (H ,E) ∈ ℵ, then (F,E) ⊆ (H ,E), (F,E) ⊆ (H ,E). Thus
1 2 1 2
T∼ (x) ≤ min T∼ (x),T∼ (x) , I∼ (x) ≤ min I∼ (x),I∼ (x)
F(e) (cid:26) H1(e) H2(e) (cid:27) F(e) (cid:26) H1(e) H2(e) (cid:27)
∼
and F∼ (x) ≤ max F∼ (x),F∼ (x) for all x ∈ X. So (F,E) ⊆
F(e) (cid:26) H1(e) H2(e) (cid:27)
∼ ∼ ∼ ∼
(H ,E)∩(H ,E) and hence (H ,E)∩(H ,E) ∈ ℵ. (cid:3)
1 2 1 2
Theorem 5. Let (X,τ,E) be a neutrosophic soft topological space over X.
The neighborhood system U(xe ,E) is a neutrosophic soft filter for every
(α,β,γ)
neutrosophic soft point xe . Also, it is called neutrosophic soft neighbor-
(α,β,γ)
hoods filter of the neutrosophic soft point xe .
(α,β,γ)
∼
Proof. (ℵ ) By 1) in Theorem 13, since xe ∈ (G,E), we obtain 0 ∈/
1 (α,β,γ) (X,E)
U(xe ,E).
(α,β,γ)
8 NAIMEDEMIRTAS¸
(ℵ ) This is clearly seen by 3) in Theorem 13.
2
(ℵ ) This is clearly seen by 2) in Theorem 13. (cid:3)
3
Now, wesetupaneutrosophicsofttopology withthehelpof aneutrosophic
soft filter.
Theorem6. If, foreveryxe , thereexistsaneutrosophic softfilterℵ(xe )=
(α,β,γ) (α,β,γ)
U(xe ,E) which satisfies the following two properties, then there exists a
(α,β,γ)
unique neutrosophic soft topology τ such that ℵ(xe ) consists of the τ-
(α,β,γ)
neutrosophic soft neighborhoods of the neutrosophic soft point xe .
(α,β,γ)
(1) Every neutrosophic soft set in the neutrosophic soft filter ℵ(xe )
(α,β,γ)
contains the neutrosophic soft point xe ,
(α,β,γ)
∼ ∼
(2) For every (G,E) ∈ ℵ(xe ) there exists a (H,E) ∈ ℵ(xe ) such
(α,β,γ) (α,β,γ)
∼ ∼
that for every ye′ ∈(H,E), (G,E) ∈ℵ(ye′ ).
(α′,β′,γ′) (α′,β′,γ′)
Proof. Since the axioms (ℵ ), (ℵ ), (ℵ ), (1) and (2) are equivalent to the
1 2 3
neighborhood axioms 1) − 4), by Theorem 15, there exists a neutrosophic
soft topology τ such that ℵ(xe ) consists of the τ-neutrosophic soft neigh-
(α,β,γ)
borhoods of the neutrosophic soft point xe . (cid:3)
(α,β,γ)
Example 2. Let (X,τ,E) be a neutrosophic soft topological space and xe
(α,β,γ)
∼
be a neutrosophic soft point over X. Since (G,E) cannot be an element of
∼ ∼ ∼
G¸(xe ,E) for every (H,E) ∈G¸(xe ,E) and (H,E) ⊆ (G,E), then the
(α,β,γ) (α,β,γ)
neutrosophic soft neighborhood base G¸(xe ,E) is not a neutrosophic soft
(α,β,γ)
filter over X.
5. Comparison of neutrosophic soft filters
Definition 14. Let ℵ and ℵ be neutrosophic soft filters over X. If ℵ ⊆ ℵ ,
1 2 1 2
then ℵ is said to be finer that ℵ or ℵ coarser than ℵ .
2 1 1 2
If also ℵ 6= ℵ , then ℵ is strictly finer than ℵ or ℵ is strictly coarser
1 2 2 1 1
than ℵ . If either ℵ ⊆ ℵ or ℵ ⊆ ℵ , then ℵ is comparable with ℵ .
2 1 2 2 1 1 2
Theorem 7. Let (ℵ ) be a family of neutrosophic soft filters over X. Then
i i∈I
ℵ = ∩ℵ is a neutrosophic soft filter over X.
i
i∈I
In fact ℵ is the greatest lower bound of the family (ℵ ) .
i i∈I
Proof. (ℵ ) Since 0 ∈/ ℵ for each i ∈ I, then 0 does not belong to
1 (X,E) i (X,E)
ℵ = ∩ℵ .
i
i∈I
NEUTROSOPHIC SOFT FILTERS 9
∼ ∼ ∼ ∼
(ℵ ) Let (F,E), (G,E) ∈ ℵ = ∩ℵ . Then (F,E), (G,E) ∈ ℵ for each
2 i i
i∈I
∼ ∼ ∼ ∼
i∈ I. Since (F,E)∩(G,E) ∈ℵ for each i ∈I, so we obtain (F,E)∩(G,E) ∈
i
ℵ= ∩ℵ .
i
i∈I
∼ ∼ ∼ ∼
(ℵ ) Let (F,E) ∈ ℵ = ∩ℵ and (F,E) ⊆ (G,E). Since (F,E) ∈ ℵ for
3 i i
i∈I
∼ ∼ ∼
each i ∈ I and (F,E) ⊆ (G,E), we get (G,E) ∈ ℵ for each i ∈ I. Hence
i
∼
(G,E) ∈ ℵ = ∩ℵ . (cid:3)
i
i∈I
Now, we investigate the least upper bound of the family of neutrosophic
soft filters over X.
Theorem 8. Let S ⊆ NSS(X,E). Then there exists a neutrosophic soft filter
ℵ which contains the family S, if S has the following property: ”The all finite
intersections of neutrosophic soft sets of S are not 0 ”.
(X,E)
∼ ∼
Proof. Let S = (F ,E) : ∀i∈ J (J is finite), ∩ (F ,E) 6= 0 . Then
i i (X,E)
(cid:26) i∈J (cid:27)
we give the family which consists of finite intersections of elements of S; β =
∼ ∼ ∼ ∼
(G,E) : ∀i∈J (J is finite), (F ,E) ∈ S and (G,E) = ∩ (F ,E) .
i i
(cid:26) i∈J (cid:27)
∼ ∼ ∼ ∼
Then the family ℵ(S) = (H,E) : (G,E) ∈ β and (G,E) ⊆ (H,E) is a
(cid:26) (cid:27)
neutrosophic soft filter over X.
∼ ∼
(ℵ ) 0 ∈ β, for every (H,E) ∈ ℵ(S), (H,E) 6= 0 and so 0 ∈/
1 (X,E) (X,E) (X,E)
ℵ(S).
∼ ∼
(ℵ ) Let (H ,E), (H ,E) ∈ ℵ(S). There exist neutrosophic soft sets
2 1 2
∼ ∼ ∼ ∼ ∼ ∼
(G ,E), (G ,E) ∈ β such that (G ,E) ⊆ (H ,E) and (G ,E) ⊆ (H ,E).
1 2 1 1 2 2
∼ ∼ ∼
From the definition of β, 0 6= (G ,E) ∩ (G ,E) ∈ β. Since (G ,E) ∩
(X,E) 1 2 1
∼ ∼ ∼ ∼ ∼
(G ,E) ⊆ (H ,E)∩(H ,E), we obtain (H ,E)∩(H ,E) ∈ℵ(S).
2 1 2 1 2
∼ ∼ ∼
(ℵ ) Let (H ,E) ∈ ℵ(S) and (H ,E) ⊆ (H ,E). Then there exists a neu-
3 1 1 2
∼ ∼ ∼ ∼
trosophic soft set (G,E) ∈ β such that (G,E) ⊆ (H ,E). Since (H ,E) ⊆
1 1
∼ ∼
(H ,E), we obtain (H ,E) ∈ ℵ(S). (cid:3)
2 2
Remark 2. The neutrosophic soft filter ℵ(S) in Theorem 26 is said to be
generated by S and S is said to be neutrosophic soft filter subbase of ℵ(S). It
is clear that S ⊆ ℵ(S).
10 NAIMEDEMIRTAS¸
Theorem 9. The neutrosophic soft filter ℵ(S) which is generated by S is the
coarsest neutrosophic soft filter which contains S.
Proof. Suppose that S ⊆ ℵ . By Theorem 26, S ⊆ β ⊆ ℵ . By Remark 27,
1 1
∼ ∼ ∼ ∼
for every (H,E) ∈ ℵ(S) there exists a (G,E) ∈ β such that (G,E) ⊆ (H,E).
∼ ∼
Sinceβ ⊆ ℵ ,then(G,E) ∈ ℵ . Sinceℵ isaneutrosophicsoftfilter,(H,E) ∈
1 1 1
ℵ by (ℵ ) in Definition 16. Hence we obtain ℵ(S)⊆ ℵ . (cid:3)
1 3 1
Theorem 10. The family (ℵ ) of neutrosophic soft filters over X has a
i i∈I
least upper bound if and only if for all finite subfamilies (ℵ ) of (ℵ )
i 1≤i≤n i i∈I
∼ ∼ ∼
and all (G ,E) ∈ℵ (1 ≤ i≤ n), (G ,E)∩ ... ∩(G ,E) 6= 0 .
i i 1 n (X,E)
Proof. =⇒: If there exists a least upper bound of the family (ℵ ) , by (ℵ )
i i∈I 1
and (ℵ ) in Definition 16, for all finite subfamilies (ℵ ) of (ℵ ) and all
2 i 1≤i≤n i i∈I
∼ ∼ ∼
(G ,E) ∈ ℵ (1 ≤ i≤ n), the intersection (G ,E)∩ ... ∩(G ,E) 6= 0 .
i i 1 n (X,E)
∼ ∼
⇐=: Let (G ,E)∩ ... ∩(G ,E) 6= 0 for all finite subfamilies (ℵ )
1 n (X,E) i 1≤i≤n
∼
of (ℵ ) and all (G ,E) ∈ ℵ (1 ≤ i ≤ n). Then the neutrosophic soft filter
i i∈I i i
ℵ(S) generated by
∼ ∼
S = ∪ℵ = (F,E) : (∃i∈ I) (F,E) ∈ ℵ
i i
i∈I (cid:26) (cid:27)
is the least upper bound of the family (ℵ ) by Theorem 28. (cid:3)
i i∈I
Definition 15. Let β ⊆ NSS(X,E), then β is said to be a neutrosophic soft
filter base on X if
(β ) β 6= ∅ and 0 ∈/ β.
1 (X,E)
(β ) The intersection of two members of β contain a member of β.
2
Remark 3. β which is in Theorem 26 is a neutrosophic soft filter base.
Remark 4. It is clear that, every neutrosophic soft filter is a neutrosophic soft
filter base.
Example 3. Let (X,τ,E) be a soft topological space and xe be a neutro-
(α,β,γ)
sophicsoftpointoverX. TheneutrosophicsoftneighborhoodbaseG¸(xe ,E)
(α,β,γ)
is a neutrosophic soft filter base over X.
∼
(β )Clearly, G¸(xe ,E) 6= ∅. For every (H,E) ∈G¸(xe ,E), xe ∈
1 (α,β,γ) (α,β,γ) (α,β,γ)
∼ ∼
(H,E). Then (H,E) 6= 0 . Hence we obtain 0 ∈/G¸(xe ,E).
(X,E) (X,E) (α,β,γ)
∼ ∼ ∼ ∼
(β )Let(G,E), (H,E) ∈ G¸(xe ,E). Since(G,E), (H,E) ∈ U(xe ,E),
2 (α,β,γ) (α,β,γ)
∼ ∼
we get (G,E) ∩ (H,E) ∈ U(xe ,E). By Definition 14, there exists a
(α,β,γ)