3. Topology of Hesitant Fuzzy Sets
A fuzzy topology is a concept that combines order structure and topology. Chang [
2] studied fuzzy topological spaces in the sense of point-like structure. Ekici [
3] showed that the concept of fuzzy topology may be relevant to quantum particle physics in connection with string theory. Silva and Landim [
4] showed that the notion of a fuzzy space topology can be used to explain the origin of the black hole entropy. Since a hesitant fuzzy topology is a more flexible variation of a fuzzy topology, we expect that a hesitant fuzzy topology may be more effectively applicable.
In this section, we introduce the concepts of hesitant fuzzy topologies, bases, and sub-bases in hesitant fuzzy topological spaces, find some of their basic properties, and give examples. One can easily see that our definition of a hesitant fuzzy topology is different from the one introduced by Mathew et al. [
25]. Moreover, in order to apply hesitant fuzzy sets to a topology, we refer mainly to [
27,
28].
Definition 8. Let X be a nonempty set, and let . Then, τ is called a hesitant topology () on X, if it satisfies the following axioms:
(HFT1),
(HFT2), for any ,
(HFT3), for each .
Especially, the pair is called a hesitant fuzzy topological space. Each member of τ is called a hesitant fuzzy open set () in X. A hesitant fuzzy set h in X is called a hesitant fuzzy closed set () in , if .
We will denote the set of all HFTson X (resp. HFOSsand HFCSsin X) as (resp. and )
Example 1. (1) Let , and consider hesitant fuzzy sets in X given by:
Then, we can easily see that .
(2) Let X be a nonempty set. Then, clearly, the class . In particular, will be called a hesitant fuzzy indiscrete topology on X and denoted by . The pair is called a hesitant fuzzy indiscrete space.
(3) Let X be a nonempty set. Then, we can easily see that the class is a hesitant fuzzy topology on X. In particular, will be called a hesitant fuzzy discrete topology on X and denoted by . The pair is called a hesitant fuzzy discrete space.
From Definition 8, we have the following result.
Proposition 1. Let be a hesitant fuzzy topological space. Then:
(1) ,
(2) , for any ,
(3) , for each .
Proposition 2. If , then .
Proof. Let . Then, clearly, , for each . Thus, . Therefore, . Hence, satisfies Axiom (HFT1).
Let . Then, , for each . Thus, , for each . Therefore, , i.e., . Hence, satisfies the axiom (HFT2).
Now, let . Then, , for each . Therefore, , i.e., . Hence, satisfies Axiom (HFT3). This completes the proof. □
For any , we say that is weaker or coarser than , if . In this case, we say that is stronger or finer than .
The following is the immediate result of Example 1 and Proposition 2.
Proposition 3. Let . Then, it forms a complete lattice with respect to the set inclusion relation of which is the smallest element and is the largest element.
Definition 9. Let be a hesitant fuzzy topological space, and let . Then, is called a base for τ, if for each , , or there is such that .
Example 2. (1) Let X be a nonempty set. Then, is a base for .
(2) Let , and consider ,
where ,
.
Suppose is a base for an HFT τ on X. Then, by the definition of a base, . Since , . It is clear that . However, for any , . Thus, by Definition 9, is not a base for some hesitant fuzzy topology on X.
Theorem 1. Let X be a nonempty set. Then, is a base for some HFT on X if and only if it satisfies the following conditions:
(1)
(2)
if and , then there is such that: Proof. Suppose is a base for some HFT on X. Since , it is clear that . Then, the condition (1) is satisfied.
Now, suppose
and
. Then, clearly,
. Thus,
and
. Since
is a base for
, there is
such that
. Since
, there is
such that:
Therefore, Condition (2) is satisfied.
Conversely, suppose the necessary conditions hold, and let:
Then, clearly, . Furthermore, we can easily see that is closed under arbitrary union. Thus, satisfies Axioms (HFT1) and (HFT3).
Let , and let . Then, by the definition of , there are such that and . Thus, by Condition (2), there is such that . Therefore, , for some . Hence, . Therefore, satisfies Axiom (HFT2). This completes the proof. □
Definition 10. If is a base for an HFT τ on X, then τ is called the hesitant fuzzy topology generated by . In fact, Example 3. Let , and consider ,
where ,
,
Then, clearly, . Furthermore, we can easily show that satisfies Condition (2)
of Theorem 1. Thus, is a base for some HFT τ on X. In fact, Definition 11. Let , and let and be bases for and , respectively. Then, and are said to be equivalent, if .
Theorem 2. Let , and let and be bases for and , respectively. Then, the following are equivalent:
(1) is finer than , i.e., ,
(2) for each and each such that , there is such that .
Proof. (1)⇒(2): Suppose ; let , and let such that . Then, clearly, . Since is generated by and , there is such that .
(2)⇒(1): Suppose the necessary condition holds; let , and let . Since is generated by , there is such that . Then, by Condition (2), there is such that . Thus, . Therefore, , for some . Hence, . Therefore, . □
Proposition 4. Let be a hesitant fuzzy topological space. Assume such that for each and each with , there is such that . Then, is a base for τ.
Proof. Let . Since , there is such that . Then, .
Suppose and . Since , . Then, there is such that . Thus, by Theorem 1, is a base for some HFT on X. By Theorem 2, is finer than . Since , arbitrary unions of members of are members of . Therefore, . Hence, . Therefore, is a base for . □
Definition 12. Let be a hesitant fuzzy topological space, and let . Then, is called a sub-base for τ, if the family: is a base for τ.
Proposition 5. Let X be a nonempty set, and let such that . Then, there is a unique HFT τ such that is a sub-base for τ.
Proof. Let
, and let:
Since , . By the definition of , . Then, satisfies Axiom (HFT1).
Let . Then, for each , there is such that . Thus, . Therefore, . Hence, satisfies Axiom (HFT3).
Now, let , and let . Then, there are such that , and . Since and are finite intersection of members of , respectively, . Thus, , for some . Therefore, . Hence, satisfies Axiom (HFT2). Therefore, . Furthermore, we can easily show that is unique. □
Example 4. Let , and consider ,
where ,
,
Then, clearly, . Thus, is a sub-base for the HFT τ on X. Furthermore, is a base for τ,
where ,
,
Therefore,
where
,
.
4. Hesitant Fuzzy Neighborhoods, Interiors, and Closures
It is well known that a neighborhood system generates a topology in a classical topological space. Then, the definition of a hesitant fuzzy neighborhood is necessary.
In this section, we define a hesitant fuzzy neighborhood of a hesitant fuzzy point, a hesitant fuzzy Q-neighborhood of a hesitant fuzzy quasi-coincident point, the hesitant fuzzy closure, interior, and exterior, study some of their properties, and give some examples.
Definition 13. Let be a hesitant fuzzy topological space; let ; and let . Then, N is called a hesitant fuzzy neighborhood () of , if there is such that .
We will denote the set of all HFNs of in as .
The following is the immediate result of Definition 13.
Theorem 3. Let be a hesitant fuzzy topological space, and let . Then, if and only if it is an HFN of each of its HFP, i.e., for each , .
From the following result, we can see that the hesitant fuzzy neighborhoods have the property of the classical neighborhoods.
Proposition 6. Let be a hesitant fuzzy topological space, and let . Then, has the following properties:
(HFN1) , for each ,
(HFN2) if , then , for each such that ,
(HFN3) if , then ,
(HFN4) if , then there is such that , for each .
Proof. (1) The proof is clear.
(2) Suppose , and let . Then, there is such that . Thus, . Therefore, .
(3) Suppose . Then, there are such that and . Thus, and . Therefore, .
(4) Suppose . Then, there is such that . Since , by Theorem 3, , for each . Thus, for each , there is such that . Therefore, . This completes the proof. □
Proposition 7. Let X be a nonempty set, and let for each there be a nonempty collection of hesitant fuzzy sets in X satisfying Properties (HFN1)–(HFN4). Then, there is a unique HFT τ on X such that for each , , where denotes the set of all HFNs of in .
Proof. Let . Then, clearly, . By (HFN2), . Thus, satisfies Axiom (HFT1).
Let , and let . Then, and . Thus, . By (HFN3), . Therefore, . Hence, satisfies Axiom (HFT2).
Let , and let . Then, there is such that . Since , Thus, by (HFN2), Therefore, . Hence, satisfies Axiom (HFT3). Therefore,
Now, we will show that . Let . Then, there is such that . Thus, by the definition of , , for each . Therefore, by (HFN2), . Hence, Suppose , and let . Then, clearly, and . By (HFN1), we can easily see that . Let us show that , i.e., , for each . Let . By the definition of U, . Since , Then, by (HFN4), there is such that , for each Thus, by the definition of U, Therefore, . Hence, by (HFN2), This completes the proof. □
Definition 14. Let X be a nonempty set; let ; and let . Then, is said to:
(i) be quasi-coincident with h, denoted by , if ,
(ii) be not quasi-coincident with h, denoted by , if .
Definition 15. Let X be a nonempty set, and let . Then, is said to be quasi-coincident with , denoted by , if there is such that .
In this case, we say that and are quasi-coincident (with each other) .
It is obvious that if and are quasi-coincident at , then .
Theorem 4. Let X be a nonempty set; let ; and let . Then, if and only if . In particular, if and only if .
Proof. The proof follows from the fact: for each
,
□
Theorem 5. Let X be a nonempty set; let ; and let . Then, if and only if there is such that .
Proof. Suppose . Then, . Thus, there is such that .
Conversely, suppose there is such that . Then, and Thus, . □
Proposition 8. Let be nonempty sets; let ; let ; and let be a mapping. If , then .
Proof. Suppose
. Then,
and
. Thus,
Therefore, . □
Definition 16. Let be a hesitant fuzzy topological space; let ; and let . Then, N is called an -neighborhood () of , if there is a such that . The set of all -neighborhoods of is called the system of of -neighborhoods of and denoted by .
Proposition 9. Let be a hesitant fuzzy topological space, and let . Then, has the following properties:
(HQN1) , for each ,
(HQN2) if , then , for each such that ,
(HQN3) if , then ,
(HQN4) if , then there is such that , for each .
Proof. The proofs are similar to Proposition 6. □
Proposition 10. Let X be a nonempty set, and let for each there be a nonempty collection of hesitant fuzzy sets in X satisfying Properties (HQN1)–(HQN4). Then, there is a unique HFT τ on X such that for each , , where denotes the set of all HQNs of in .
Proof. The proof is similar to Proposition 7. □
Theorem 6. Let be a hesitant fuzzy topological space, and let . Then, is a base for τ if and only if for each and each such that , there is such that .
Proof. The necessary condition follows directly from the definition of a base and Proposition 8. Let us show that the sufficient condition. Assume that is not a base for . Then, there is such that . Thus, there is such that . Let . Then, clearly, . Thus, . Therefore, .
On the other hand, for any such that , . Then, . Thus, . Since , . Therefore, this contradicts the assumption. This completes the proof. □
Definition 17. Let be a hesitant fuzzy topological space, and let . Then, the hesitant fuzzy interior points of h, denoted by , are a hesitant fuzzy set in X defined by: Example 5. Let , and let ,
where ,
,
,
.
Let h be the hesitant fuzzy set in X given by: Then, clearly and . Thus, .
From Definition 17, we have the following result.
Proposition 11. Let be a hesitant fuzzy topological space, and let . Then,
(1) ,
(2) is the largest hesitant fuzzy open set contained in h,
(3) if and only if ,
(4) if , then .
Theorem 7. Let be a hesitant fuzzy topological space; let ; and let . Then, if and only if there is such that .
Proof. The proof is straightforward. □
Proposition 12. (Hesitant fuzzy interior axioms). Let be a hesitant fuzzy topological space, and let . Then:
(HFI1) ,
(HFI2) ,
(HFI3) ,
(HFI4) .
Proof. From Definition 17 and Proposition 11 (1), the proofs of (HFI1) and (HFI2) are clear.
(HFI3) By (HFI2), it is obvious that . Let Then, there is such that . Thus, by Proposition 11 (2) and (3) and (HFI2), . Therefore, . Hence, . Therefore, .
(HFI4) It is clear that and . Then, by Proposition 11 (3), and . Thus, . Let . Then, and . Thus, there are such that and . Therefore, and . Hence, , i.e., . Therefore, . □
Definition 18. Let X be a nonempty set. Then, a mapping is called a hesitant fuzzy interior operator on X, if it satisfies Properties (HFI1)–(HFI4) of Proposition 12.
The following result shows that a hesitant fuzzy interior operator completely determines a hesitant fuzzy topology on X and that in this topology, the operator is the hesitant fuzzy interior.
Proposition 13. Let be hesitant fuzzy interior operator on a set X. Let: Then, , and if is the hesitant fuzzy interior defined by τ, then Proof. By (HFI1) and (HFI2), and . Then, . Thus, satisfies Axiom (HFT1).
Let
. Then,
and
. Thus, by (HFI4),
Therefore, . Hence, satisfies Axiom (HFT2).
Let , and let . Then clearly, , for each . By (HFI2), , for each . Thus, . Since , , for each . Therefore, , i.e., . Hence, satisfies Axiom (HFT3). Therefore, .
Suppose is the hesitant fuzzy interior defined by , and let . Then clearly, is the largest hesitant fuzzy open set in X contained in h. Thus, . By (HFI3), . Therefore, . By (HFI2), . Hence, . On the other hand, by (HFI2), Then, . Thus, . Therefore, □
Definition 19. Let be a hesitant fuzzy topological space, and let . Then, the hesitant fuzzy closure of h, denoted by , is a hesitant fuzzy set in X defined by: Example 6. Let , and let ,
where ,
,
,
.
Then,
where
,
,
,
.
Let h be the hesitant fuzzy set in X given by: Then clearly, and . Thus, .
From Definition 19, we have the following result.
Proposition 14. Let be a hesitant fuzzy topological space, and let . Then,
(1) ,
(2) is the smallest hesitant fuzzy closed set containing h,
(3) if and only if ,
(4) if , then .
Theorem 8. Let be a hesitant fuzzy topological space; let ; and let . Then, if and only if for each , .
In this case,
is called a hesitant fuzzy closure point of
A. In fact,
Proof. iff for each with , , i.e.,
for each with ,
for each with ,
by Theorem 4,
for each , . □
Proposition 15. Let be a hesitant fuzzy topological space, and let . Then: Proof. Let . Then clearly, such that . Thus, , i.e., . Since , . Therefore, . Hence, .
Now, let . Then, . Thus, , i.e., . Since , there is such that and . Therefore, . Hence, , i.e., . Therefore, . □
From Definition 19 and Proposition 15, we have the following result.
Proposition 16. (Hesitant fuzzy Kuratowski closure axioms). Let be a hesitant fuzzy topological space, and let . Then:
(HFC1) ,
(HFC2) ,
(HFC3) ,
(HFC4) .
Definition 20. Let X be a nonempty set. Then, a mapping is called a hesitant fuzzy closure operator on X, if it satisfies Properties (HFC1)–(HFC4) of Proposition 16.
As expected, a result analogous to Proposition 13 holds for the hesitant fuzzy closure operator. Then, a hesitant fuzzy closure operator completely determines a hesitant fuzzy topology, and in that topology, the hesitant fuzzy closure operator is the hesitant fuzzy closure.
Proposition 17. Let be a hesitant fuzzy closure operator on a set X. Let , and let . Then, , and if is the hesitant fuzzy closure defined by τ, then: Proof. It is similar to Proposition 13. □
Definition 21. Let X be a nonempty set, and let . Then, the difference of and , denoted by , is a hesitant fuzzy set in X defined by: Definition 22. Let be a hesitant fuzzy topological space, and let . Then, is called a hesitant fuzzy limit point or an accumulation point of h, if , for each with , where The union of all hesitant fuzzy limit points of h will be called the hesitant fuzzy derived set of h and will be denoted by . Then clearly, Example 7. (1) Let X be the hesitant fuzzy discrete space. Then, , for each .
(2) Let X be the hesitant fuzzy indiscrete space. Then, , for each .
From Definition 22, we have the following result.
Proposition 18. Let be a hesitant fuzzy topological space, and let . Then:
(HFD1)
(HFD2) if , then ,
(HFD3) if , then ,
(HFD4) .
Definition 23. Let be a hesitant fuzzy topological space, and let . Then, the hesitant fuzzy exterior of h, denoted by , is a hesitant fuzzy set in X defined by .
It is obvious that .
By the above definition, we have the following result.
Proposition 19. Let be a hesitant fuzzy topological space, and let . Then:
(HFD1)
(HFD2) ,
(HFD3) ,
(HFD4) .
Definition 24. Let be a hesitant fuzzy topological space, and let . Then, the hesitant fuzzy boundary of h, denoted by , is a hesitant fuzzy set in X defined by .
5. Hesitant Fuzzy Continuous Mappings
We define a hesitant fuzzy continuous mapping and a hesitant fuzzy open (resp. closed) mapping and prove that each concept has similar properties in classical topological spaces.
Definition 25. Let be hesitant fuzzy topological spaces. Then, a mapping is said to be hesitant fuzzy continuous, if , for each .
From Result 2 and Definition 25, we have the following result.
Proposition 20. Let be hesitant fuzzy topological spaces.
(1) The identity mapping is continuous.
(2) If and are continuous, then is continuous.
Remark 2. From Proposition 20, we can see that the class of all hesitant fuzzy topological spaces and hesitant fuzzy continuous mappings forms a concrete category.
Definition 26. Let be hesitant fuzzy topological spaces. Then, a mapping is said to be hesitant fuzzy continuous at , if , for each (see Remark 1).
The following is the immediate result of Result 2 and Definitions 25 and 26.
Theorem 9. Let be hesitant fuzzy topological spaces. Then, a mapping is hesitant fuzzy continuous if and only if f is hesitant fuzzy continuous at each hesitant fuzzy point of X.
Theorem 10. Let be hesitant fuzzy topological spaces, and let be a mapping. Then, the following are equivalent:
(1) f is continuous,
(2) , for each ,
(3) , for each , where is the sub-base for σ,
(4) f is continuous at each ,
(5) for each and each , there is such that ,
(6) , for each ,
(7) , for each
Proof. (1)⇒ (2): The proof is clear from Definitions 8 and 25.
(2)⇒ (3): Suppose , for each , and let . Then, clearly, . Thus, by the hypothesis, . It is obvious that . Therefore, . Hence, .
(3) ⇒ (4): Suppose (3); and let , and let . Then, there is such that , where is the sub-base for . Thus, by (3), . Since , . Therefore, . Hence, f is continuous at each .
(4) ⇒ (5): The proof is obvious.
(5)⇒ (6): Suppose (5), and let , for each . Then, there is such that and . Let . Then, by (5), there is such that . Since , . Thus, . Therefore, . Hence, . Therefore, .
(6) ⇒ (7): Suppose (6), and let , for each . Then, by (6), . Thus, Therefore, .
(7) ⇒ (2): The proof is clear. □
The following is the immediate result of Theorem 10.
Corollary 1. is continuous if and only if , for each .
Definition 27. Let be hesitant fuzzy topological spaces. Then, a mapping is said to be hesitant fuzzy open (resp. closed), if , for each (resp. , for each ).
Proposition 21. Let be hesitant fuzzy topological spaces. If and are open (resp. closed), then is open (resp. closed).
Proof. Let . Since is open, . Since is open, . Then, . Thus, is open. The proof of the second part is similar. □
Theorem 11. Let be hesitant fuzzy topological spaces, and let be a mapping. Then, f is open if and only if , for each
Proof. Suppose f is open, and let Then, clearly, . Thus, by the hypothesis, . Since , . Since is the largest hesitant fuzzy open set contained in , .
Conversely, assume the necessary condition holds, and let . Then, clearly, . Thus, by the hypothesis, . Since , . Therefore, . Hence, f is open. □
Proposition 22. Let be hesitant fuzzy topological spaces, and let be an injective mapping. If is continuous, then , for each
Proof. Suppose f is continuous, and let Then, clearly, . Thus, by the hypothesis, . Since f is injective, by Result 2 (9), . Therefore, . Hence, . □
From Theorem 11 and Proposition 22, we have the following result.
Corollary 2. Let be hesitant fuzzy topological spaces. If is continuous, open, and injective, then , for each
Theorem 12. Let be hesitant fuzzy topological spaces, and let be a mapping. Then, f is closed if and only if , for each
Proof. Suppose f is closed, and let Then, clearly, . Thus, . Since f is closed and is closed in X, is closed in Y. Therefore, .
Conversely, assume the necessary condition holds, and let
. Then, clearly,
. Thus, by the hypothesis,
Therefore, . Hence, is closed in Y. Therefore, f is closed. □
The following is the immediate result of Theorems 10 and 12.
Corollary 3. Let be hesitant fuzzy topological spaces. Then, a mapping is continuous and closed if and only if , for each
Definition 28. Let be hesitant fuzzy topological spaces. Then, a mapping is called a hesitant fuzzy homeomorphism, if it is bijective, continuous, and open.
Remark 3. For any hesitant fuzzy discrete spaces X and Y, is a hesitant fuzzy homeomorphism if and only it is bijective.
6. Hesitant Fuzzy Subspaces
We define a hesitant fuzzy subspace topology, and we obtain some of its similar properties in classical topological spaces. Moreover, we prove that the “Pasting lemma” holds in hesitant fuzzy topological spaces.
Definition 29. Let be a hesitant fuzzy topological space, and let . Then, the collection is called a hesitant fuzzy subspace topology or hesitant fuzzy relative topology on h. The pair is called a hesitant fuzzy subspace, and each member of is called a hesitant fuzzy open set in h.
Example 8. (1) Let , and let ,
where ,
,
,
.
Let h be the hesitant fuzzy set in X given by:
.
Then, .
(2) If X is a hesitant fuzzy discrete space and , then is a hesitant fuzzy discrete space.
(3) If X is a hesitant fuzzy indiscrete space and , then is a hesitant fuzzy indiscrete space.
Proposition 23. Let be a hesitant fuzzy topological space, and let . Then, is a hesitant fuzzy topology on h.
Proof. Since , and .
Let . Then, there are such that and . Thus, . Since , .
Now, let . Then, there is such that , for each . Thus, . Since , . Therefore, is a hesitant fuzzy topology on h. □
Proposition 24. Let be a hesitant fuzzy topological space, and let such that . Then, .
Proof. Let . Then, such that . Since , Thus, . Since , . Therefore, . Let . Then, there is such that . Since , there is such that . Thus, . Since , . Therefore, . Hence, , i.e., . Therefore, . □
Proposition 25. Let be a hesitant fuzzy topological space; let ; and let be a base for τ. Then, is a base for .
Proof. Let , and let . Then, there is such that . Thus, . Therefore, by Proposition 4, is a base for . □
The following gives a special situation in which every member of the the hesitant fuzzy space topology is also a member of the hesitant topology on X.
Proposition 26. Let be a hesitant fuzzy topological space, and let . If , then .
Proof. Suppose . Then, there is such that . Since , . Thus, . □
Theorem 13. Let be a hesitant fuzzy topological space; let ; and such that . Then, A is closed in h if and only if there is such that .
Proof. Assume A is closed in h. Then clearly, . Thus, there is such that . Since , . Let . Since , is closed in X. Therefore, F is closed in X. Hence, the necessary condition holds.
Conversely suppose the necessary condition holds, and let such that . Then, there is such that . Thus, and . Therefore, . Therefore, A is closed in h. □
There is also a criterion for a hesitant fuzzy closed set in a hesitant fuzzy subspace to be closed in the hesitant fuzzy topological space. The proof is similar to Proposition 26.
Proposition 27. Let be a hesitant fuzzy topological space, and let . If A is closed in , then .
Proposition 28. Let be a hesitant fuzzy topological space; let ; and let . Then, , where denotes the closure of A in .
Proof. It is clear that
is closed in
X. Then, by Theorem 13,
is closed in
. Since
and
,
. Since:
. Since is closed in , by Theorem 13, there is such that . Since , . Since , Thus, Therefore, This completes the proof. □
From Definitions 13 and 29, we have the following result.
Theorem 14. Let be a hesitant fuzzy topological space; let ; let ; and let . Then, if and only if there is such that , where denotes the set of all neighborhoods of in .
Let
X be a nonempty set, and let
A be a subset of
X. Then, we can consider
A as the mapping
defined by: for each
,
In this case, A is also a hesitant fuzzy set, and we will write A as .
Remark 4. Let be a hesitant fuzzy topological space, and let . Then, we can easily see that the collection is a hesitant fuzzy subspace topology on A. Furthermore, we can see that all the propositions and all theorems obtained in the above hold in .
Proposition 29. Let be hesitant fuzzy topological spaces, and let , .
(1) The inclusion mapping is continuous.
(2) If is continuous, then is continuous.
(3) If is continuous, then the mapping defined by for each is continuous.
(4) If and , then the mapping defined by for each is continuous.
Proof. (1) Let . Then, clearly, . Thus, . Therefore, i is continuous.
(2) Let . Then clearly, . Since f is continuous, . Thus, . Therefore, is continuous.
(3) Let
. Then clearly,
. Since
f is continuous,
Thus, by the definition of g, . Therefore, Hence, g is continuous.
(4) Suppose
and
. Let
. Then, there is
such that
. Thus, by the hypothesis,
. Since
, by the definition of
g,
Therefore, . Hence, g is continuous. □
Proposition 30. Let be hesitant fuzzy topological spaces, and let be a mapping. Let be any family of subsets of X such that , and let . If is continuous, for each , then is continuous.
Proof. Let . Then, by the hypothesis, . Since , by Proposition 26, . On the other hand, . Thus, . Therefore, f is continuous. □
Proposition 31. (Pasting lemma). Let be hesitant fuzzy topological spaces, and let such that and . Let and be continuous mappings such that for each . Then, the mapping defined by for each and for each is continuous.
Proof. Let . Since f and g are continuous, by Theorem 10, is closed in , and is closed in . Since , by Proposition 26, . Then, . Thus, by Theorem 10, h is continuous. □
7. Hesitant Fuzzy Product Topologies and Initial Topologies
We define a hesitant fuzzy product topology and prove that there exists an initial structure in hesitant fuzzy topological spaces (See Theorem 16).
Definition 30. Let be a family of hesitant fuzzy topological spaces; let ; and let be a family of projections. For each , let , and let . Then, is a sub-base for a hesitant fuzzy topology τ on X induced by .
In this case, τ is called the hesitant fuzzy product topology on X and will be denoted by . The pair is called a hesitant fuzzy product space.
Proposition 32. Let be a family of hesitant fuzzy topological spaces. Then, the hesitant fuzzy product topology is the coarsest hesitant fuzzy topology on for which each is continuous.
Proof. Let be the sub-base for the hesitant fuzzy product topology . For each , let . Then clearly, . Since , . Thus, by Theorem 10 (3), is continuous, for each .
Suppose is any hesitant fuzzy topology on for which each is continuous, where is the projection mapping. Let . Then, there are and such that . Thus, by the hypothesis, , i.e., . Therefore, □
Theorem 15. Let be a family of hesitant fuzzy topological spaces, and let be a hesitant fuzzy topological space. Then, a mapping is continuous if and only if is continuous, for each .
Proof. Suppose f is continuous, and let . Then, by Proposition 32, is continuous, where is the projection mapping. Thus, by Proposition 20, is continuous.
Conversely, suppose the necessary condition holds, and let be the sub-base for the hesitant fuzzy product topology on given by Definition 30. For each and each , let . Then, . Thus, by the hypothesis, . Therefore, . Hence, by Theorem 10 (3), f is continuous. □
The following is the immediate result of Theorem 15.
Corollary 4. Let be a family of hesitant fuzzy topological spaces; let be a hesitant fuzzy topological space and for each ; let be a mapping. We define a mapping as follows: Then, f continuous if and only if is continuous, for each .
Proposition 32 is the motivation of the following definition.
Definition 31. Let X be a nonempty set; let be a family of hesitant fuzzy topological spaces; and let be a family of mappings. For each , let , and let . Then, the coarsest hesitant fuzzy topology τ on X with the sub-base for which each is continuous.
Especially, τ is called the hesitant fuzzy initial (or weak) topology on X induced by .
From Proposition 32, we can easily see that the hesitant fuzzy product topology on is the hesitant fuzzy initial topology induced by the family of projection mappings. By Theorem 15, we obtain the following theorem.
Theorem 16. Let be a family of hesitant fuzzy topological spaces; let x be a set and for each ; let be a mapping. Let τ be the hesitant fuzzy initial topology on X with the sub-base induced by , where . Let be a hesitant fuzzy topological space. Then, a mapping is continuous if and only if is continuous, for each .
Proof. Suppose f is continuous, and let . Then, by the definition of the hesitant fuzzy initial topology, is continuous. Thus, by Proposition 20, is continuous.
Conversely, suppose the necessary condition holds, and let be the sub-base for the hesitant fuzzy initial topology on X given by Definition 31. For each and each , let . Then, . Thus, by the hypothesis, . Therefore, . Hence, by Theorem 10 (3), f is continuous. □
Proposition 33. Let be a family of hesitant fuzzy topological spaces; let X be a set and for each ; let be a mapping. Let τ be the hesitant fuzzy initial topology on X with the sub-base induced by , and let . Then, is the hesitant fuzzy initial topology on A induced by .
Proof. Let be the hesitant fuzzy initial topology on A induced by , and let Then, clearly, is a sub-base for . In order to prove that for each and , let . Then, clearly, . Since is the hesitant fuzzy initial topology on X induced by , , thus, by Remark 4, Therefore, Hence,
Now, let us show that . For each and , let . Then, clearly, and . Thus, Therefore, . This completes the proof. □