H., Jude Immaculate

K, Sivaranjani

P., Evanzalin Ebenanjar

English

University of New Mexico

Neutrosophic Sets and Systems 
Volume 31 Article 21 
5-2-2020 
Introduction to neutrosophic soft topological spatial region 
Evanzalin Ebenanjar P. 
Jude Immaculate H. 
Sivaranjani K 
Follow this and additional works at: https://digitalrepository.unm.edu/nss_journal 
Recommended Citation 
P., Evanzalin Ebenanjar; Jude Immaculate H.; and Sivaranjani K. "Introduction to neutrosophic soft 
topological spatial region." Neutrosophic Sets and Systems 31, 1 (2020). 
https://digitalrepository.unm.edu/nss_journal/vol31/iss1/21 
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Neutrosophic Sets and Systems, Vol. 31, 2020
University of New Mexico
Introduction to neutrosophic soft topological spatial region
Evanzalin Ebenanjar P.1∗, Jude Immaculate H.2 and Sivaranjani K.3
1Department of Mathematics, Karunya Institute of Technology and Sciences, Coimbatore, India-641114; email
evanzalin86@yahoo.com
2Department of Mathematics, Karunya Institute of Technology and Sciences, Coimbatore, India-641114; email
judeh@karunya.edu
3Department of Mathematics, Karunya Institute of Technology and Sciences, Coimbatore, India-641114; e-
mail sivaranjani@karunya.edu
∗Correspondence: evanzalin86@yahoo.com, evenzalin@karunya.edu;
Abstract. Spatial information often deals with regions which are vague or incompletely determined. Under-
standing vagueness, indeterminacy and imprecision are the most important in GIS. Smarandache’s neutrosophic
set is a computational method to tackle problems involving incomplete, infinite and reliable data. The definition
of soft sets was introduced by Molodtsov as a new mathematical method to tackle uncertainty. Maji presented
the Neutrosophic Soft Set theory. This paper provides concepts of a neurtrosophic soft spatial region for its
possible application in GIS. The notions of neutrosophic soft α-open, neutrosophic soft pre-open, neutrosophic
soft semi-open and neutrosophic soft β-open sets are introduced.
Keywords: Neutrosophic soft set; neutrosophic soft topology; neutrosophic soft connected; neutrosophic soft
spatial region; GIS.
—————————————————————————————————————————-
1. Introduction
Many real-life issues deal with uncertainties in economics, engineering, environment, so-
cial sciences, medical sciences, and business management. There are difficulties with classical
mathematical modeling in solving the uncertainties in these data. Theories such as fuzzy
set[1], rough set[2] and intuitionist fuzzy set[3] are used to prevent difficulties in dealing with
uncertainty. But all of these hypotheses have some difficulties in addressing the indeterminate
or contradictory data problems. Smarandache[4] described the neutrosophical set as a math-
ematical method for dealing with indeterminate and inaccurate problems in nature. There is
a lot of use in all fields, such as IT, information systems and decision support systems.
Evanzalin P., Jude Immaculate H. and Sivaranjani K., Introduction to neutrosophic soft topological spatial
region
Neutrosophic Sets and Systems, Vol. 31, 2020 298
Abdel-Basset[5] has developed a Novel Intelligent Medical Decision Support Model based
on soft computing and IoT as the use of neutrosophical sets for decision-making. In[6] the
researchers developed neutrosophic multi-criterion approach to help healthcare professionals
predict illness. In[7] a solution is proposed to Neutrosophic Linear Fractional Programming
Problem (NLFP) in the case of triangular neutrosophic number costs of the objective function,
capital and engineering coefficients. In[8] the researchers suggest the method to help the patient
and doctor know whether the patient is having a heart failure through neutrophic multi-criteria
decision making (NMCDM).
The neutrosophical topological space theory was proposed in [9]. Further neutrosophic
topological space was studied in [10]. Subsequently, the sets were added similar to the neutro-
sophic open and neutrosophic closed sets. Neutrosophic semi-open set[NSO] and neutrosophic
semi-closed sets[NSC] have been introduced by Iswaraya et.al.[11]. Imran et.al.[12] proposed
neutrosophic semi-α open sets and analysed their basic properties. Arokiarani et.al.[13] stud-
ied about neutrosophic semi-open (resp. pre-open and α-open) functions and examined their
relations. Rao et.al.[14] proposed neutrosophic pre-open sets.
In [15] the researchers investigate new kind of neutrosophic continuity in neutrosophic topo-
logical spaces known as Neutrosophicαgs continuity maps and also the properties and char-
acterization Neutrosophic αgs Irresolute Maps were examined. Anitha et.al.[16] proposed the
concept of NGSR-closed sets and NGSR-open sets. NGSR continuous and NGSR-contra con-
tinuous mappings are also further studied. Dhavaseelan et.al. [17] introduced neutrosophic
almost α-contra-continuous function and studied their properties. In [18] the authors intro-
duced neutrosophic generalized b–closed sets and Neutrosophic generalized b-continuity in
Neutrosophic topological spaces.
Molodstov[19] introduced the soft set theory as a computational method for tackling inse-
curity. Maji[20] combined the concept of soft set and neutrosophic set together by introducing
the current mathematical framework called neutrosophic soft set. In[21] neutrosophic soft
set was applied in making decision. Several researchers[22, 23, 24, 25, 26] applied in various
mathematical systems the concept of neutrosophic soft sets. Bera[27] introduced neurosophic
soft topological spaces. Neutrosophic spatial region as introduced by A.A.Salama[28]. This
paper explores the theory and some of its features of neutrosophic soft topological space. The
notions of neutrosophic soft α-open, neutrosophic soft pre-open, neutrosophic soft semi-open
and neutrosophic soft β-open sets are introduced. Furthermore, for possible application in
GIS, the simple neutrosophic soft region is introduced.
Evanzalin, Jude and Sivaranjani, Introduction to neutrosophic soft topological spatial region
Neutrosophic Sets and Systems, Vol. 31, 2020 299
2. Preliminaries
Definition 2.1. ([19]). (F,E) is a soft set in X where F : E → P(Y ) is a mapping where
P(Y ) is a power set of Y . We express (F,E) by F .˜ F˜ = {(e, F (e)) : e ∈ E}.
Definition 2.2. ([4]). A neutrosophic set(NS) A on Y is defined as: A = {<
y, TA(y), IA(y), FA(y) >: y ∈ Y } where T, I, F : Y −→]−0, 1+[ and −0 ≤ TA(y) + IA(y) +
FA(y) ≤ 3+
Definition 2.3. Let Y be an set and E be parameter set. Let P(Y ) denotes the set of all
neutrosophic soft set(NSS) of Y . Then (F,E) is called a NSS over Y where F : E → P(Y ) is
a mapping. We express the NSS (F,E) by FN .˜
That is, F˜N = {(e, {< y, TF˜N (e)(y), IF˜N (e)(y), FF˜N (e)(y) >: y ∈ Y })e ∈ E}
Definition 2.4. The complement of the NSS F˜N is denoted by (F˜N )
c and is defined by
F˜ cN = {(e, {< y, FF˜N (e)(y), IF˜N (e)(y), TF˜N (e)(y) >: y ∈ Y })e ∈ E}
Definition 2.5. For any two NSS F˜N and G˜N over Y , F˜N is a neutrosophic soft subset of
G˜N if TF˜N (e)
(y) ≤ T
G˜N (e)
(y) ; I
F˜N (e)
(y) ≤ I
G˜N (e)
(y) ; F
F˜N (e)
(y) ≥ F
G˜N (e)
(y); for all e ∈ E and
y ∈ Y .
Definition 2.6. A NSS F˜N over Y is said to be null NSS if TF˜N (e)
(y) = 0 ; I
F˜N (e)
(y) = 0 ;
F
F˜N (e)
(y) = 1; for all e ∈ E and y ∈ Y . It is denoted by Φ˜N .
Definition 2.7. A NSS F˜N over Y is said to be absolute NSS if TF˜N (e)
(y) = 1 ; I
F˜N (e)
(y) = 1
; F
F˜N (e)
(y) = 0; for all e ∈ E and y ∈ Y . It is denoted by Y˜N
Definition 2.8. The union of two NSS F˜N and G˜N is denoted by F˜N ∪ G˜N and is defined
by H˜N = F˜N ∪ G˜N , where the truth-membership, indeterminacy-membership and falsity
membership of H˜N are as follows
T
H˜N (e)
(y) =

T
F˜N (e)
(y) if e ∈ A−B
T
G˜N (e)
(y) if e ∈ B −A
max{T
F˜N (e)
(y), T
G˜N (e)
(y)} if e ∈ A ∩B
I
H˜N (e)
(y) =

I
F˜N (e)
(y) if e ∈ A−B
I
G˜N (e)
(y) if e ∈ B −A
I
F˜N (e)
(y)+I
G˜N (e)
(y)}
2 if e ∈ A ∩B
Evanzalin, Jude and Sivaranjani, Introduction to neutrosophic soft topological spatial region
Neutrosophic Sets and Systems, Vol. 31, 2020 300
F
H˜N (e)
(y) =

F
F˜N (e)
(y) if e ∈ A−B
F
G˜N (e)
(y) if e ∈ B −A
min{F
F˜N (e)
(y), F
G˜N (e)
(y)} if e ∈ A ∩B
Definition 2.9. The intersection of two NSS F˜N and G˜N is denoted by F˜N ∩ G˜N and is
defined by H˜N = F˜N ∩ G˜N , where the truth-membership, indeterminacy-membership and
falsity membership of H˜N are as follows
T
H˜N (e)
(y) = min{T
F˜N (e)
(y), T
G˜N (e)
(y)},
I
H˜N (e)
(y) =
I
F˜N (e)
(y) + I
G˜N (e)
(y)}
2
,
F
H˜N (e)
(y) = may{F
F˜N (e)
(y), F
G˜N (e)
(y)}
3. Neutrosophic soft topological space
Definition 3.1. Let NSS(Y,E) be the family of all NSS over Y and τN ⊂ NSS(Y,E). Then˜
τ˜N is called neutrosophic soft topology(NST) on (Y,E) if the following conditions are satisfied:
(i) Φ˜N , Y˜N ∈ τ˜N
(ii) τ˜N is closed under arbitrary union.
(iii) τ˜N is closed under finite intersection.
Then the triplet (Y, τ˜N , E) is called neutrosophic soft topological space(NSTS). The mem-
bers of τ˜N are called neutrosophic soft open sets in (Y, τ˜N , E). A NSS F˜N in NSS(Y,E) is
soft closed in (Y, τ˜N , E) if its complement (F˜N )
c is neutrosophic soft open set in (Y, τ˜N , E).
The neutrosophic soft closure of F˜N is the NSS, Nscl(F˜N ) = ∩{G˜N : G˜N is neutrosophic
soft closed and F˜N ⊆ G˜N}.
The neutrosophic soft interior of F˜N is the NSS, Nsint(F˜N ) = ∪{O˜N : O˜N is neutrosophic
soft closed and O˜N ⊆ F˜N}.
It is easy to see that F˜N is neutrosophic soft open if and only if F˜N = Nsint(F˜N ) and
neutrosophic soft closed if and only if F˜N = Nscl(F˜N ).
Theorem 3.2. Let (Y, τ˜N , E) be a NSTS over (Y,E) and F˜N and G˜N ∈ NSS(Y,E) then
(i) Nsint(F˜N ) ⊂ F˜N and Nsint(F˜N ) is the largest open set.
(ii) F˜N ⊂ F˜N implies Nsint(F˜N ) ⊂ Nsint(F˜N )
(iii) Nsint(F˜N ) is an neutrosophic soft open set. That is Nsint(F˜N ) ∈ τ˜N
(iv) F˜N is neutrosophic soft open iff Nsint(F˜N ) = F˜N
(v) Nsint(Nsint(F˜N )) = Nsint(F˜N )
(vi) Nsint(Φ˜N ) = Φ˜N and Nsint(Y˜N ) = Y˜N
(vii) Nsint(F˜N ∩ G˜N ) = Nsint(F˜N ) ∩Nsint(G˜N )
Evanzalin, Jude and Sivaranjani, Introduction to neutrosophic soft topological spatial region
Neutrosophic Sets and Systems, Vol. 31, 2020 301
(viii) Nsint(F˜N ) ∪Nsint(G˜N ) ⊂ Nsint(F˜N ∪ G˜N )
Theorem 3.3. Let (Y, τ˜N , E) be a NSTS (Y,E) and F˜N and G˜N ∈ NSS(Y,E) then
(i) F˜N ⊂ Nscl(F˜N ) and Nscl(F˜N ) is the smallest closed set
(ii) F˜N ⊂ F˜N implies Nscl(F˜N ) ⊂ Nscl(F˜N )
(iii) Nscl(F˜N ) is neutrosophic soft closed set. That is Nscl(F˜N ) ∈ (τ˜N )c
(iv) F˜N is neutrosophic soft closed iff Nscl(F˜N ) = F˜N
(v) Nscl(Nscl(F˜N )) = Nscl(F˜N )
(vi) Nscl(Φ˜N ) = Φ˜N and Nscl(Y˜N ) = Y˜N
(vii) Nscl(F˜N ∪ G˜N ) = Nscl(F˜N ) ∪Nscl(G˜N )
(viii) Nscl(F˜N ) ∩Nscl(G˜N ) ⊂ Nscl(F˜N ∩ G˜N )
4. Neutrosophic soft nearly open sets
Definition 4.1. Let (Y, τ˜N , E) be a NSTS and F˜N be a neutrosophic soft open set in (Y,E),
then F˜N is called
(i) Neutrosophic soft α-open iff F˜N ⊆ Nsint(Nscl(Nsint(F˜N )))
(ii) Neutrosophic soft pre-open iff F˜N ⊆ Nsint(Nscl(F˜N ))
(iii) Neutrosophic soft semi-open iff F˜N ⊆ Nscl(Nsint(F˜N ))
(iv) Neutrosophic soft β-open iff F˜N ⊆ Nscl(Nsint(Nscl(F˜N )))
(v) Neutrosophic soft regular-open iff F˜N = Nsint(Nscl(F˜N ))
Definition 4.2. Let (Y, τ˜N , E) be a NSTS and F˜N ∈ NSS(Y,E), then F˜N is called
(i) Neutrosophic soft α-closed iff Nscl(Nsint(Nscl(F˜N ))) ⊆ F˜N
(ii) Neutrosophic soft pre-closed iff Nscl(Nsint(F˜N )) ⊆ F˜N
(iii) Neutrosophic soft semi-clsed iff Nsint(Nscl(F˜N )) ⊆ F˜N
(iv) Neutrosophic soft β-closed iff Nsint(Nscl(Nsint(F˜N ))) ⊆ F˜N
(v) Neutrosophic soft regular-closed iff F˜N = Nscl(Nsint(F˜N ))
5. Neutrosophic soft region
Topological relationships have played a significant role during space search, analysis and
reasoning through Geographical information systems (GIS) and Geospatial databases. The
topological relations between smooth, unstable and fuzzy spatial regions have been developed
on the basis of the nine-intersection model. In the past couple of decades a lot of emphasis has
been given to the topological relationship research issue, particularly between uncertain spa-
tial regions. Nevertheless, formal representation and calculation of topological links between
unknown regions remains an open issue and needs further investigation. We discuss further
Evanzalin, Jude and Sivaranjani, Introduction to neutrosophic soft topological spatial region
Neutrosophic Sets and Systems, Vol. 31, 2020 302
definitions and proposals for a neutrosophic soft topological region, which provide an theo-
retical framework for the modeling of neutrosophic soft topology relations among uncertain
regions.
Definition 5.1. Let (Y, τ˜N , E) be a NSTS over (Y,E) and F˜N ∈ NSS(Y,E). Then neutro-
sophic soft boundary of F˜N is defined by ∂F˜N = Nscl(F˜N ) ∩Nscl((F˜N )c)
Definition 5.2. Let (Y, τ˜N , E) be a NSTS over (Y,E). Then the neutrosophic soft exterior
of F˜N ∈ NSS(Y,E) is denoted by (F˜N )o and is defined by (F˜N )o = Nsint((F˜N )c)
Theorem 5.3. Let F˜N and G˜N be two NSS over (Y,E). Then
(i) (F˜N )o = Nsint((F˜N )
c)
(ii) (F˜N ∪ G˜N )o = (F˜N )o ∩ (G˜N )o
(iii) (F˜N )o ∪ (G˜N )o ⊂ (F˜N ∩ G˜N )o
Theorem 5.4. Let (Y, τ˜N , E) be a NSTS over (Y,E) and F˜N , G˜N ∈ NSS(Y,E). Then
(i) (∂F˜N )
c = Nsint(F˜N ) ∪Nsint((F˜N )c)
(ii) Nscl(F˜N ) = Nsint(F˜N ) ∪ ∂F˜N
(iii) ∂F˜N = Nscl(F˜N ) ∩Nscl((F˜N )c)
(iv) ∂F˜N ∩Nsint(F˜N ) = Φ˜N
(v) ∂(∂(∂(F˜N ))) = ∂(∂(F˜N ))
Definition 5.5. Let (Y, τ˜N , E) be a NSTS over (Y,E). Then a pair of non-empty neutrosophic
soft open sets F˜N , G˜N is called a neutrosophic soft separation of (Y, τ˜N , E) if Y˜N = F˜N ∪ G˜N
and F˜N ∩ G˜N = Φ˜N
Definition 5.6. A NSTS (Y, τ˜N , E) is said to be neutrosophic soft connected if there does not
exist a neutrsophic soft separation of (Y, τ˜N , E). Otherwise (Y, τ˜N , E) is said to be neutrosophic
soft disconnected.
Now we shall describe a model for basic spatial neutrosophic soft region based on neutro-
sophic soft connectedness.
Definition 5.7. Let (Y, τ˜N , E) be a NSTS. A spatial neutrosophic soft region in (Y,E) is a
non empty neutrosophic soft subset F˜N such that
(i) Nsint(F˜N ) is neutrosophic soft connected.
(ii) F˜N = Nscl(Nsint(F˜N ))
6. Conclusion
The neutrosophic soft4-intersection model can be implemented as an application to GIS
for neutrosophic soft topological relationships between neutrosophic soft regions with sharp
Evanzalin, Jude and Sivaranjani, Introduction to neutrosophic soft topological spatial region
Neutrosophic Sets and Systems, Vol. 31, 2020 303
neutrosophical soft boundaries and for neutrosophic soft regions with broad neutrosophical soft
boundaries. These models can be used to formulate spatial database consistency constraints
and can also be used in information systems such as mobile robots and route navigation
systems.
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Evanzalin, Jude and Sivaranjani, Introduction to neutrosophic soft topological spatial region


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