1. Introduction
Fuzzy set theory is a mathematical framework that deals with sets whose elements have degrees of membership. This departure from traditional set theory, where an element either belongs or does not belong to a set, allows for more nuanced and flexible modelling, particularly in situations where boundaries between categories are ambiguous or uncertain. Zadeh’s [
1] seminal paper, “Fuzzy Sets”, published in 1965, laid the foundation for this field of study. This whole new field of study has influenced many other scientific fields since that time. Numerous advancements have emerged subsequent to Zadeh’s introduction of fuzzy sets in 1965. For instance, Atanassov [
2] introduced intuitionistic fuzzy sets, Smarandache [
3] proposed neutrosophic sets, and in 2023, Al-Shami and Mhemdi [
4] introduced orthopair fuzzy sets. These foundational contributions paved the way for the development of various related concepts, including the introduction of fuzzy metric spaces by Kramosil and Michálek [
5] in 1975 and its modification by George and Veeramani [
6] in 1994, intuitionistic fuzzy topological spaces by Coker [
7] in 1997, and intuitionistic fuzzy metric spaces by Park [
8] in 2004.
The introduction of all these concepts and ideologies has impacted many other research works by several mathematicians, such as: in 1984, Kaleva and Seikkala [
9] defined fuzzy metric space as a distance between two points and expressed it as a positive fuzzy number; in 2006, Smarandache [
10] gave the notion of neutrosophic sets as a generalization of intuitionistic fuzzy sets; Salama and Alblowi [
11] in 2012 extended the concepts of fuzzy topological space and intuitionistic fuzzy topological space to the case of neutrosophic sets; Ejegwa [
12], in 2014, introduced some algebraic operations such as modal operator, and normalization to intuitionistic fuzzy sets; and Majumdar [
13] conducted an exploration into the practical applications of neutrosophic sets within decision-making contexts. Apart from these, several researchers delved into various generalizations of fuzzy metric space such as intuitionistic fuzzy metric space, soft fuzzy metric space, fuzzy soft metric space, orthogonal fuzzy metric space, bipolar fuzzy metric space, etc. [
14,
15,
16,
17,
18,
19,
20,
21,
22,
23,
24]. Furthermore, Broumi [
25] introduced several novel concepts related to neutrosophic sets, including refined neutrosophic sets, bipolar neutrosophic sets, neutrosophic hesitant sets, multi-valued neutrosophic sets, rough neutrosophic sets, and rough bipolar neutrosophic sets. Additionally, Jeyaraman [
26] applied multi-criteria decision-making (MCDM) techniques in conjunction with neutrosophic sets to address a real-life problem involving wind turbines.
In 2020, Kirisci and Şimşek [
27] introduced the notion of neutrosophic metric spaces. Further, several fixed-point results have been established in this generalization [
28] with some generalizations of neutrosophic fuzzy metric space were introduced by researchers such as orthogonal neutrosophic metric spaces by Ishtiaq et al. [
29], neutrosophic 2-metric spaces by Asghar et al. [
30], orthogonal neutrosophic 2-metric spaces by Janardhanan et al. [
31], neutrosophic pentagonal metric spaces by Mani et al. [
32], neutrosophic b-metric space [
33], etc. Further, Das et al. [
34] introduced the conception of neutrosophic fuzzy sets. It is quite noticeable how the concept of fuzzy sets has been used to establish the concept of fuzzy metric space. In a similar way, the ideology of neutrosophic sets has been used to define the conception of neutrosophic metric space. Keeping these in mind, the primary objective of this research is to elucidate the conceptual framework of neutrosophic fuzzy metric space. The concept of neutrosophic fuzzy sets has been utilised for this purpose, in a similar manner as described above.
The motivation behind introducing neutrosophic fuzzy metric spaces (NFMS) lies in their ability to model uncertainty and indeterminacy in real-world phenomena more accurately than traditional fuzzy or crisp sets. Neutrosophic sets allow for the representation of elements with three components, truth, indeterminacy, and falsehood, providing a more nuanced description of uncertainty. An example that highlights the need for NFMS could be in the field of medical diagnosis. Consider a scenario where a patient presents symptoms that are not clearly indicative of a single disease. Traditional fuzzy sets could represent the likelihood of the patient having a certain disease on a continuum between 0 and 1. However, this approach might not adequately capture the uncertainty associated with the diagnosis, especially if the symptoms are ambiguous or conflicting. In contrast, neutrosophic sets can represent the uncertainty more comprehensively by explicitly accounting for the degree of truth, indeterminacy, and falsehood associated with each symptom and potential diagnosis. NFMS can then be used to define a metric space where the distance between two diagnoses reflects not only their similarity but also the degree of uncertainty or indeterminacy associated with each.
By introducing NFMS, we provide a mathematical framework that aligns more closely with the complex and uncertain nature of real-world phenomena, such as medical diagnosis, decision making under uncertainty, or pattern recognition in ambiguous data sets. This motivates the development of NFMS theory to enhance our ability to model, analyze, and make decisions in uncertain environments more effectively.
The following describes the structure of the paper. Some characteristics and fundamental ideas of neutrosophic fuzzy sets and neutrosophic metric spaces are provided in
Section 2.
Section 3 introduces the conception of neutrosophic fuzzy metric space, followed by examples of neutrosophic fuzzy metric space. The topological characteristics of the established generalization of metric space have been demonstrated in this section. In this section, we have also presented various results describing distinct properties of the established neutrosophic fuzzy metric space, such as the Hausdorff property, compactness, completeness, and nowhere denseness.
Section 4 of this work contains the conclusion. Further, this study can be extended to investigate neutrosophic fuzzy metric space in connection with other concepts such as best proximity point results, optimization theory, approximation theory, etc. Moreover, concepts like soft sets, soft equality, soft lattices, etc., can be connected to the established theory and results [
35,
36].
3. Neutrosophic Fuzzy Metric Space and Its Topological Properties
This section introduces the concept of neutrosophic fuzzy metric space and explores various topological characteristics of it. Firstly, we present the definition of a neutrosophic fuzzy metric space.
Definition 7. A 7-tuple is known as a Neutrophic Fuzzy Metric Space (NFMS) if G is an arbitary set, ★ is a continuous t-norm, ⋄ is a continuous t-conorm, and S, A, M, and R are fuzzy sets on satisfying the following conditions:
and
- 1.
;
- 2.
;
- 3.
;
- 4.
if ;
- 5.
;
- 6.
;
- 7.
is continuous;
- 8.
;
- 9.
if ;
- 10.
;
- 11.
;
- 12.
is continuous;
- 13.
;
- 14.
if
- 15.
;
- 16.
;
- 17.
is continuous;
- 18.
;
- 19.
if ;
- 20.
;
- 21.
;
- 22.
is continuous;
- 23.
For , and .
In this context, represents the certainity that distance between x and y is less than t, represents the degree of nearness, stadns for the degree of neutralness, and denotes the degree of non-nearness between x and y with respect to t, respectively.
Example 1. Let be a metric space where and . Define the t-norm and t-conorm, ★ and ⋄ as and . And let the fuzzy sets on be defined as:Note that, - (1)
- (2)
Since , we have ,
- (3)
if , and
- (4)
for all and .
- (5)
for all , .
Similarlly, all the conditions for , and R can be verified. Therefore, is an NFMS induced by a metric d, called the standard neutrosophic fuzzy metric.
Example 2. Take , ★ be a TN and ⋄ be a TC defined as and . Also, , we define:Note that, - (1)
- (2)
Since , we have
,
- (3)
, if ,
- (4)
, for all and .
- (5)
for all , .
Therefore, forms an NFMS.
Remark 1. The 7-tuple defined in above Example 1 would not be a NFMS if t-norm and t-conorm .
Definition 8. Consider as an NFMS. For , , and , denote the set:as an open ball (OB), where p serves as the center and ε as the radius with respect to t. Definition 9. Consider as an NFMS. A subset H of G is said to be an open set if for each , there exists an open ball such that .
Theorem 1. Each OB, within an NFMS constitutes an open set (OS).
Proof. We consider an open ball and choose . Then, .
There exists such that,
.
Taking for , so that .
For any given and so that . Then, s.t,
.
Choosing and considering the open ball , our motive is to show that .
Now taking we get,
.
Also,
,
,
,
.
Hence, and . □
Remark 2. From Definition 9 and Theorem 1, it can be said that,is a topology on G. In this case every neutrosophic fuzzy metric on G produces a topology on G which has a base as the family of open sets . Theorem 2. Let be an NFMS. Then,
- (i)
ϕ and G are open sets in .
- (ii)
The union of any finite, countable, or uncountable family of open sets is open.
- (iii)
The intersection of a finite family of open sets is open.
Proof. (i) As an empty set contains no points, the necessity that each point in is the centre of an open ball contained in it is satisfied automatically.
The whole space G is open since every open ball centred at any of its points is contained in G.
(ii) Let be a family of open sets and . If then it is open [by (i)]. So let . Taking , it can be said that for some . Since is open, such that (where ). Thus, for each such that . Implies that H is open.
(iii) Let be a finite family of open sets in G, and let . If M is empty, then it is open. (by (i)).
Suppose , and . Then, . Since is open, such that where and .
Let
. Then
and
where
and
. Therefore,
centred at
x satisfies
This completes the proof. □
Theorem 3. A subset X in a NFMS is open if it is the union of all open balls contained in X.
Proof. Let X be open. If , then there are no open balls contained in it. Thus, the union of all open balls contained in X is the union of an empty class, which is empty and therefore equal to X.
Now if , then since X is open, each of its points is the centre of an open ball entirely contained in X. So X is the union of all open balls contained in it. The converse follows from Theorem 3.1 and Theorem 3.3. □
Definition 10. Let A be a subset of an NFMS . A point is an interior point of A if for some . That is, .
Theorem 4. Let A be a subset of an NFMS . Then,
- (i)
is an open subset of A that contains every open subset of A.
- (ii)
A is open if .
Proof. Let be arbitary. Then, by Definition 3.4, for some . But since is an open set [by Theorem 3.1], each point of it is the centre of an open ball contained in and consequently also in A. Therefore, each point of is an interior point of A. That is, . Thus, x is the centre of an open ball contained in . Since is arbitary, it can be said that each has the property of being the centre of an open ball contained in . Hence, is open.
Now it is to be shown that contains every open subset . Let . Since X is open, . So . This shows that . In other words, .
(ii) is immediate after (i). □
Definition 11. Let be an NFMS and let be the collection of all non empty compact subsets of G. Consider and .
Define and as follows:The 6-tuple is called Hausdorff NFMS, or shortly HNFMS. Theorem 5. Every NFMS possesses the property of being Hausdorff.
Proof. We consider as an NFMS. Let be different. Then, .
Take
and .
Now, taking such that,
and .
Taking if we consider the open balls and . Then, it is clear that, .
Now, if we choose a point
then,
which shows a contradiction. Therefore, every NFMS is Hausdorff. □
Definition 12. We consider as an NFMS. A set is known as nuetrosophic bounded (NB) if, there remains a positive real number t and ε in satisfying, and .
Definition 13. Let an NFMS be given, then
- (1)
Consider as an assembly of open sets and . Then, the assembly is said to be an open cover(OC) of W.
- (2)
A subspace W of G is compact, if every open cover of W has a finite subcover.
- (3)
A subspace W of G is said to be sequentially compact if each sequence in W possesses a convergent subsequence in W.
Theorem 6. In an NFMS, every compact subset is neutrosophic bounded.
Proof. We consider as an NFMS, and W be a compact subset of G. Also consider an open cover of W where and . W being a compact set, there exists in such a way that . Further, for there remains some such that and . Then, we can write
and
Now, take ,
, .
Then,
. From here, we can have some
satisfying,
Taking
and
, we have
,
. Therefore, the set
W is Neutrosophic bounded. □
If is a NFMS produced by a metric d on G and , then W is Neutrosophic bounded if it is bounded. As a result with Theorems 5 and 6 it can be written that:
Corollary 1. In an NFMS, each compact set is both closed set and bounded set.
Theorem 7. We consider as an NFMS and to be the topology on G produced by the FM, then if and only if, Proof. We consider
. Suppose
. For any given
there remains
such that,
Thus,
.
In these cases, we express it as follows:
Conversely, suppose
as
, for each
. Then for any
there exists
so that
. By this, we obtain
. Thus,
. Hence the proof. □
Definition 14. We consider as an NFMS. Any sequence in G is said to be a Cauchy sequence if for any positive real numbers ε and t, there subsists so that, the following conditions hold:for all . Moreover, is said to be a complete NFMS if each Cauchy sequence converges with respect to the topology in G.
Theorem 8. We consider as an NFMS. Under the assumption that each Cauchy sequence of G possesses a converging subsequence, the NFMS is considered complete.
Proof. We consider
as a Cauchy sequence in
G and assume a subsequence
of
coverges to a point
. We complete the proof by showing
. Let
and
. Taking
so that
,
. By the Cauchyness of
, there exists
so that
,
Since
, there remains
satisfying
,
. Then, for
,
Thus, we have
. Therefore, NFMS
is complete. □
Theorem 9. Consider as an NFMS and W be a subset of G with the subspace NFM . Then is complete if and only if is closed.
Proof. Let is closed. Suppose be Cauchy in . Since is Cauchy in G, there exists such that . It is clear that, and so converges in W.
In contrary, let
be complete. Also, suppose
W is open. If we choose a point
, then there exists
of points belonging to
W converging to
p and thus,
is Cauchy. This gives, for
, each
and
,
satisfying
Now, since
is in
W, we can write
. Therefore,
is Cauchy in
W. Since
is complete,
such that
. Hence,
such that
for all
, each
and
. Since
is in
W and
, we can write
. This concludes that
converges both to
p and
q in
. Since
and
, this implies
. This contradiction leads to the desired result. □
Lemma 1. Let be a NFMS. If and satisfy , , then .
Proof. We take
and
be an open ball with radius
and centered at
q. Since
,
. Then we obtain,
Hence,
and thus
. □
Theorem 10. For an NFMS , a subset is nowhere dense if and only if each non-empty OS in G consists of an OB such that the closure of the OB and W are disjoint.
Proof. We consider , as open. Then, there exists a non-empty OS such that , . If we take , then and such that . Now, choose such that and . By Lemma 1, we obtain . In this scenario, we can assert that and .
On the contrary, presume that W is not nowhere dense. Hence, , indicating the existence of an OS such that . Assume as an OB, so as . Consequently, , contradicts the assumption. □
Now, we present the Baire Category Theorem in NFMS.
Theorem 11. In a complete NFMS, the intersection of countably many dense open sets is dense.
Proof. Consider a sequence of dense open subsets
in a complete NFMS
. Then, it is to be proved that the intersection
is dense within
G. Consider
as a non-empty open set of
G. As
is dense,
. Let
. Now,
being open,
,
so that
. Choose
and
such that
. As
is dense,
. Choose
. As
is open,
and
so that
. Choose
and
so that
. By proceeding in this manner, we establish a sequence
in
G along with another sequence
where
, and
Next, we demonstrate that
is Cauchy. For any
, we select
such that
and
. Thus, for
and
, we have
This shows that
is Cauchy.
G being complete, there exists
so that
. As
, we obtain
. Hence, it can be written that
,
. Then,
. Then
is dense in G. □
Definition 15. Consider an NFMS . We define a collection , where , to possess neutrosophic diameter zero (NDZ) if for every and , there exists such that , , , and hold for all .
Theorem 12. The NFMS is complete if and only if each sequence of nested non-empty closed sets, each possessing NDZ, has a nonempty intersection.
Proof. Assuming the given condition is satisfied, i.e., every sequence
of nested nonempty closed sets, each possessing NDZ, has a nonempty intersection. It remains to be demonstrated that
is complete. Consider the Cauchy sequence
. Define
and
. It can then be asserted that
possesses NDZ. Given
and
, choose
such that
and
. As
is Cauchy, there exists a natural number
N satisfying
So, we obtain
[where
, the NFM on
G].
Select
. Then, sequences
and
exist such that
and
. Hence, for sufficiently large
m,
and
. Consequently, we obtain
Therefore,
,
,
,
. This implies that
possesses NDZ, hence, by the assumption,
is non-empty. Let
. Then, for
in
and
t, a positive real number,
such that
. That is,
for each
as
. Hence,
, which means
is complete.
Conversely, assume is complete. Consider a nested sequence of nonempty closed sets having NDZ. For each , let be a point in . Our aim is to demonstrate that is a Cauchy sequence. Since possesses NDZ, for and , there exists such that , , , hold for all . Due to the fact that is nested, it can be written that . Thus, forms a Cauchy sequence. As is complete, converges to some . This implies that , for all m, and therefore, p belongs to . □
Theorem 13. Every separable NFMS possesses second countability.
Proof. Let
be separable. Consider
as countable dense. Define a collection
. It is evident that
is countable. Our objective is to demonstrate that
serves as a basis for the assembly of all OS within
G. Take
as an open set in
G containing
p. Then,
s.t
. Due to this fact,
,
s.t
and
. Chosse
so that
. As
is dense,
such that
. If
, we can write
Then
, thus
forms a basis. □
Remark 3. It is worth noting that second countability implies separability, and second countability is an inherited property. Hence, it follows that every subspace of a separable NFMS is also separable.
Definition 16. Consider a set and a NFMS . We say that the sequence of functions uniformly converges to a function if corresponding to any , , there remains a natural number N satisfyingfor all and . Now we present uniform convergence theorem for NFMS.
Theorem 14. Let W be a topological space, be a NFMS, and be a sequence of continuous functions. If converges uniformly to then is considered continuous.
Proof. Assume
as an OS in
G,
. Since
is open,
so that
. Since
, we select
such that
and
. Given that
is uniformly convergent to
,for any
and
so that
for all
. Since
are continuous, we have for each
n that there remains
, a neighborhood of
, so as
. Therefore,
. Now,
Implies that
. Therefore,
and thus,
is continuous. □