Jafari, S.

Karthika, M.

Parimala, M.

Smarandache, Florentin

English

University of New Mexico

Neutrosophic Sets and Systems, Vol. 24, 2019
University of New Mexico 
Neutrosophic Nano ideal topological structure
M. Parimala 1, M. Karthika 2, S. Jafari3 , F. Smarandache 4, R.Udhayakumar5∗
1Department of Mathematics, Bannari Amman Institute of Technology,Sathyamangalam-638401, Tamil Nadu, India
E-mail: rishwanthpari@gmail.com
2Department of Mathematics, Bannari Amman Institute of Technology,Sathyamangalam-638401, Tamil Nadu, India
E-mail: karthikamuthusamy1991@gmail.com
3College of Vestsjaelland South, Herrestraede 11, 4200 Slagelse, Denmark
E-mail: jafaripersia@gmail.com
4Mathematics & Science Department, University of New Maxico, 705 Gurley Ave, Gallup, NM 87301, USA
E-mail: fsmarandache@gmail.com
5School of Advanced Sciences, Department of Mathematics, VIT, Vellore, TN, India
E-mail: udhayaram v@yahoo.co.in
∗Correspondence: Author (udhayaram v@yahoo.co.in)
Abstract: This paper addressed the concept of Neutrosophic nano ideal topology which is induced by the two litere-
ture, they are nano topology and ideal topological spaces. We defined its local function, closed set and also defined
and give new dimnesion to codense ideal by incorporating it to ideal topological structures. we investigate some
properties of neutrosophic nano topology with ideal.
Keywords: neutrosophic nano ideal, neutrosophic nano local function, topological ideal, neutrosophic nano topolog-
ical ideal.
1 Introduction and Preliminaries
In 1983, K. Atanassov [1] proposed the concept of IFS(intuitionstic fuzzy set) which is a generalization of
FS(fuzzy set) [17], where each element has true and false membership degree. Smarandache [15] coined the
concept of NS (neutrosophic set) which is new dimension to the sets. Neutrosophic set is classified into three
independently related functions namely, membership, indeterminacy function and non-membership function.
Lellis Thivagar [8], introduced the new notion of neutrosophic nano topology, which consist of upper, lower
approximation and boundary region of a subset of a universal set using an equivalence class on it. There have
been wide range of studies on neutrosophic sets, ideals and nano ideals [9, 10, 11,12,13,14]. Kuratowski [7]
and Vaidyanathaswamy [16] introduced the new concept in topological spaces, called ideal topological spaces
and also local function in ideal topological space was defined by them. Afterwards the properties of ideal
topological spaces studied by Hamlett and Jankovic[5,6].
M. Parimala, M. Karthika, S. Jafari, F. Samrandache and R. Udhayakumar, Neutrosophic Nano ideal
topolgical structure
70
71 Neutrosophic Sets and Systems, Vol.24, 2019
In this paper, we introduce the new concept of neutrosophic nano ideal topological structures, which is a
generalized concept of neutrosophic nano and ideal topological structure. Also defined the codense ideal in
neutrosophic nano topological structure.
We recall some relevant basic definitions which are useful for the sequel and in particular, the work of M. L.
Thivagar [8], Parimala et al [9], F. Smarandache [15].
Definition 1.1. Let U be universe of discourse andR be an indiscernibility relation on U. Then U is divided
into disjoint equivalence classes. The pair (U,R) is said to be the approximation space. Let F be a NS in U
with the true µF , the indeterminancy σF and the false function νF . Then,
(i) The lower approximation of F with respect to equivalence class R is the set denoted by N(F ) and
defined as follows
N(F ) =
{〈
a, µR(F )(a), σR(F )(a), νR(F )(a)
〉
|y ∈ [a]R, a ∈ U
}
(ii) The higher approximation of F with respect to equivalence class R is the set is denoted by N(F ) and
defined as follows, N(F ) =
{〈
a, µR(F )(a), σR(F )(a), νR(F )(a)
〉
|y ∈ [a]R, a ∈ U
}
(iii) The boundary region of F with respect to equivalence classR is the set of all objects is denoted byB(F )
and defined byB(F ) = N(F )−N(F ).
where,
µR(F )(a) =
⋃
y1∈[a]R
µF (y1), σR(F )(a) =
⋃
y1∈[a]R
σF (y1),
νR(F )(a) =
⋂
y1∈[a]R
νF (y1). µR(F )(a) =
⋂
y1∈[a]R
µF (y1),
σR(F )(a) =
⋂
y1∈[a]R
σF (y1), νR(F )(a) =
⋂
y1∈[a]R
νF (y1).
Definition 1.2. LetU be a nonempty set and the neutrosophic setsX and Y in the formX = {〈a, µX(a), σX(a), νX(a)〉 , a ∈ U}
and Y = {〈a, µY (a), σY (a), νY (a)〉 , a ∈ U}. Then the following statements hold:
(i) 0N = {〈a, 0, 0, 1〉 , a ∈ U} and 1N = {〈a, 1, 1, 0〉 , a ∈ U}.
(ii) X ⊆ y if and only if µX(a) ≤ µY (a), σX(a) ≤ σY (a), νX(a) ≥ νY (a) for all a ∈ U .
(iii) X = Y if and only if X ⊆ Y and Y ⊆ X .
(iv) XC = {〈a, νX(a), 1− σX(a), µX(a)〉 , a ∈ U}.
(v) X ∩ Y if and only if µX(a) ∧ µX(a), σX(a) ∧ σY (a), νY (a) ∨ νY (a) for all a ∈ U .
(vi) X ∪ Y if and only if µY (a) ∨ µY (a), σX(a) ∨ σY (a), νX(a) ∧ νY (a) for all a ∈ U .
(vii) X − Y if and only if µX(a) ∧ νY (a), σX(a) ∧ 1− σY (a), νX(a) ∨ µY (a) for all a ∈ U .
Definition 1.3. Let X be a non-empty set and I is a neutrosophic ideal (NI for short) on X if
M. Parimala, M. Karthika, S. Jafari, F. Samrandache and R. Udhayakumar, Neutrosophic Nano ideal
topolgical structure
Neutrosophic Sets and Systems, Vol. 24, 2019 72
(i) A1 ∈ I and B1 ⊆ A1⇒ B1 ∈ I [heredity],
(ii) A1 ∈ I and B1 ∈ I ⇒ A1 ∪B1 ∈ I [finite additivity].
2 Neutrosophic nano ideal topological spaces
In this section we introduce a new type of local function in neutrosophic nano topological space. Before that
we shall consider the following concepts.
Neutrosophic nano ideal topological space(in short NNI) is denoted by (U, τN (F ), I), where (U, τN (F ), I)
is a neutrosophic nano topological space(in short NNT) (U, τN (F )) with an ideal I on U
Definition 2.1. Let (U, τN (F ), I) be a NNI with an ideal I on U and (.)∗N be a set of operator from P (U) to
P (U)× P (U) (P (U) is the set of all subsets of U ). For a subset X ⊂ U , the neutrosophic nano local function
X∗N (I, τN (F )) of X is the union of all neutrosophic nano points (NNP, for short) C(α, β, γ) such that
X∗N (I, τN (F )) = ∨{C(α, β, γ) ∈ U : X ∩ G /∈ Ifor all G ∈ N(C(α, β, γ))}. We will simply write X∗N for
X∗N (I, τN (F )).
Example 2.2. Let (U, τN (F )) be a neutrosophic nano topological space with an ideal I on U and for every
X ⊆ U .
(i) If I = {0∼}, then X∗N = N cl(X),
(ii) If I = P (U), then X∗N = 0∼.
Theorem 2.3. Let (U, τN (F )) be a NNT with ideals I , I ′ on U and X,B be subsets of U . Then
(i) X ⊆ B ⇒ X∗N ⊆ B∗N ,
(ii) I ⊆ I ′ ⇒ X∗N (I ′) ⊆ X∗N (I),
(iii) X∗N = N cl(X∗N ) ⊆ N cl(X) (X∗N is a neutrosophic nano closed subset of N cl(X)),
(iv) (X∗N )
∗
N ⊆ X∗N ,
(v) X∗N ∪B∗N = (X ∪B)∗N ,
(vi) X∗N −B∗N = (X −B)∗N −B∗N ⊆ (X −B)∗N ,
(vii) V ∈ τN (F )⇒ V ∩X∗N = V ∩ (V ∩X)∗N ⊆ (V ∩X)∗N and
(viii) J ∈ I ⇒ (X ∪ J)∗N = X∗N = (X − J)∗N .
Proof. (i) Let X ⊂ B and a ∈ X∗N . Assume that a /∈ B∗N . We have GN ∩ B ∈ I for some GN ∈ GN (a).
Since GN ∩ X ⊆ GN ∩ B and GN ∩ B ∈ I , we obtain GN ∩ X ∈ I from the definition of ideal. Thus, we
have a /∈ X∗N . This is a contradiction. Clearly, X∗N ⊆ B∗N .
(ii) Let I ⊆ I ′ and a ∈ X∗N (I ′). Then we have GN ∩ X /∈ I ′ for every GN ∈ GN (a). By hypothesis, we
obtain GN ∩X /∈ I . So a ∈ X∗N (I).
(iii) Let a ∈ X∗N . Then for every GN ∈ GN (a), GN ∩ X /∈ I . This implies that GN ∩ X 6= 0∼. Hence
M. Parimala, M. Karthika, S. Jafari, F. Samrandache and R. Udhayakumar, Neutrosophic Nano ideal
topolgical structure
73 Neutrosophic Sets and Systems, Vol.24, 2019
a ∈ N cl(X).
(iv) From (iii), (X∗N )
∗
N ⊆ N cl(X∗N ) = X∗N , since X∗N is a neutrosophic nano closed set.
The proofs of the other conditions are also obvious.
Theorem 2.4. If (U, τN (F ), I) is a NNT with an ideal I and X ⊆ X∗N , then X∗N = N cl(X∗N ) = N cl(X).
Proof. For every subset X of U , we have X∗N = N cl(X∗) ⊆ N cl(X), by Theorem 2.3. (iii) X ⊆ X∗N implies
that N cl(X) ⊆ Ncl(X∗N ) and so X∗N = N cl(X∗N ) = N cl(X).
Definition 2.5. Let (U, τN (F )) be a NNT with an ideal I on U . The set operator N cl∗ is called a neutro-
sophic nano∗-closure and is defined as N cl∗(X) = X ∪X∗N for X ⊆ a.
Theorem 2.6. The set operator N cl∗ satisfies the following conditions:
(i) X ⊆ N cl∗(X),
(ii) N cl∗(0∼) = 0∼ and N cl∗(1∼) = 1∼,
(iii) If X ⊂ B, then N cl∗(X) ⊆ N cl∗(B),
(iv) N cl∗(X) ∪N cl∗(B) = N cl∗(X ∪B).
(v) N cl∗(N cl∗(X)) = N cl∗(X).
Proof. The proofs are clear from Theorem 2.3 and the definition of N cl∗.
Now, τN (F )∗(I, τN (F )) = {V ⊂ U : N cl∗(U − V ) = U − V }. τN (F )∗(I, τN (F )) is called neutrosophic
nano∗-topology which is finer than τN (F ) (we simply write τN (F )∗ for τN (F )∗(I, τN (F ))). The elements of
τN (F )
∗(I, τN (F )) are called neutrosophic nano∗-open (briefly,N∗-open) and the complement of anN∗-open
set is called neutrosophic nano∗-closed (briefly, N∗-closed). Here N cl∗(X) and N int∗(X) will denote the
closure and interior of X respectively in (U, τN (F )∗).
Remark 2.7. (i) We know from Example 2.2 that if I = {0∼} then X∗N = N cl(X). In this case, N cl∗(X) =
N cl(X).
(ii) If (U, τN (F ), I) is a NNI with I = {0∼}, then τN (F )∗ = τN (F ).
Definition 2.8. A basis β(I, τN (F )) for τN (F )∗ can be described as follows:
β(I, τN (F )) = {X −B : X ∈ τN (F ), B ∈ I}.
Theorem 2.9. Let (U, τN (F )) be a NNT and I be an ideal on U . Then β(I, τN (F )) is a basis for τN (F )∗.
Proof. We have to show that for a given space (U, τN (F )) and an ideal I on U , β(I, τN (F )) is a basis for
τN (F )
∗. If β(I, τN (F )) is itself a neutrosophic nano topology, then we have β(I, τN (F )) = τN (F )∗ and all the
open sets of τN (F )∗ are of simple form X −B where X ∈ τN (F ) and B ∈ I .
Theorem 2.10. Let (U, τN (F ), I) be a NNT with an ideal I on U and X ⊆ U . If X ⊆ X∗N , then
(i) N cl(X) = N cl∗(X),
(ii) N int(U −X) = N int∗(U −X).
Proof. (i) Follows immediately from Theorem 2.4.
(ii) If X ⊆ X∗N , then N cl(X) = N cl∗(X) by (i) and so U − N cl(X) = U − N cl∗(X). Therefore,
N int(U −X) = N int∗(U −X).
M. Parimala, M. Karthika, S. Jafari, F. Samrandache and R. Udhayakumar, Neutrosophic Nano ideal
topolgical structure
Neutrosophic Sets and Systems, Vol. 24, 2019 74
Theorem 2.11. Let (U, τN (F ), I) be a NNT with an ideal I on U and X ⊆ X . If X ⊆ X∗N , then
X∗N = N cl(X∗N ) = n-cl(X) = N cl∗(X).
Definition 2.12. A subset A of a neutrosophic nano ideal topological space (U, τN (F ), I) is N∗-dense in
itself (resp. N∗-perfect) if X ⊆ X∗N (resp. X = X∗N ).
Remark 2.13. A subset X of a neutrosophic nano ideal topological space (U, τN (F ), I) is N ∗-closed if
and only if X∗N ⊆ X .
For the relationship related to several sets defined in this paper, we have the following implication:
N∗-dense in itself⇐N∗-perfect⇒N ∗-closed
The converse implication are not satisfied asthe following shows.
Example 2.14. Let U be the universe, X = {P1, P2, P3, P4, P5} ⊂ U , U/R = {{P1, P2}, {P3}, {P4, P5}}
and τN (F ) = {1∼, 0∼,N ,N , B} and the ideal I = 0∼, 1∼. For X = {< P1, (.5, .4, .7) >,< P2, (.6, .4, .5) >
,< P3, (.4, .5, .4) >,< P4, (.7, .3, .4) >,< P5, (.8, .5, .2) >}, N(X) = { P1,P2.5,.4,.7 ,
P3
.4,.5,.4
, P4,P5
.7,.3,.4
},
N(X) = { P1,P2
.6,.4,.5
, P3
.4,.5,.4
, P4,P5
.8,.5,.2
}, B(X) = { P1,P2
.6,.4,.5
, P3
.4,.5,.4
, P4,P5
.4,.3,.7
} . If I = 0∼ then X∗N = Ncl(a). Thus
X ⊆ X∗N . Hence X is N ∗-dense but not N ∗-perfect.
If I = 1∼ then X∗N = 0∼. Thus X ⊇ X∗N . Hence X∗N is N ∗-closed but not N ∗-perfect.
Lemma 2.15. Let (U, τN (F ), I) be a NNI andX ⊆ U . IfX isN∗-dense in itself, thenX∗N = N cl(X∗N ) =
N cl(X) = N cl∗(X).
Proof. Let X be N∗-dense in itself. Then we have X ⊆ X∗N and using Theorem 2.11 we getX∗N =
N cl(X∗N ) = N cl(X) = N cl∗(X).
Lemma 2.16. If (U, τN (F ), I) is a NNT with an ideal I andX ⊆ U , thenX∗N (I, τN (F )) = X∗N (I, τN (F )∗)
and hence τN (F )∗ = τN (F )∗∗.
3 τN (F )-codence ideal
n this section we incorporated codence ideal [5] in ideal topological space and introduce similar concept in
neutrosophic nano ideal topological spaces.
Definition 3.1. An ideal I in a space (U, τN (F ), I) is called τN (F )-codense ideal if τN (F ) ∩ I = {0∼}.
Following theorems are related to τN (F )-codense ideal.
Theorem 3.2. Let (U, τN (F ), I) be an NNI and I is τN (F )-codense with τN (F ). Then U = U∗N .
Proof. It is obvious that U∗N ⊆ U . For converse, suppose a ∈ U but a 6∈ U∗N . Then there existsGx ∈ τN (F )(a)
such that Gx ∩ U ∈ I . That is Gx ∈ I , a contradiction to the fact that τN (F ) ∩ I = {0∼}. Hence U = U∗N .
Theorem 3.3. Let (U, τN (F ), I) be a NNI. Then the following conditions are equivalent:
(i) U = U∗N .
M. Parimala, M. Karthika, S. Jafari, F. Samrandache and R. Udhayakumar, Neutrosophic Nano ideal
topolgical structure
75 Neutrosophic Sets and Systems, Vol.24, 2019
(ii) τN (F ) ∩ I = {0∼}.
(iii) If J ∈ I , then N int(J) = 0∼.
(iv) For every X ∈ τN (F ), X ⊆ X∗N .
Proof. By Lemma 2.16, we may replace ‘τN (F )’ by ‘τN (F )∗’ in (ii), ‘N int(J) = 0∼’ by ‘N int∗(J) = 0∼’
in (iii) and ‘X ∈ τN (F )’ by ‘X ∈ τN (F )∗’ in (iv).
4 Conclusions
this paper, we introduced the notion of neutrosophic nano ideal topological structures and investigated some
relations over neutrosophic nano topology and neutrosophic nano ideal topological structures and studied some
of its basic properties. In future, it motivates to apply this concepts in graph structures.
References
[1] K. T. Atanassov Intuitionstic fuzzy sets, Fuzzy sets and systems, 20(1), (1986), 87-96.
[2] M. E. Abd El-Monsef, E. F. Lashien and A. A. Nasef On I-open sets and I-continuous functions. Kyungpook Math. J., 32,
(1992), 21-30.
[3] T.R. Hamlett and D. Jankovic Ideals in topological spaces and the set operator ψ, Bull. U.M.I., 7( 4-B), (1990), 863-874.
[4] E. Hayashi Topologies defined by local properties, Math. Ann., 156(3), (1964), 205 - 215.
[5] D. Jankovic and T. R. Hamlett Compatible extensions of ideals, Boll. Un. Mat. Ital., B(7)6, (1992), 453-465.
[6] D. Jankovic and T. R. Hamlett New Topologies from old via Ideals, Amer. Math. Monthly, 97(4), (1990), 295 - 310.
[7] K. Kuratowski Topology, Vol. I, Academic Press (New York, 1966).
[8] M. Lellis Thivagar, S. Jafari, V. Sutha Devi, V. Antonysamy A novel approach to nano topology via neutrosophic sets,
Neutrosophic Sets and Systems, 20, (2018),86-94.
[9] M. Parimala and R. Perumal Weaker form of open sets in nano ideal topological spaces, Global Journal of Pure and Applied
Mathematics, 12(1), (2016), 302-305.
[10] M.Parimala, R.Jeevitha and A.Selvakumar. A New Type of Weakly Closed Set in Ideal Topological Spaces, International
Journal of Mathematics and its Applications, 5(4-C), (2017), 301-312.
[11] M. Parimala, S. Jafari, and S. Murali Nano Ideal Generalized Closed Sets in Nano Ideal Topological Spaces, Annales Univ.
Sci. Budapest., 60, (2017), 3-11.
[12] M. Parimala, M. Karthika, R. Dhavaseelan, S. Jafari. On neutrosophic supra pre-continuous functions in neutrosophic topo-
logical spaces, New Trends in Neutrosophic Theory and Applications , 2, (2018), 371-383.
[13] M. Parimala, M. Karthika, S. Jafari, F. Smarandache and R. Udhayakumar Decision-Making via Neutrosophic Support Soft
Topological Space, Symmetry, 10(6), (2018), 217, 1-10.
M. Parimala, M. Karthika, S. Jafari, F. Samrandache and R. Udhayakumar, Neutrosophic Nano ideal
topolgical structure
Neutrosophic Sets and Systems, Vol. 24, 2019 76
[14] M. Parimala, F. Smarandache, S. Jafari and R. Udhayakumar On Neutrosophic αψ -Closed Sets, Information, 9, (2018), 103,
1-7 .
[15] F. Smarandache A Unifying Field in Logics. Neutrosophic Logic: Neutrosophy, Neutrosophic Set, Neutrosophic Probability,
Rehoboth: American Research Press. (1999).
[16] R. Vaidyanathaswamy The localization theory in set topology, Proc. Indian Acad. Sci., 20(1), (1944), 51 - 61.
[17] L. A. Zadeh Fuzzy sets, Information and Control 8(1965), 338-353 .
Received: January 02, 2019. Accepted: February 28, 2019.
M. Parimala, M. Karthika, S. Jafari, F. Samrandache and R. Udhayakumar, Neutrosophic Nano ideal
topolgical structure


Neutrosophic Nano ideal topological structure

https://digitalrepository.unm.edu/cgi/viewcontent.cgi?article=1321&amp;context=nss_journal

Neutrosophic Nano ideal topological structure

Abstract

Similar works

Full text

Available Versions

University of New Mexico