1. Introduction
Zadeh [
1] introduced the notion of fuzzy sets. After that, there have been a number of generalizations of this fundamental concept. The study of fuzzy topological spaces was first initiated by Chang [
2,
3] in 1968. Atanassov [
4] introduced the notion of intuitionistic fuzzy sets (IFs). This notion was extended to intuitionistic
L-fuzzy setting by Atanassov and Stoeva [
5], which currently has the name “intuitionistic
L-topological spaces”. Coker [
6] introduced the notion of intuitionistic fuzzy topological space by using the notion of (IFs). The concept of generalized fuzzy closed set was introduced by Balasubramanian and Sundaram [
7]. In various recent papers, Smarandache generalizes intuitionistic fuzzy sets and different types of sets to neutrosophic sets
. On the non-standard interval, Smarandache, Peide and Lupianez defined the notion of neutrosophic topology [
8,
9,
10]. In addition, Zhang et al. [
11] introduced the notion of an interval neutrosophic set, which is a sample of a neutrosophic set and studied various properties.
Recently, Al-Omeri and Smarandache [
12,
13] introduced and studied a number of the definitions of neutrosophic closed sets, neutrosophic mapping, and obtained several preservation properties and some characterizations about neutrosophic of connectedness and neutrosophic connectedness continuity.
This paper is arranged as follows. In
Section 2, we will recall some notions that will be used throughout this paper. In
Section 3, we mention some notions in order to present neutrosophic generalized pre-closed sets and investigate its basic properties. In
Section 4 and
Section 5, we study the neutrosophic generalized pre-open sets and present some of their properties. In addition, we provide an application of neutrosophic generalized pre-open sets. Finally, the concepts of generalized neutrosophic connected space, generalized neutrosophic compact space and generalized neutrosophic extremally disconnected spaces are introduced and established in
Section 6 and some of their properties in neutrosophic topological spaces are studied.
This class of sets belongs to the important class of neutrosophic generalized open sets which is very useful not only in the deepening of our understanding of some special features of the already well-known notions of neutrosophic topology but also proves useful in neutrosophic multifunction theory in neutrosophic economy and also in neutrosophic control theory. The applications are vast and the researchers in the field are exploring these realms of research.
2. Preliminaries
Definition 1. Let be a non-empty set. A neutrosophic set ( for short) is an object having the form , where , , and the degree of non-membership (namely ), the degree of indeterminacy (namely ), and the degree of membership function (namely ), of each element to the set , see [14]. A neutrosophic set can be identified as in on .
Definition 2. Let be an on . [15] The complement of the set may be defined as follows: - (i)
,
- (ii)
,
- (iii)
.
Neutrosophic sets (
)
and
[
14] in
are introduced as follows:
can be defined as four types:
- (i)
,
- (ii)
,
- (iii)
,
- (iv)
.
2- can be defined as four types:
- (i)
,
- (ii)
,
- (iii)
,
- (iv)
.
Definition 3. Let k be a non-empty set, and generalized neutrosophic sets and be in the form , . Then, we may consider two possible definitions for subsets [14]: - (i)
,
- (ii)
.
Definition 4. Let be an arbitrary family of in . Then,
- (i)
can defined as two types:
,
.
- (ii)
can defined as two types:
,
, see [14].
Definition 5. A neutrosophic topology ( for short) [16] and a non empty set is a family of neutrosophic subsets of satisfying the following axioms: - (i)
,
- (ii)
for any ,
- (iii)
, .
In this case, the pair is called a neutrosophic topological space ( for short) and any neutrosophic set in Γ is known as neutrosophic open set . The elements of Γ are called neutrosophic open sets. A closed neutrosophic set if and only if its is neutrosophic open.
Note that, for any in , we have and .
Definition 6. Let be a neutrosophic open set and a neutrosophic set on a neutrosophic topological space . Then,
- (i)
is called neutrosophic regular open [14] iff . - (ii)
If then B is called neutrosophic regular closed [14] iff .
Definition 7. Let be and an in . Then,
- (i)
- (ii)
, see [14].
It can be also shown that is an and is an in . We have
- (i)
is in iff .
- (ii)
is an in iff .
Definition 8. Let be an and an . Then,
- (i)
Neutrosophic semiopen set [12] if , - (ii)
Neutrosophic preopen set [12] if , - (iii)
Neutrosophic α-open set [12] if - (iv)
Neutrosophic β-open set [12] if
The complement of is an NSOS, NOS, NPOS, and NROS, which is called NSCS, NCS, NPCS, and NRCS, resp.
Definition 9. Let be an and an . Then, the *-neutrosophic closure of ( for short [12]) and *-neutrosophic interior for short [12]) of are defined by - (i)
,
- (ii)
,
- (iii)
,
- (iv)
,
- (v)
,
- (vi)
,
- (vii)
,
- (viii)
,
- (ix)
,
- (x)
.
Definition 10. An of an is called a generalized neutrosophic closed set [17] ( in short) if wherever and is a neutrosophic closed set in . Definition 11. An in an is said to be a neutrosophic α generalized closed set ( [18]) if whensoever and is an in . The complement of an is an in . 3. Neutrosophic Generalized Connected Spaces, Neutrosophic Generalized Compact Spaces and Generalized Neutrosophic Extremally Disconnected Spaces
Definition 12. Let and be any two neutrosophic topological spaces.
- (i)
A function is called generalized neutrosophic continuous( -continuous) of every closed set in is -closed in .
Equivalently, if the inverse image of every open set in is -open in :
- (ii)
A function is called generalized neutrosophic irresolute of every -closed set in is -closed in .
Equivalently of every -open set in is -open in
- (iii)
A function is said to be strongly neutrosophic continuous if is both neutrosophic open and neutrosophic closed in for each neutrosophic set in .
- (iv)
A function is said to be strongly -continuous if the inverse image of every -open set in is neutrosophic open in , see ([17] for more details).
Definition 13. An is said to be neutrosophic- ( in short) space if every in is an in .
Definition 14. Let be any neutrosophic topological space. is said to be generalized neutrosophic disconnected (in shortly -disconnected) if there exists a generalized neutrosophic open and generalized neutrosophic closed set such that and . is said to be generalized neutrosophic connected if it is not generalized neutrosophic disconnected.
Proposition 1. Every -connected space is neutrosophic connected. However, the converse is not true.
Proof. For a -connected space and let not be neutrosophic connected. Hence, there exists a proper neutrosophic set, , such that is both neutrosophic open and neutrosophic closed in . Since every neutrosophic open set is -open and neutrosophic closed set is -closed, is not -connected. Therefore, is neutrosophic connected. □
Example 1. Let . Define the neutrosophic sets and in as follows: , . Then, the family is neutrosophic topology on . It is obvious that is . Now, is neutrosophic connected. However, it is not a -connected for is open and closed in .
Theorem 1. Let be a neutrosophic space; then, is neutrosophic connected iff is -connected.
Proof. Suppose that is not -connected, and there exists a neutrosophic set which is both -open and -closed. Since is neutrosophic , is both neutrosophic open and neutrosophic closed. Hence, is -connected. Conversely, let is -connected. Suppose that is not neutrosophic connected, and there exists a neutrosophic set such that is both and . Since the neutrosophic open set is -open and the neutrosophic closed set is -closed, is not -connected. Hence, is neutrosophic connected. □
Proposition 2. Suppose and are any two . If is -continuous surjection and is -connected, then is neutrosophic connected.
Proof. Suppose that is not neutrosophic connected, such that the neutrosophic set is both neutrosophic open and neutrosophic closed in . Since g is -continuous, is -open and -closed in (. Thus, is not connected. Hence, is neutrosophic connected. □
Definition 15. Let be an . If a family of open sets in satisfies the condition , then it is called a open cover of . A finite subfamily of a open cover of , which is also a open cover of is called a finite subcover of Definition 16. An is called compact iff every open cover of has a finite subcover.
Theorem 2. Let and be any two , and be continuous surjection. If is -compact, hence so is
Proof. Let
be a neutrosophic open cover in
with
Since
g is
continuous,
is
open cover of
. Now,
Since
is
compact, there exists a finite subcover
, such that
Therefore, is neutrosophic compact. □
Definition 17. Let be an and K be a neutrosophic set in . If a family of open sets in satisfies the condition , then it is called a open cover of K. A finite subfamily of a open cover of K, which is also a open cover of K is called a finite subcover of .
Definition 18. An is called compact iff every open cover of K has a finite subcover.
Theorem 3. Let and be any two , and be an continuous function. If K is -compact, then so is in .
Proof. Let
be a neutrosophicopen cover of
in
. That is,
Since
g is
continuous,
is
open cover of K in
. Now,
Since
K is
is
compact, there exists a finite subcover
, such that
Therefore, is neutrosophic compact. □
Proposition 3. Let be a neutrosophic compact space and suppose that K is a -closed set of . Then, K is a neutrosophic compact set.
Proof. Let
be a family of neutrosophic open set in
such that
Since K is -closed, . Since is a neutrosophic compact space, there exists a finite subcover . Now, . Hence, . Therefore, K is a neutrosophic compact set. □
Definition 19. Let be any neutrosophic topological space. is said to be extremally disconnected if neutrosophic open and K is open.
Proposition 4. For any neutrosophic topological space , the following are equivalent:
- (i)
is extremally disconnected.
- (ii)
For each closed set K, is a closed set.
- (iii)
For each open set K, we have .
- (iv)
For each pair of open sets K and M in , , we have .
4. Generalized Neutrosophic Pre-Closed Set
Definition 20. An is said to be a neutrosophic generalized pre-closed set ( in short) in if whensoever and is an in . The family of all of an is defined by .
Example 2. Let and be a neutrosophic topology on , where . Then, the is .
Theorem 4. Every is a , but the converse is not true.
Proof. Let be an in , and is in . Since and is in , . Therefore, is . □
Example 3. Let and be a neutrosophic topology on , where . Then, the is a in but not an .
Theorem 5. Every is , but the converse is not true.
Proof. Let be an in and let and is an in . Now, . Since , . Hence, . Therefore, is . □
Example 4. Let and let is a neutrosophic topology on , where . Then, the is a in but not in since .
Theorem 6. Every is a , but the converse is not true.
Proof. Let be , , be an in . By Definition 6, . This implies and . Therefore, . Hence, is . □
Example 5. Let and be a neutrosophic topology on , where . Then, the is in but not in since .
Definition 21. An is said to be a neutrosophic generalized pre-closed set ( ) in if whensoever and is an in . The family of all of an is defined by .
Proposition 5. Let be a two of an . and are independent.
Example 6. Let , be a neutrosophic topology on , where . Then, the is but not in since but
Example 7. Let , be a neutrosophic topology on , where . Then, the is GNPC but not GNsC in since .
Proposition 6. NSC and GNPC are independent.
Example 8. Let , be a neutrosophic topology on , where . Then, the is an but not in since but .
Example 9. Let , be a neutrosophic topology on , where . Then, the is but not an in since .
The following
Figure 1 shows the implication relations between
set and the other existed ones.
Remark 1. Let be a two of an . Then, the union of any two is not a in general—see the following example.
Example 10. Let be a neutrosophic topology set on , where , . Then, is neutrosophic topology on and the , are but is not a GNPC in .
5. Generalized Neutrosophic Pre-Open Sets
In this section, we present generalized neutrosophic pre-open sets and investigate some of their properties.
Definition 22. An is said to be a generalized neutrosophic pre-open set ( ) in if the complement is a in . The family of all of is denoted by .
Example 11. Let and be a neutrosophic topology on , where . Then, the is .
Theorem 7. Let be an . Then, for every and for every , implies .
Proof. By Theorem . Let and be . Since . However, is a , . In addition, (by theorem). Therefore, . Hence, is . This implies that is a of . □
Remark 2. Let be two of an . The intersection of any two is not a in general.
Example 12. Let and be a neutrosophic topology on , where . Then, the , and is , but is not .
Theorem 8. For any an , the following hold:
- (i)
Every is ,
- (ii)
Every is ,
- (iii)
Every is ,
- (iv)
Every is .
Proof. The proof is clear, so it has been omitted. □
The converses are not true in general.
Example 13. Let and . Then, is a neutrosophic topology on , an is an in but not an .
Example 14. Let and be neutrosophic topology on , where . Then, an is but not an .
Example 15. Let and be a neutrosophic topology on , where . Then, an is but not an .
Example 16. Let and be a neutrosophic topology on , where . Then, an is but not an .
Theorem 9. Let be an . If , then whensoever and is an in .
Proof. Let . Then, is in . Therefore, whensoever and is an in . That is, . This implies whensoever and is in . Replacing , by , we get whensoever and is an in . □
Theorem 10. For , is an and in if and only if is an in .
Proof. ⟹ Let be an and a in . Then, . This implies . Since is an , it is an . Hence, . Therefore, . Hence, is an in .
⟸ Let be an in . Therefore, . Let and be an in . This implies . Hence, is in . □
Theorem 11. An of an is a iff , whensoever H is an and .
Proof. ⟹ Let be in . Let H be an and . Then, is an in such that . Since is , we have . Hence, . Therefore, .
⟸ Suppose is an of and let whensoever H is an and . Then, and is an . By assumption, , which implies . Therefore, is of . Hence, is a of . □
Corollary 1. An of an is iff , whensoever H is an and .
Proof. ⟹ Let is a in . Let H be an and . Then, is an in such that . Since is , we have . Therefore, . Hence, . This implies .
⟸ Suppose be an of and , whensoever H is an and . Then, and is an . By assumption, . Hence, . This implies . Hence, is a of . □
6. Applications of Generalized Neutrosophic Pre-Closed Sets
Definition 23. An is said to be neutrosophic- ( in short) space if every in is an .
Definition 24. An is said to be neutrosophic- ( in short) space if every in is an .
Theorem 12. Every space is an space.
Proof. Let be an space and be . By assumption, is in . Since every is an , is an in . Hence, is an space. □
The converse is not true.
Example 17. Let , and . Then, is an space, but it is not since an is but not an .
Theorem 13. Let be an and is an space; then,
- (i)
the union of is ,
- (ii)
the intersection of is .
Proof. (i) Let be a collection of in an space . Thus, every is an . However, the union of an is an . Therefore, the Union of is in .
(ii) Proved by taking complement in (i). □
Theorem 14. An is an space iff .
Proof. ⟹ Let be a in ; then, is in . By assumption, is an in . Thus, is in . Hence, .
⟸ Let be . Then, is in . By assumption, is an in . Thus, is an . Therefore, is an space. □
Theorem 15. For an the following are equivalent:
- (i)
is a neutrosophic pre- space.
- (ii)
Every non-empty set of is either an or .
Proof. . Suppose that is a neutrosophic pre- space. Suppose that is not an for some . Then, is not an and hence is the only an containing . Hence, is an in . Since is a neutrosophic pre- space, then is an or equivalently is an . . Let every singleton set of be either or . Let be an of . Let . We show that in two cases.
Case (i): Suppose that is . If , then . Now, contains a non—empty . Since is , by Theorem 7, we arrived to a contradiction. Hence, .
Case (ii): Let be . Since , then . Thus, . Thus, in any case . Thus, . Hence, or equivalently is an . Thus, every is an . Therefore, is neutrosophic pre- space. □