1. Introduction
Neutrosophic set theory has a very powerful influence given that is a recent section of philosophy that is presented as the study of origin, nature and scope of neutralities. The idea of neutrosophy is initiated by Smarandache [
1] in 1999 as a new mathematical approach that corresponds to degree of indeterminacy (uncertainty, etc.). Moreover, the soft set theory was successfully applied to several directions, such as smoothness of functions and architecture-based neuro-linguistic programming (NLP) in the papers of Bakbak et al. [
2] and Mishra et al. [
3]. The concept of continuous mappings plays a crucial role in many branches of mathematics, such as, fuzzy set theory, algebra and quantum gravity (see [
4]). El-Naschie also has shown that both string theory and
theory are kind of some applications in quantum particle physics especially in relation to heterotic strings and were influenced by the fuzzy topology in
ostak sense. [
5].
In current times, the theory of neutrosophy has been recycled at various junctions of mathematics. More precisely, this theory has made an exceptional advancement in the field of topological spaces. Salama et al. [
6,
7,
8] dispatched their works of neutrosophic topological spaces, following the method of Chang [
9] in the situation of fuzzy topological spaces
. Afterward, Hur et al. [
10,
11] presented NSet(H) and NCSet. Smarandache [
12] defined the idea of neutrosophic topology on the non-standard interval. One can simply detect that the fuzzy topology familiarized by Chang is a crisp group of fuzzy subsets.
Šostak [
13] determined that Chang’s style is crisp in nature and so he redefined the idea of fuzzy topology, frequently mentioned as smooth fuzzy topology, as a mapping from the group of all fuzzy subsets of
to
. Fang Jin-ming et al. and Zahran et al. [
14,
15] discussed the notion of foundation as a function from an appropriate collection of fuzzy subsets of
X to
. Saber et al. [
16] found a parallel theory in the context fuzzy ideal topological space.
Wang [
17], in 2010, established the idea of a single-valued neutrosophic set. In 2016, Gayyar [
18] presented the notion of fuzzy neutrosophic topological spaces in a Šostak sense. The concept of the foundation for an ordinary single-valued neutrosophic topology was explored by Kim [
19]. Several authors [
20,
21,
22,
23,
24,
25] posted their efforts for the idea of single-valued neutrosophic topological spaces
. Others focusing their works on single valued neutrosophic relations, see [
26,
27]. Last but not least, in the sense that not only the objects are fuzzified, but also the axiomatics, the single-valued neutrsophic ideal theory was introduced in 1985 by Šostak [
13] as a generalization of classical topological structures and as an extension of both crisp topology and Changs fuzzy topology.
In this article, preliminaries of single-value neutrosophic sets and single-valued neutrosophic topology are reviewed in
Section 2. In
Section 3, we define the notions of a single-valued neutrosophic semi-closure space. Some of their characteristic properties are considered. Further, we present and explore the properties and characterizations of the single-valued neutrosophic operators, namely
-closure
and
-interior
in the single-valued neutrosophic ideal topological space
. The concepts of single-valued neutrosophic (almost, faintly, weakly)
-continuous mappings are introduced and studied in
Section 4. In
Section 5, we introduce a new improved single-valued neutrosophic lower and single-valued neutrosophic upper sets by which we obtain a more reliable single-valued neutrosophic boundary region set of a single-valued neutrosophic set
. From these single-valued neutrosophic lower and fuzzy upper sets, we define new single-valued neutrosophic interior and single-valued neutrosophic closure operators associated with a specific single-valued neutrosophic set
.
2. Preliminaries
This section is devoted to bring a complete survey, some previous studies and important related notions to this work. Let us have a fixed universe to be a finite set of objects and a closed unit interval . We will also let to denote the set of all single-valued neutrosophic subsets of .
Definition 1 ([
12]).
Let be a non-empty set. A neutrosophic set (briefly, ) in is an object having the formwhereandrepresent the degree of membership (), the degree of indeterminacy () and the degree of non-membership (), respectively, of any to the set . Definition 2 ([
17]).
Suppose that is a universal set a space of points (objects), with a generic element in denoted by υ. Then, is called a single-valued neutrosophic set (briefly, ) in , if has the formNow, indicate the degree of non-membership, the degree of indeterminacy and the degree of membership, respectively, of any element to the set .
Definition 3 ([
17]).
Let be an SVNS on . The complement of the set (briefly ) defined as follows: Definition 4 ([
9]).
Let be a non-empty set, be in the form: and on then,(a) for every; (b) iffand.
(c) and.
Definition 5 ([
26]).
Let . Then,(a) is an SVNS, if for every ,where, and , for all , (b) is an SVNS, if for every , Definition 6 ([
6]).
For any arbitrary family of SVNS the union and intersection are given by (a) ,
(b) .
Definition 7 ([
18]).
A single-valued neutrosophic topological spaces is an ordered where is a mapping satisfying the following axioms: (SVNT1) and .
(SVNT2) , ,
, for every ,
(SVNT3) , ,
, for every .
The quadruple is called a single-valued neutrosophic topological space (briefly, , for short). Occasionally we write for and it will cause no ambiguity.
Definition 8 ([
21]).
Let be an SVNTS. Then, for every and . Then the single-valued neutrosophic closure and single-valued neutrosophic interior of are defined by: Definition 9 ([
24,
25]).
Let be an and , . Then,- (1)
is said to be r-single-valued neutrosophic semi-open (briefly, r-SVNSO) iff - (2)
is said to be r-single-valued neutrosophic β-open (briefly, r-SVNβO) iff - (3)
is said to be r-single-valued neutrosophic regular open (briefly, r-SVNRO) iff
The complement of r-SVNSO (resp, r-SVNO) are said to be r-SVNSC(resp, r-SVNC)), respectively.
Definition 10 ([
21]).
Let be a non-empty set and , let , and , then the single-valued neutrosophic point in given byWe say that, iff , and . We indicate the set of all single-valued neutrosophic points in as . A single-valued neutrosophic set is said to be quasi-coincident with another single-valued neutrosophic set , denoted by , if there exists an element such that Definition 11 ([
21]).
A mapping is called single-valued neutrosophic ideal () on if it satisfies the following conditions: () and .
() If then ,
and , for every .
() ,
and , for every .
The triable is called a single-valued neutrosophic ideal topological space in the Šostak sense (briefly, ).
Definition 12 ([
21]).
Let be an SVNITS for each . Then, the single-valued neutrosophic ideal open local function of is the union of all single-valued neutrosophic points such that if and , , , then there is at least one for which Occasionally, we will write for and it will have no ambiguity.
Remark 1 ([
21]).
Let be an SVNITS and , we can defineClearly, is a single-valued neutrosophic closure operator and is the single-valued neutrosophic topology generated by , i.e., Definition 13 (25). An SVNS δ in is called a single-valued neutrosophic relation (SVNR) in , denoted by , where , and denote the truth-membership function, indeterminacy membership function and falsity-membership function of δ, respectively. In what follows, SVNR() will denote the family of all single-valued neutrosophic relations in .
3. Single-Valued Neutrosophic Semi-Closure Spaces in Šostak Sense
We begin this section by defining the notion of single-valued neutrosophic semi-closure space. Some of its characteristic properties are considered. Further, we present and explore the properties and characterizations of the single-valued neutrosophic operators, namely -closure and -interior in the single-valued neutrosophic ideal topological space .
Definition 14. A mapping is called a single-valued neutrosophic semi-closure operator on if, for every and , the following axioms are satisfied:
() ,
() ,
() ,
() if ,
() .
The pair is a single-valued neutrosophic semi-closure space ().
If and are single-valued neutrosophic closure operators on . Then, is finer than , denoted by iff , for every and .
Theorem 1. Let be an SVNTS. Then, for any and , we define an operator as follows: Then, is an .
Proof. Suppose that is an . Then, , and () follows directly from the definition of .
(
) Since
we obtain
and
, therefore,
Let
be an
. From (
), we have
It implies that
and
Hence,
; therefore,
(
) Suppose that there exists
,
and
such that
By the definition of
, there exists an
with
and
that is
r-SVNSC such that
Since
and
is
r-SVNSC, by the definition of
, we have
It is a contradiction. Thus, . Hence, is a single-valued neutrosophic semi-closure operator on . □
Theorem 2. Let be an SVNSCS and . Define the mapping on by Then,
- (1)
is an SVNTS on ;
- (2)
is finer than .
Proof. (SVNT1) Let be an . Since and for every ,
(SVNT2) Let
be an
. Suppose that there exists
such that
There exists
such that
For each
, there exists
with
such that
In addition, since
by
and
of Definition 13, for any
,
It follows that , and . It is a contradiction. Thus, for every , , and .
(SVNT3) Suppose that there exists
such that
There exists
such that
For every
, there exists
and
such that
In addition, since
, by
of Definition 13,
It implies, for all
,
Thus, , that is, , and . It is a contradiction. Hence, is an on .
Since
,
From of Definition 9, we have . Thus, is finer than . □
Example 1. Let . Define as follows: We define the mapping as follows: Then, is a single-valued neutrosophic closure operator.
From Theorem 2, we have a single-valued neutrosophic topology on as follows: Thus, the is a single-valued neutrosophic topology on .
Theorem 3. Let be an SVNTS. Then, for any and , we define an operator as follows: For each and the operator satisfies the following conditions:
- (1)
,
- (2)
,
- (3)
,
- (4)
if ,
- (5)
.
Proof. From the Definition of , the proof can be performed □
Definition 15. Let be an SVNTS, , and . Then,
- (1)
is called r-single-valued neutrosophic -neighborhood of if with - (2)
is called r-single-valued neutrosophic θ-cluster point (r--cluster point) of if for any , we have ,
- (3)
r-θ-closure operator is a mapping defined as: - (4)
is said to be r--closed iff . We define
Theorem 4. Let be an SVNTS. For and . The following properties hold:
- (1)
If ,
- (2)
If , then
- (3)
, , , }.
- (4)
,
- (5)
is r--closed,
- (6)
.
Proof. (1) and (2) are easily proved from Definition 14.
(3)
,
,
,
}. Suppose that
, then there exists
and
such that
Then
is not
r-
θ-cluster point of
. So, there exists
, and
. Thus,
and
,
,
. Hence,
It is a contradiction for Equation (
1). Thus
.
Suppose that
, then there exists
r-
θ-cluster point of
of
such that
By definition of
, there exists
with
and
,
,
such that
Then
. Furthermore,
which implies
. Hence
is not an
r-
θ-cluster point of
. It is a contradiction for Equation (
2). Thus
.
(4) Let
for each
. Then
So, . Hence, .
(5) It is directly obtained from (4).
(6) Since , by (5), we have . □
Definition 16. Let be an SVNITS and , . Then, is said to be r-single-valued neutrosophic -open (briefly, r-SVNO) iff . The complement of r-SVNβO is said to be r-SVNβC.
Remark 2. Let be an SVNITS. For , and . Then, is called r-open -neighborhood of if with is r-SVNO set, denoted as: Definition 17. Let be an SVNITS. For each and , we define the operators as follows: Theorem 5. Let be an SVNITS and . Then,
- (1)
,
- (2)
,
- (3)
,
- (4)
,
- (5)
,
- (6)
isr-SVNCiff ,
- (7)
,
- (8)
.
Proof. (1), (2), (3), (4), (5) and (6) are easily proved from the definitions of and .
(7) From (4) we only show
. Suppose that
there exist
and
such that
Since
and
by definition
there exists
r-SVNC set
with
such that
Since
,
Again, by the definition of
we have
. Hence
It is a contradiction for Equation (
1).
Hence,
is an
r-SVNO set contained in
and so
. Since
is an
r-SVNO set, we have
So,
. Hence
□
4. New Terms of Single-Valued Neutrosophic Continuity
In this section, we introduce and characterize new classes of mappings called single-valued neutrosophic almost , faintly , weakly and -continuous mappings. These findings lead to many theorems and consequences. Using these different attributes, we provide an example at the end of this section to show the difference between these kinds of mappings.
Definition 18. Let be a mapping. Then, f is called single-valued neutrosophic almost -continuous mapping (SVNAC, for short) iff for each there exists such that .
Lemma 1. For any single-valued neutrosophic set in an SVNTS and , if , then .
Lemma 2. For any single-valued neutrosophic set in an SVNTS and , if is r-SVNβO, then .
Theorem 6. Let be a mapping for each and . Then the following statements are equivalent:
- (1)
f is SVNAC,
- (2)
For every ,
- (3)
for every , , .
Proof. (1)⇒(2): Let
. Then by (1), there exists
such that
. Since,
,
,
. by Lemma 1,
. Hence,
. Since
is
r-SVNO set,
Since, we obtain .
(2)⇒(3): Let
and
,
,
, by (2), we have
Since
we obtain
Hence, by Theorem 5(8),
Thus,
implies
. Hence
(3)⇒(1): Let
. and since
. Then by (3),
. Since,
is
r-SVNO set,
. Moreover,
by Lemma 1,
. □
Theorem 7. Let be a mapping for each , and . Then the following statements are equivalent:
- (1)
f is SVNAC,
- (2)
is an r-SVNO set in , for every , , ,
- (3)
is an r-SVNC set in , for each , , ,
- (4)
is an r-SVNO set in , for each r-SVNRO set ,
- (5)
is an r-SVNC set in , for each r-SVNRC set ,
- (6)
For each there exists such that ,
- (7)
for each , , ,
- (8)
for each , , ,
- (9)
for each , , ,
- (10)
for each r-SVNβO set ,
- (11)
for each r-SVNSO set .
Proof. (1)⇒(2): Let
and
is
r-SVNRO set in
. Then
. Since
f is
SVNAC, then there exists
such that
. So,
Thus,
implies
, Hence,
Therefore is an r-SVNO set in .
(2)⇒(3): Let , , . Then, by (2), is r-SVNO set. This yields to be an r-SVNC set in .
(3)⇒(4): Let be an r-SVNRO set in . Then, , , . From (3), we have is r-SVNC set. Hence, is r-SVNO set in .
(4)⇒(5): It is easily proved from (4) and the fact that .
(5)⇒(6): Let
. Then
and
is
r-SVNRO, which implies that
is
r-SVNRC. By (5),
is an
r-SVNC set. Then,
is
r-SVNO. Put
. Then
and
Since
,
,
and by Lemma 1, we have
(6)⇒(7): Let
and
,
,
. Then,
by (6), there exists
such that
. Thus,
Since
is
r-SVNO set,
Thus,
implies
. Hence,
(7)⇒(8): Let
,
,
. Then by (7),
Then, .
(8)⇒(9): Since
,
,
, by (8) and Lemma 1, we have
(9)⇒(10): Let
be an
r-SVNβO set in
Then by Lemma 2,
and hence
,
,
by (9), we have
Since
we obtain
(10)⇒(11): It is easily proved from Definition 9.
(11)⇒(1): Obvious. □
Theorem 8. Let be a mapping for each , and . Then the following statements are equivalent:
- (1)
f is SVNAC,
- (2)
,
- (3)
, for every , , ,
- (4)
, for every , , ,
- (5)
, for every , , ,
- (6)
, for every , , ,
- (7)
, for every , , .
Proof. (1)⇒(2): Let
and
Then,
and since
,
,
. Then,
, By
SVNAC of
, there exists
such that
It implies that
. Since
is
SVNAO set in
Thus,
implies
. Hence,
(2)⇒(3): It is trivial.
(3)⇒(4): Since
,
,
, we have
By (3), we have
(4)⇒(5): Since
,
,
, we have
,
,
. and by Lemma 1, we have
. From (4), we have
It implies that .
(5)⇒(6): Let
,
,
. Then by (5), we have
(6)⇒(7): It is easily proved from Lemma 1.
(7)⇒(1): Let , then and , , . From (7), . Since , , , and By Theorem 6(2), we have f is SVNAC. □
Definition 19. Let be a mapping. Then,
- (1)
f is called single-valued neutrosophic faintly -continuous (SVNFC, for short) iff for every , there exists such that ,
- (2)
f is called single-valued neutrosophic weakly -continuous (SVNWC, for short) iff for every , there exists such that ,
- (3)
f is called single-valued neutrosophic -continuous (SVNC, for short) iff is r-SVNO, for every , , .
Remark 3. From the above definition we obtain the following diagram: Some supporting examples will be shown after the following two theorems.
Theorem 9. Let be a mapping. Then the following statements are equivalent:
- (1)
f is SVNWC,
- (2)
, for each ,
- (3)
for each ,
- (4)
is r-SVNC set in for each r-θ-closed set,
- (5)
isr-SVNO set in for each r-θ-open set.
Proof. (1)⇒(2) Suppose there exists
and
such that
. Then there exists
and
such that
If
, provides a contradiction that
. If
, there exists
such that
Since
,
and
. Then,
is not
r-
θ-cluster point of
, there exists
such that
. By
SVNWC of
f, there exists
such that
. Thus,
implies
. Hence
It is a contradiction for Equation (
2).
(2)⇒(3), (3)⇒(4) and (4)⇒(5): are obvious.
(5)⇒(1): Let . Then and is r- -open set. By (5), we have is r-SVNO set in . Since we obtain and hence from . Then, f is SVNWC. □
Theorem 10. A mapping is SVNFC iff for eachr--closed set is r-SVNC.
Example 2. Let . Define as follows: We define the mapping as follows: From Theorems 4 and 5, we obtain as follows: By Theorem 9(2), the identity mapping is SVNWC but is not SVNAC, because by Theorem 8(5), for each , , and , 5. Single-Valued Neutrosophic Approximation Space
In this section, and for symmetrical purposes, we establish the definition of the single-valued neutrosophic upper, single-valued neutrosophic lower and single-valued neutrosophic boundary sets of a rough single-valued neutrosophic set in a single-valued neutrosophic approximation space . Based on and , we introduce the single-valued neutrosophic approximation interior operator and the single-valued neutrosophic approximation closure operator .
Definition 20. Assume that an SVNR δ is defined so that for every , , , for every and , and for every . That is, δ is a single-valued neutrosophic equivalence relation on . Then is called a single-valued neutrosophic approximation space based on the single-valued neutrosophic equivalence relation (briefly, SVN-equivalence relation) δ on .
Definition 21. For each , define a single-valued neutrosophic coset by: All elements with SVNR value are elements having a membership value in the single-valued neutrosophic coset , and any element with is not included in the single-valued neutrosophic coset . Any single-valued neutrosophic coset surely include the element , and consequently Further,such that . Clearly, if , then the single-valued neutrosophic cosets (as SVNSs) are containing the same elements of with some non-zero membership values, and moreover, if and , then it must be that and whenever . That is, any two single-valued neutrosophic cosets are either two single-valued neutrosophic sets containing the same elements of with some non-zero membership values or containing completely different elements of with some non-zero membership values. Definition 22. Let and δ be SVN-equivalence relation on and the single-valued neutrosophic cosets. Then, the single-valued neutrosophic lower set (briefly, SVN-lower) , the single-valued neutrosophic upper set (briefly, SVN-upper) and the single-valued neutrosophic boundary region set (briefly, SVN-boundary region) are defined as follows: for , , and are then called SVN-lower, SVN-upper and SVN-boundary region sets associated with the SVNS and based on the SVN-equivalence relation δ in a single-valued neutrosophic approximation space .
From (5) and (6), we obtain that , and for each . Whenever so that , and we obtain that , and , and then from (7), we obtain , and . Otherwise, , and .
Theorem 11. For any SVNS we find that
- (1)
and ,
- (2)
,
- (3)
,
- (4)
, implies that and ,
- (5)
,
- (6)
,
- (7)
and ,
- (8)
,
- (9)
.
Example 3. Let δ be an SVNR on a set as shown below.
| | | |
| | | |
| | | |
| | | |
Assume that . Then, the single-valued neutrosophic cosets are as follows: Hence, and Hence, . Similarly, we can obtain ; therefore, , and then .
Definition 23. The single-valued neutrosophic approximation interior operator is defined as follows: That is associated with an SVNS in a single-valued neutrosophic approximation space .
Theorem 12. The following conditions are satisfied
- (1)
,
- (2)
,
- (3)
⇒ , ,
- (4)
,
- (5)
.
Proof. For (1): .
For (2): .
For (3): then ⇒.
For (4): .
For (5): Similarly to (4). □
Thus, this is called a single-valued neutrosophic interior associated with
in the single-valued neutrosophic approximation space
generating a single-valued neutrosophic topology defined by:
Definition 24. The single-valued neutrosophic approximation closure operator is defined as follows: Theorem 13. The single-valued neutrosophic approximation closure operator satisfies the following conditions:
- (1)
,
- (2)
,
- (3)
⇒, ,
- (4)
and ,
- (5)
.
Proof. Similar to the proof of Theorem 12. □