Bipolar fuzzy metric spaces with application

Abstract

The concept of a bipolar metric space is presented in this article. In bipolar fuzzy metric space, the notions of continuous t-norms and bipolar continuous symmetry $t_s$-conorms employs important role. Requirements in metric space, triangular norms are used to extrapolate with the probability density function of triangle inequality. Dual processes of triangular norms are known as triangular conorms. Continuous t-norms and continuous symmetry $t_s$-conorms are used to define bipolar metric space in this paper. Besides, a number of topological and functional properties of the bipolar metric space have been studied. Afterwards, Baire Category and Uniform Convergence Theorems for bipolar metric spaces are presented. After that, an application on determining the appropriate type of vaccine in the treatment process of COVID-19 is given using similarity measure between bipolar fuzzy metric spaces. For vaccine selection and supply chain management, an advanced multi-attribute decision-making (MADM) algorithm is being developed. Finally, the validity of the proposed MADM method for selecting the best proper form of vaccine is also established.

1 Introduction

Being a member of elements in a set is determined in binary terms according to a bivalent condition in classical set theory, which means that an element either member of the set or does not member of the set. Fuzzy set theory, on the other hand, allows for a gradual assessment of the membership of elements in a set, which is defined using a membership function with a real unit interval [0, 1]. Since the indicator functions of classical sets are special cases of the membership functions of fuzzy sets, if the latter only take values 0 or 1, fuzzy sets generalize classical sets. Crisp sets are widely used in fuzzy set theory to refer to classical bivalent sets. Many researchers have been interested in fuzzy sets, which were first advanced by Zadeh (1965) in 1965 to convey indefiniteness in data. A membership function $\mu :X:\rightarrow [0,1]$ defines the a fuzzy set description, which assigns a real number in the unit closed interval [0, 1] to each object in the universe. The fuzzy set theory can be applied to a broad variety of domains, including bioinformatics, where knowledge is incomplete or imprecise. Chang (1968) developed the notion of fuzzy topological spaces. Atanassov (1984), Atanassov and Stoeva (1983) suggested the intuitionistic fuzzy set (IFS) as a direct extension of the fuzzy set, based on the definition of membership grade (MG) and non-membership grade (NMG), with the restriction that the number of MG and NMG not exceed 1. Çoker (1997) introduced the concept of the intuitionistic fuzzy topological space. Smarandache (2005) described the Neutrosophic Set (NS) as a modern version of the classical set definition. Neutrosophy has opened the way for many other modern mathematical ideas that extend both classical and fuzzy equivalents. Balasubramanian and Sundaram (1997) presented the definition of a generalized fuzzy closed set. Pythagorean fuzzy set and Pythagorean fuzzy membership grades were introduced by Yager (2014, 2017) and Yager and Abbasov (2013), respectively. Yager and Abbasov (2013), Park (2004), Yager (2014), Yager (2017) introduced Pythagorean fuzzy sets (PFSs), also known as intuitionistic fuzzy sets of type-2, as an extension of IFSs and presented Pythagorean membership grades with requirements, by changing the constraint on the parameters. In Gulistana et al. (2021) authors study complex bipolar fuzzy sets. The notion of rough bipolar fuzzy $\varGamma$r-hyperideals in $\varGamma$r -semihypergroups is introduced by Yaqoob et al. (2018). Besides, Yaqoob (2014) deal with on bipolar fuzzy sets. The concept of bipolar-valued fuzzification of ordered AG-groupoids is studied in Yousafzai et al. (2012). The definitions of probabilistic metric space and fuzzy sets are used to construct the fuzzy metric space (FMS) in Kramosil and Michàle (1975). Afterwards FMS was described by Kaleva and Seikkala (1984) as a non-negative fuzzy number that measures the distance between two points. Following that, FMS has been extended to a variety of fields, fixed point theory, image and signal processing, medical imaging, decision-making, and so on are some of the topics covered. After it was defined, the intuitionistic fuzzy set (IFS) was described and used in all areas where FS theory was explored. In addition these, the concept of intuitionistic fuzzy metric space was invented by Park (2004). He demonstrated that the topology created by the intuitionistic fuzzy metric (M, N) coincides with the topology generated by the fuzzy metric M for each intuitionistic fuzzy metric space $(X,M,N,*,\blacklozenge )$. A hesitant fuzzy linear regression model (HFLRM) is investigated in Sultan et al. (2021) to account for multicriteria decision-making (MCDM) problems in a hesitant environment. In Kizielewicz et al. (2021), a new modified TOPSIS technique for dealing with data uncertainty that is based on TFN similarity metrics is offered.

Based on the theory of NSs, a new metric space called Neutrosophic metric Spaces (NMS) was defined in Kirişci and Şimşek (2020). Some properties of NMS are investigated in Kirişci and Şimşek (2020), including open set, Hausdorff, neutrosophic bounded, compactness and completeness. For NMSs, they also include the Baire Category Theorem and the Uniform Convergence Theorem. As extensions of Partial Algebra, Smarandache (2005) generalized classical Algebraic Structures to NeutroAlgebraic Structures with partially real, partially indeterminate, and partially false operations and axioms. Recently, as a generalization of fuzzy sets, Zhang (1994, 1998) initiated the concept of bipolar fuzzy sets in 1994. For more information you can see (Abdullah et al. 2014; Hussain et al. 2019; Qurashi and Shabir 2018; Riaz et al. 2020). In this research, fundamental concepts about the subject was given in the second part. In addition, in this section, the concepts of bipolar continuous symmetry $t_s$-conorm and bipolar fuzzy metric space are introduced and relevant examples are discussed. In the third chapter, the topology of bipolar fuzzy sets has been constructed and some theorems and proofs are given. For example, it is showed that BFMS is $T_2$-space. Additionally, in this part BF bounded, complete BFMS and convergence of a BFMS concepts are defined. It is demonstrated in Chapter fourth that a BFMS is complete. The Baire Category and Uniform Limit theorems, as well as their proofs, were discovered. In the fifth section, a new similarity measure method for bipolar fuzzy metric spaces is presented. A multiple attribute decision making (MADM) algorithm is also provided, which was designed to determine the best form of vaccine for COVID-19. In the future work, we plan to give definitions of bipolar normed space and statistical convergence on bipolar fuzzy sets. Besides, we will make an application about medicine image processing.

2 Preliminaries

This section contains some definitions for fuzziness, bipolar fuzziness, and bipolar fuzzy metric space.

Definition 1

(Zhang 1994). BFSs are a type of fuzzy set whose membership value range has been extended from [0, 1] to $[-1,1]$. In a bipolar fuzzy set, a membership value of 0 denotes that elements are irrelevant to the corresponding property, a membership value of $[-1,0]$ denotes that elements satisfy the assumed counter-property, and a membership value of (0, 1) denotes that elements satisfy the property. The accepted representation of a bipolar fuzzy set A on the domain X is as follows:

$$\begin{aligned} A=\{(x,(\mu _{A}^{r}(a),\mu _{A}^{\ell }(a))):a\in X\}. \end{aligned}$$

Here, the positive membership value $\mu _{A}^{r}(a):X\rightarrow [0,1]$ specifies the value of satisfaction of some element $a\in X$ to the property according to the bipolar fuzzy set A, and the negative membership value $\mu _{A}^{\ell }(a):X\rightarrow [-1,0]$ describes the value to which a certain element $a\in X$ satisfies some implied counter-property corresponding to A.

Now let’s include some basic definitions defined on bipolar fuzzy sets.

Definition 2

(Zhang 1994). Let’s admit that $A=\{a,(\mu _A^{\ell }(a), \mu _A^r(a)):a\in X\}$ and $B=\{a,(\mu _B^{\ell }, \mu _B^r):a\in X\}$ are two bipolar fuzzy sets defined on X. In this case, $A\subseteq B$, if

• $\mu _A^{r}(a)\le \mu _B^{r}(a)$

• $\mu _A^{\ell }(a)\ge \mu _B^{\ell }(a)$ for all $a\in X$.

Additionally,

$$\begin{aligned} A\cup B= & {} \left\{ a,\left( \max \{\mu _A^r(a),\mu _B^r(a)\},\min \{\mu _A^{\ell }(a),\mu _B^{\ell }(a)\}\right) :a\in X\right\} (\mathrm{Union}),\\ A\cap B= & {} \left\{ a,\left( \min \{\mu _A^r(a),\mu _B^r(a)\},\max \{\mu _A^{\ell }(a),\mu _B^{\ell }(a)\}\right) :a\in X\right\} (\mathrm{Intersection}),\\ A^c= & {} \left\{ a,\left( 1-\mu _A^r(a),-1-\mu _A^{\ell }(a)\right) :a\in X\right\} (\mathrm{Complement}). \end{aligned}$$

The positive membership degree $\mu _{A}^{r}(a)$, in BFSs, denotes the value to which the element a supplies the property of A, while the negative membership value $\mu _{A}^{\ell }(a)$ denotes the value to which the element a supplies an implied counter-property of A. However, the membership value $\mu _{A}(a)$ denotes the value to which the element a satisfies the property A, while the membership value $\nu _{A}(a)$ denotes the value to which a satisfies the not-property of A, in IFSs. BFSs and IFSs are different extensions of FSs since a counter-property is not always equal to not-property. Their differences can be seen in how they view an element a with a membership value of (0, 0). It is considered that the element a does not satisfy both the property A and its implicit counter-property by taking into account the perspective of bipolar fuzzy set A. It implies that it is unaffected by the property and its implicit counter-property. When this situation is considered in terms of NS A, the variable a is noted as not satisfying the property and its not-property. Menger (1942) established the use of triangular norms (t-norms). Menger discussed using probability distributions instead of numbers to solve the problem of calculating the distance between two elements in space. In metric space conditions, t-norms are used to make generalisations with the probability distribution of triangle inequality. Dual operations of TNs are triangular conorms (t-conorms).

Definition 3

(Menger 1942). Continuous t-norms are binary operations on the interval [0, 1], it means that $\varDelta :[0,1]\times [0,1]\rightarrow [0,1]$ satisfies the following situations with neutral element 1:

• 1. $\varDelta$ is commutative and associative

• 2. $\varDelta$ is continuous

• 3. $a\varDelta 1=a$ for all $a\in [0,1]$

• 4. $a\varDelta b \le c\varDelta d$ where $a\le c, b\le d$ for all $a,b,c,d\in [0,1].$

Definition 4

A doubled arithmetic operation $\nabla :[-1,0]\times [-1,0]\rightarrow [-1,1]$ is referred to as bipolar continuous symmetry $t_s$-conorm if $\nabla$ applies the required specifications:

• 1. $\nabla$ is commutative and associative

• 2. $\nabla$ is continuous

• 3. $a\nabla 0=a$ for all $a\in [-1,0]$

• 4. $a\nabla b \le c\nabla d$ where $a\le c, b\le d$ for all $a,b,c,d\in [-1,0].$

Definition 5

The tetra set $(A,{\mathscr {B}}, \varDelta , \nabla )$ is called bipolar fuzzy metric space (BFMS) when the following conditions are satisfied for all $u,v,z\in A$, where A be arbitrary set, $\{<u,\mu _{A}^{r}(u),\mu _{A}^{\ell }(u)>:u\in A\}$ be BFS such that $\mu _A^r,\mu _{A}^{\ell }$ are defined as fuzzy sets on $A \times A\times (0,\infty )$, ${\mathscr {B}}:A \times A\times (0,\infty )\rightarrow [-1,1]$, $\alpha ,\gamma >0$ and $\varDelta$, $\nabla$ show the continuous t-norm and continuous symmetry $t_s$-conorm, respectively:

• 1. $0\le \mu _{A}^{r}(u,v,\gamma )\le 1, -1\le \mu _{A}^{\ell }(u,v,\gamma )\le 0$

• 2. $\mu _{A}^{r}(u,v,\gamma )+\mu _{A}^{\ell }(u,v,\gamma )\le 1$

• 3. $\mu _{A}^{r}(u,v,\gamma )=1\Leftrightarrow u=v$

• 4. $\mu _{A}^{r}(u,v,\gamma )=\mu _{A}^{r}(v,u,\gamma )$

• 5. $\mu _{A}^{r}(u,v,\gamma )\varDelta \mu _{A}^{r}(v,z,\alpha )\le \mu _{A}^{r}(u,z,\gamma +\alpha )$

• 7. $\mu _{A}^{r}(u,v,.):[0,\infty )\rightarrow [-1,1]$ is continuous

• 8. $\lim _{k}\mu _{A}^{r}(u,v,\gamma )=1, (\forall \gamma >0)$

• 9. $\mu _{A}^{\ell }(u,v,\gamma )=-1\Leftrightarrow u=v$

• 10. $\mu _{A}^{\ell }(u,v,\gamma )=\mu _{A}^{\ell }(v,u,\gamma )$

• 11. $\mu _{A}^{\ell }(u,v,\gamma )\nabla \mu _{A}^{\ell }(v,z,\alpha )\ge \mu _{A}^{\ell }(u,z,\gamma +\alpha )$

• 12. $\mu _{A}^{\ell }(u,v,.):[0,\infty )\rightarrow [-1,1]$ is continuous

• 13. $\lim _{k}\mu _{A}^{\ell }(u,v,\gamma )=-1, (\forall \gamma >0)$.

The functions $\mu _{A}^{r}(u,v,\gamma )$ and $\mu _{A}^{\ell }(u,v,\gamma )$ represent the value of closeness and the value of distance between u and v with respect to $\gamma$, respectively.

Remark 1

In case of $\mu _{A}^{r}(u)=\mu _{A}^{\ell }(u)=0$ for all $u\in A$ every fuzzy metric space is bipolar fuzzy metric space.

We can see from the definitions above that if we select $\delta _1,\delta _2\in (0,1)$ taken together $\delta _1>\delta _2$, there is $\delta _3,\delta _4\in (0,1)$ such that $\delta _1\varDelta \delta _3\ge \delta _2$ and $\delta _1\ge \delta _4\nabla \delta _2$. Also, in case of $\delta _5\in (0,1)$ there is $\delta _6,\delta _7\in (0,1)$ such that $\delta _6\varDelta \delta _6\ge \delta _5$ and $\delta _7\nabla \delta _7\le \delta _5$.

Example 1

Let (A, d) be a crisp metric space. Give the functions $\varDelta$ and $\nabla$ as default $u\varDelta v=\min \{u,v\}$, $u\nabla v=\max \{u,v\}$. $\mu _{A}^{r}(u)=\frac{\gamma }{\gamma +d(u,v)}, \mu _{A}^{\ell }(u)=\frac{-\gamma }{\gamma +d(u,v)}$, for all $\gamma >0, u,v\in A$. Then, $(A,{\mathscr {B}}, \varDelta , \nabla )$ be bipolar fuzzy metric space by considering ${\mathscr {B}}:A \times A\times (0,\infty )\rightarrow [-1,1]$.

Example 2

If we take $A={\mathbb {Z}}-\{0\}$, $a\varDelta b=\max \{0,a+b-1\}$, $a\nabla b=\max \{0,a+b-1\}$ for all $a,b\in [-1,1]$ and define fuzzy sets $\mu _{A}^{r},\mu _{A}^{\ell }$ on $A\times A\times (0,\infty )$ as given below:

$$\begin{aligned} \mu _A^r(u,v,\gamma )&=\left\{ \begin{array}{rl} \frac{u}{v},&{} u\le v\\ \frac{v}{u},&{} v\le u\\ \end{array}\right. \end{aligned}$$

(1)

$$\begin{aligned} \mu _{A}^{\ell }(u,v,\gamma )&=\left\{ \begin{array}{rl} -\frac{u}{v},&{} u\le v\\ -\frac{v}{u},&{} v\le u\\ \end{array}\right. \end{aligned}$$

(2)

for all $u,v\in A$ and $\gamma >0$. Then, $(A,{\mathscr {B}}, \varDelta , \nabla )$ is a bipolar fuzzy metric space.

3 Investigation of topology induced by bipolar fuzzy metric

In this section, the topological structure of bipolar fuzzy sets is examined. In this way, we deal with the structural qualities of BFMS, such as open ball, open set, $T_2$ space, completeness, and nowhere dense.

Definition 6

Given $(A,{\mathscr {B}},\varDelta , \nabla )$ be a BFMS, $0<\delta <1, \gamma >0$ and $u\in A$. The set $\phi (u,\delta , \gamma )=\{v\in A:\mu _{A}^{r}(u,v,\gamma )>1-\delta , \mu _{A}^{\ell }(u,v,\gamma )<\delta -1 \}$ is called an open ball because it has a middle point u and a radius $\delta$ in relation to $\gamma$.

Theorem 1

Any open ball, $\phi (u,\delta , \gamma )$ in $(A,{\mathscr {B}},\varDelta , \nabla )$ is an open set.

Proof

Consider $\phi (u,\delta , \gamma )$ is a open ball together u as the center and $\delta$ as the radius. Let us take $v\in \phi (u,\delta , \gamma )$ after that $\mu _{A}^{r}(u,v,\gamma )>1-\delta , \mu _{A}^{\ell }(u,v,\gamma )<\delta -1$. There exists $\gamma _0 \in (0,\gamma )$ so as $\mu _{A}^{r}(u,v,\gamma _0)>1-\delta$, $\mu _{A}^{\ell }(u,v,\gamma _0)<\delta -1$ because of the fact that $\mu _{A}^{r}(u,v,\gamma )>1-\delta$. If we choose $\delta _0=\mu _{A}^{r}(u,v,\gamma _0)$ then for $\delta _0>1-\delta$, there is $\sigma \in (0,1)$ such that $\delta _0>1-\sigma >1-\delta$. Receive $\delta _0$ and $\sigma$ as $\delta _0>1-\sigma$. After, $\delta _1, \delta _2\in (0,1)$ will exist such that $\delta _0\varDelta \delta _1 >1-\sigma$ and $(\delta _0-1)\triangledown (\delta _2-1)\le \sigma -1$. And choose $\delta _3=\max \{\delta _1,\delta _2\}$. Now, take into consideration the open ball $\phi (v,1-\delta _3, \gamma -\gamma _0)$. We will show that $\phi (v,1-\delta _3, \gamma -\gamma _0)\subset \phi (u,\delta ,\gamma )$. If we get $z\in \phi (v,1-\delta _3, \gamma -\gamma _0)$ then, $\mu _{A}^{r}(v,z,\gamma -\gamma _0)>\delta _3$ and $\mu _{A}^{\ell }(v,z,\gamma -\gamma _0)<\delta _3$. Also, $\mu _{A}^{r}(u,z,\gamma ) \ge \mu _{A}^{r}(u,v,\gamma _0)\vartriangle \mu _{A}^{r}(v,z,\gamma -\gamma _0)\ge \delta _0\vartriangle \delta _3\ge \delta _0\vartriangle \delta _1\ge 1-\sigma >1-\delta$, $\mu _{A}^{\ell }(u,z,\gamma )\le \mu _{A}^{\ell }(u,v,\gamma _0)\triangledown \mu _{A}^{\ell }(v,z,\gamma -\gamma _0)\le (\delta _0-1)\triangledown (\delta _3-1)\le (\delta _0-1)\triangledown (\delta _2-1)\le \sigma -1<\delta -1$. It means that $z\in \phi (u,\delta , \gamma )$ and from here we deduced $\phi (v,1-\delta _3, \gamma -\gamma _0)\subset \phi (u,\delta ,\gamma )$. The proof is now complete. $\square$

Remark 2

$(A,{\mathscr {B}}, \varDelta , \nabla )$ be bipolar fuzzy metric space (BFMS). On A $\tau _{{\mathscr {B}}}=\{X\subset A: \gamma >0~~ \text {and}~~ \delta \in (0,1)\ni \phi (u,v, \gamma )\subset X~~ \text {for each}~ u\in X \}$ is a topology. Any BFM ${\mathscr {B}}$ on A provides a topology $\tau _{{\mathscr {B}}}$ in that case.

Theorem 2

Any BFMS is $T_2$ space.

Proof

Let us $(A,{\mathscr {B}}, \varDelta , \nabla )$ be bipolar fuzzy metric space (BFMS) and u, v two discrete points in A. From here, $0<\mu _{A}^{r}(u,v,\gamma )<1, -1<\mu _{A}^{\ell }(u,v,\gamma )<0$. Take $\delta _1=\mu _{A}^{r}(u,v,\gamma ), \delta _2=\mu _{A}^{\ell }(u,v,\gamma )$ and $\delta =\max \{\delta _1, -\delta _2\}$. If we take $\delta _0\in (\delta ,1)$ then there exist $\delta _3$ and $\delta _4$ such that $\delta _3\varDelta \delta _3\ge \delta _0$ and $(\delta _4-1)\nabla (\delta _4-1)\le (\delta _0-1)$. Put $\delta _5=\max \{\delta _3,\delta _4\}$ and consider the open balls $\phi (u,1-\delta _5, \frac{\gamma }{2})$ and $\phi (v,1-\delta _5, \frac{\gamma }{2})$. Clearly, $\phi (u,1-\delta _5, \frac{\gamma }{2})\cap \phi (v,1-\delta _5, \frac{\gamma }{2})=\emptyset$. If there exists $z\in \phi (u,1-\delta _5, \frac{\gamma }{2})\cap \phi (v,1-\delta _5, \frac{\gamma }{2})$ then $\delta _1=\mu _{A}^{r}(u,v,\gamma )\ge \mu _{A}^{r}(u,z,\frac{\gamma }{2})\varDelta \mu _{A}^{r}(z,v,\frac{\gamma }{2})\ge \delta _5 \varDelta \delta _5\ge \delta _3 \varDelta \delta _3\ge \delta _0>\delta _1$ and $\delta _2=\mu _{A}^{\ell }(u,v,\gamma )\le \mu _{A}^{\ell }(u,z,\frac{\gamma }{2})\nabla \mu _{A}^{\ell }(z,v,\frac{\gamma }{2})\le (\delta _5-1)\nabla (\delta _5-1)\le (\delta _4-1) \nabla (\delta _4-1)\le (\delta _0-1)\le \delta _2$. This leads in a contradiction. As a result $(A,{\mathscr {B}},\varDelta ,\nabla )$ is $T_2$ space. $\square$

Theorem 3

Let $(A, {\mathscr {B}},\varDelta ,\nabla )$ be a BFMS and $\tau _{{\mathscr {B}}}$ is topology on A. Then a sequence $(u_k)$ of A is convergent to u if and only if $\mu _{A}^{r}(u_k,u,\gamma )\rightarrow 1$ and $\mu _{A}^{\ell }(u_k,u,\gamma )\rightarrow -1$ for $k\rightarrow \infty$.

Proof

Let $\gamma >0$ and $u_k\rightarrow u$. Then for $\delta \in (0,1)$ there exists $k_0\in {\mathbb {N}}$ such that $u_k\in \phi (u,\delta ,\gamma )$ for all $k\ge k_0$. Then $\mu _{A}^{r}(u_k,u,\gamma )>1-\delta$ and $\mu _{A}^{\ell }(u_k,u,\gamma )>\delta -1$. From here, we can deduced that $\mu _{A}^{r}(u_k,u,\gamma )\rightarrow 1$ and $\mu _{A}^{\ell }(u_k,u,\gamma )\rightarrow -1$ for $k\rightarrow \infty$.

Conversely, for each $\gamma >0$, if $\mu _{A}^{r}(u_k,u,\gamma )\rightarrow 1$ and $\mu _{A}^{\ell }(u_k,u,\gamma )\rightarrow -1$ for $k\rightarrow \infty$, then for $\delta \in (0,1)$ there exists $k_0\in {\mathbb {N}}$ such that $1-\mu _{A}^{r}(u_k,u,\gamma )<\delta$ and $-1-\mu _{A}^{\ell }(u_k,u,\gamma )<\delta$ for all $k\ge k_0$. It follows that $\mu _{A}^{r}(u_k,u,\gamma )>1-\delta$ and $\mu _{A}^{\ell }(u_k,u,\gamma )<\delta -1$ for all $k\ge k_0$. Thus, $u_k\in \phi (u,\delta ,\gamma )$ for all $k\ge k_0$ and hence $u_k\rightarrow u$. $\square$

Definition 7

A BFMS is defined as $(A, {\mathscr {B}},\varDelta ,\nabla )$. If there is one $M>0$ and $\delta \in (0,1)$ so that $\mu _{A}^{r}(u,v,M)>1-\delta$ and $\mu _{A}^{\ell }(u,v,M)<\delta -1$ for all $u,v\in A$, the subset F of A is called bipolar fuzzy bounded (BFB).

Definition 8

Let $(A, {\mathscr {B}},\varDelta ,\nabla )$ be a BFMS and $(u_k)$ is a sequence of A. If for all $\delta >0$ there exist $n_0\in {\mathbb {N}} \ni$ for all $n,m\ge n_0$ $\mu _{A}^{r}(u,v,\gamma )>1-\delta$ and $\mu _{A}^{\ell }(u,v,\gamma )<\delta -1$ then $(u_k)$ is a Cauchy sequence. Additionally, if all Cauchy sequence is convergent in respect of $\tau _{{\mathscr {B}}}$, $(A, {\mathscr {B}},\varDelta ,\nabla )$ be a complete bipolar fuzzy metric space.

Remark 3

Allow $(A, {\mathscr {B}},\varDelta ,\nabla )$ to be a BFMS produced by a metric d on A. If and only if $X\subset A$ is bounded, it is bipolar fuzzy bounded.

Remark 4

Any compact set in bipolar fuzzy metric space is closed and bounded.

Theorem 4

Any bipolar fuzzy metric space $(A, {\mathscr {B}},\varDelta ,\nabla )$ has a compact subset A that is bipolar fuzzy bounded.

Proof

Assume that X is a compact subset of A, a bipolar fuzzy metric space. Let $\gamma >0, \delta \in (0,1)$ be the case. Take into account the $\{\phi (u,\delta ,\gamma ):u\in X\}$ is a open cover. Because of the fact that X is compact, there exist $u_1,u_2,...,u_k\in X\ni X\subseteq \cup _{i=1}^{k}\phi (u_i,\delta ,\gamma )$. Consider $u,n\in X$. After that $u\in \phi (u_i,\delta ,\gamma )$ and $v\in \phi (u_{\ell },\delta ,\gamma )$ for some $i,\ell$. It means that, we have $\mu _{A}^{r}(u,u_{i},\gamma )>1-\varepsilon , \mu _{A}^{\ell }(u,u_{i},\gamma )<\varepsilon -1$, $\mu _{A}^{r}(v,u_{\ell },\gamma )>1-\varepsilon , \mu _{A}^{\ell }(v,u_{\ell },\gamma )<\varepsilon -1$. Here, get $\lambda =\min \{\mu _{A}^{r}(u_i,u_{\ell },\gamma ):1\le i,\ell \le k\}$ and $\lambda ^{'}=\max \{\mu _{A}^{\ell }(u_i,u_{\ell },\gamma ):1\le i,\ell \le k\}$. Then, $\lambda \ge 0, \lambda ^{'}\le 0$. Now we have $\mu _{A}^{r}(u,v,3\gamma )\ge \mu _{A}^{r}(u,u_i,3\gamma )\varDelta \mu _{A}^{r}(u_i,u_{\ell },\gamma ) \varDelta \mu _{A}^{r}(u_{\ell },v,\gamma )\ge (1-\delta )\varDelta (1-\delta )\varDelta \lambda >1-\sigma ^{'}$ where $0<\sigma ^{'}<1$ and $\mu _{A}^{\ell }(u,v,3\gamma )\le \mu _{A}^{\ell }(u,u_i,3\gamma )\nabla \mu _{A}^{\ell }(u_i,u_{\ell },\gamma )\nabla \mu _{A}^{\ell }(u_{\ell },v,\gamma )\le (\delta -1)\nabla (\delta -1)\nabla \lambda ^{'}<1-\sigma ^{''}$ where $-1<\sigma ^{''}<0$. Considering $\sigma =\max \{\sigma ^{'},\sigma ^{''}\}$ and $\mu _{A}^{r}(u,v,3\gamma )>1-\sigma , \mu _{A}^{\ell }(u,v,3\gamma ) <\sigma -1$ for all $u,v\in X$. As a result, X is bipolar fuzzy bounded. $\square$

Theorem 5

Let $(A, {\mathscr {B}},\varDelta ,\nabla )$ be a BFMS with a convergent subsequence for every Cauchy sequence in A. After that, $(A, {\mathscr {B}},\varDelta ,\nabla )$ is complete.

Proof

Let $(u_k)$ be a Cauchy sequence and let $(u_{k_n})$ be a subsequence of $(u_k)$ that converges to u. We must prove that $u_k\rightarrow u$. Let $\gamma >0$ and $\varepsilon \in (0,1)$. Choose $\delta \in (0,1)$ such that $(1-\delta )\nabla (1-\delta )\ge 1-\varepsilon$ and $(\delta -1)\varDelta (\delta -1)\le \varepsilon -1$. Since $(u_k)$ is Cauchy sequence, there is $k_0 \in {\mathbb {N}}$ such that $\mu _{A}^{r}(u_n,u_k,\frac{\gamma }{2})>1-\delta$ and $\mu _{A}^{\ell }(u_n,u_k,\frac{\gamma }{2})>\delta -1$ for all $n,k\ge k_0$. Since $u_{k_n}\rightarrow u$, there is $n_i\in {\mathbb {N}}$ such that $n_i>n_0, \mu _{A}^{r}(u_{n_i},u,\frac{\gamma }{2})>1-\delta$ and $\mu _{A}^{\ell }(u_{n_i},u,\frac{\gamma }{2})>\delta -1$. Then, if $k>k_0$, $\mu _{A}^{r}(u_k,u,\gamma )\ge \mu _{A}^{r}(u_k,u_{n_i},\frac{\gamma }{2})>\varDelta \mu _{A}^{r}(u_{n_i},u,\frac{\gamma }{2})>(1-\delta )\varDelta (1-\delta )\ge 1-\varepsilon$ and $\mu _{A}^{\ell }(u_k,u,\gamma )\le \mu _{A}^{\ell }(u_k,u_{n_i},\frac{\gamma }{2})\nabla \mu _{A}^{\ell }(u_{n_i},u,\frac{\gamma }{2})<(\delta -1)\nabla (\delta -1)\le \varepsilon -1$. Therefore $u_k\rightarrow u$ and as a result $(A, {\mathscr {B}},\varDelta ,\nabla )$ is complete. $\square$

Theorem 6

Let $(A, {\mathscr {B}},\varDelta ,\nabla )$ be a BFMS and $X\subset A$ with the $(\mu _X^r,\mu _X^\ell )$ subspace bipolar fuzzy metric. If and only if X is a closed subset of A, then $(X, \mu _X^r,\mu _X^\ell ,\varDelta ,\nabla )$ is complete.

Proof

Let $(u_k)$ be a Cauchy sequence in $(X,\mu _X^r,\mu _X^\ell ,\varDelta ,\nabla )$, and $X\subset A$ be a closed. And so, $(u_k)$ in A is a Cauchy sequence, and there is a point u in A where $u_k\rightarrow u$. As a result, $u\in \overline{X}=X$, and $(u_k)$ converges in X. From here, $(X, \mu _X^r,\mu _X^\ell ,\varDelta ,\nabla )$ is complete.

Consider the case, where $(X, \mu _X^r,\mu _X^\ell ,\varDelta ,\nabla )$ is complete but X is not closed. Take $u\in \overline{X}\setminus X$. Then there will be a sequence of points in X called $(u_k)$ that converges to u, and $(u_k)$ is a Cauchy sequence. It means that, for each $\delta \in (0,1)$ and $\gamma >0$, there is $k_0\in {\mathbb {N}}\ni \mu _{A}^{r}(u_k,u_\ell ,\gamma )>1-\delta$ and $\mu _{A}^{\ell }(u_k,u_\ell ,\gamma )>\delta -1$ for all $k,\ell \ge k_0$. So, $(u_k)$ is a sequence in X, $\mu _{A}^{r}(u_k,u_\ell ,\gamma )=\mu _{X}^r(u_k,u_\ell ,\gamma )$ and $\mu _{A}^{\ell }(u_k,u_\ell ,\gamma )=\mu _{X}^r(u_k,u_\ell ,\gamma )$. As a result, in X, $(u_k)$ is a Cauchy sequence. There is a $v\in X\ni u_k\rightarrow v$ because $(X,\mu _X^r,\mu _X^\ell ,\varDelta ,\nabla )$ is complete. For all $k\ge k_0$, there is $k_0\in {\mathbb {N}}\ni \mu _{X}^r(v,u_k,\gamma )>1-\delta$ and $\mu _{X}^-(v,u_k,\gamma )<\delta -1$ for all $k\ge k_0$ for each $\delta \in (0,1)$ and each $\gamma >0$. However, because $(u_k)$ is a sequence in X and $v\in X$, $\mu _{A}^{r}(v,u_k,\gamma )=\mu _{X}^r(v,u_k,\gamma )$ and $\mu _{A}^{\ell }(v,u_k,\gamma )=\mu _{X}^\ell (v,u_k,\gamma )$ are valid. As a result, $(u_k)$ converges to both u and v in $(A,{\mathscr {B}},\varDelta ,\nabla )$. Because of the fact that $u\notin X$ and $v\in X, u\ne v$. This is an inconsistency. $\square$

Lemma 1

A BFMS is defined as $(A,{\mathscr {B}},\varDelta ,\nabla )$. In the case of $\gamma >0$, $\delta , \varepsilon \in (0,1)$ and $(1-\varepsilon )\varDelta (1-\varepsilon )\ge 1-\delta$ and $(\varepsilon -1)\nabla (\varepsilon -1)\le \delta -1$, so $\overline{\phi (u,\varepsilon ,\frac{\gamma }{2})}\subset \phi (u,\delta ,\gamma )$ is valid.

Proof

Let us take $v\in \overline{\phi (u,\varepsilon ,\frac{\gamma }{2})}$ and $\phi (v,\varepsilon ,\frac{\gamma }{2})$ be an open ball with center u and radius $\varepsilon$. Because of the fact that $\phi (v,\varepsilon ,\frac{\gamma }{2})\cap \phi (u,\varepsilon ,\frac{\gamma }{2})\ne \emptyset$, there exist a $z\in \phi (v,\varepsilon ,\frac{\gamma }{2})\cap \phi (u,\varepsilon ,\frac{\gamma }{2})$. Next, there’s $\mu _{A}^{r}(u,v,\gamma )\ge \mu _{A}^{r}(u,z,\frac{\gamma }{2})\varDelta \mu _{A}^{r}(v,z,\frac{\gamma }{2})>(1-\varepsilon )\varDelta (1-\varepsilon )\ge 1-\delta$ and $\mu _{A}^{\ell }(u,v,\gamma )\le \mu _{A}^{\ell }(u,z,\frac{\gamma }{2})\nabla \mu _{A}^{\ell }(v,z,\frac{\gamma }{2})<(\varepsilon -1)\nabla (\varepsilon -1)\le \delta -1$. Hence $z\in \phi (u,\delta ,\gamma )$ and thus $\overline{\phi (u,\varepsilon ,\frac{\gamma }{2})}\subset \phi (u,\delta ,\gamma )$. $\square$

Theorem 7

If and only if a open set in $A\ne \emptyset$ includes an open ball which closure is disjoint from X, then a subset X of a bipolar fuzzy metric space $(A,{\mathscr {B}},\varDelta ,\nabla )$ is dense in nowhere.

Proof

Let $\emptyset \ne W\subset X$ be the case. Then there is a open set $Z\ne \emptyset$ in which $Z\subset W$ and $Z\cap \overline{X}\ne \emptyset$ are valid. Let $u\in Z$. After that, there’s $\delta \in (0,1)$ and $\gamma >0\ni \phi (u,\delta ,\gamma )\subset Z$. Consider $\varepsilon \in (0,1)$ so that $(1-\varepsilon )\varDelta (1-\varepsilon )\ge 1-\delta$ and $(\varepsilon -1)\nabla (\varepsilon -1)\le \delta -1$. By taking into account Lemma 1$\overline{\phi (u,\varepsilon ,\frac{\gamma }{2})}\subset \phi (u,\delta ,\gamma )$. It means that, $\phi (u,\varepsilon ,\frac{\gamma }{2})\subset W$ and $\overline{\phi (u,\varepsilon ,\frac{\gamma }{2})}\cap X=\emptyset$.

Assume, on the other hand, that X is not dense anywhere. Then, $int(\overline{X})\ne \emptyset$, implying that there is a nonempty open set W such that $W\subset \overline{X}$. Let $\phi (u,\delta ,\gamma )$ be an open ball with the property that $\phi (u,\delta ,\gamma )$ is a subset of W. Then $\overline{\phi (u,\delta ,\gamma )}\cap X\ne \emptyset$ comes into play. This is an inconsistency. $\square$

For bipolar fuzzy metric space, we can prove Baire’s theorem.

Theorem 8

Allow $\{W_k:k\in {\mathbb {N}}\}$ represent a sequence of dense open subsets of a complete bipolar fuzzy metric space $(A,{\mathscr {B}},\varDelta ,\nabla )$. And after that $\cap _{k\in {\mathbb {N}}}W_k$ is dense in A.

Proof

Let’s get this part with $Z\ne \emptyset$ be a open set of A. Because $W_1$ is dense in A, $Z\cap W_1\ne \emptyset$. Take $u_1\in Z\cap W_1$. Since $Z\cap W_1$ is open, there is $\delta _1\in (0,1)$ and $\gamma _1>0$ so that $\phi (u_1,\delta _1,\gamma _1)\subset Z\cap W_1$. Consider $\delta _1^{'}<\delta$ and $\gamma _1^{'}=\min (\gamma _1,1)$ such that $\overline{\phi (u_1,\delta _1^{'},\gamma _1^{'})}\subset Z\cap W_1$. Since $W_2$ is dense in A, $\phi (u_1,\delta _1^{'},\gamma _1^{'})\cap W_2\ne \emptyset$. Let $u_2\in \phi (u_1,\delta _1^{'},\gamma _1^{'})\cap W_2$. Since $\phi (u_1,\delta _1^{'},\gamma _1^{'})\cap W_2$ is open there exist $\delta _2\in (0,\frac{1}{2})$ and $\gamma _2>0 \ni$ $\phi (u_2,\delta _2,\gamma _2)\subset \phi (u_1,\delta _1^{'},\gamma _1^{'})\cap W_2$. Consider $\delta _2^{'}<\delta _2$ and $\gamma _2^{'}=\min (\gamma _2,\frac{1}{2})$ such that $\overline{\phi (u_2,\delta _2^{'},\gamma _2^{'})}\subset \phi (u_1,\delta _1^{'},\gamma _1^{'})\cap W_2$. If we keep going this way, we obtain a sequence $(u_k)$ in A and a sequence $(\gamma _k^{'})$ such that $0<\gamma _k^{'}<\frac{1}{k}$ and $\overline{\phi (u_k,\delta _k^{'},\gamma _k^{'})}\subset \phi (u_{k-1},\delta _{k-1}^{'},\gamma _{k-1}^{'}) \cap W_k$. Now we will take into account that $(u_k)$ is a Cauchy sequence. For $\gamma >0$ and $\varepsilon >0$, select $k_0\in {\mathbb {N}}$ such that $\frac{1}{k_0}<\gamma$ and $\frac{1}{k_0}<\varepsilon$. Then for $k\ge k_0$ and $\ell \ge k$, $\mu _{A}^{r}(u_k,u_{\ell },\gamma )\ge \mu _{A}^{r}(u_k,u_{\ell },\frac{1}{k})\ge 1-\frac{1}{k}>1-\varepsilon , \mu _{A}^{\ell }(u_k,u_{\ell },\gamma )\le \mu _{A}^{\ell }(u_k,u_{\ell },\frac{1}{k})\le \frac{1}{k}-1<\varepsilon -1$. As a result, the sequence $(u_k)$ is a Cauchy sequence. Because of the fact that A is complete, $u\in A$ arises such that $u_k\rightarrow u$. We offer $u\in \overline{\phi (u_k,\delta _k^{'},\gamma _k^{'})}$ for $n\ge k$ because $u_n\in \phi (u_k,\delta _k^{'},\gamma _k^{'})$. Consequently, $u\in \overline{\phi (u_k,\delta _k^{'},\gamma _k^{'})}\subset \phi (u_{k-1},\delta _{k-1}^{'},\gamma _{k-1}^{'})\cap W_k$ for all k. Therefore, $Z\cap \left( \cap _{k\in {\mathbb {N}}}W_k\right) \ne \emptyset$. from here, $\cap _{k\in {\mathbb {N}}} W_k$ is dense in A. $\square$

Note 1

The first group does not include any complete bipolar fuzzy metric space that cannot be described as the union of a sequence of nowhere dense sets. As a result, any complete bipolar fuzzy metric space belongs to the second group.

Remark 5

Since every metric induces a bipolar fuzzy metric, and bipolar fuzzy metric is a general statement of fuzzy metric, Baire’s theorem for complete metric space (Zhang 1998) and Baire’s theorem for complete fuzzy metric space (Qurashi and Shabir 2018) are special cases of the above theorem.

4 Some properties of complete bipolar fuzzy metric spaces

The topological structure of bipolar fuzzy sets is investigated in further depth in this section, as well as the Uniform Limit Theorem.

Definition 9

A BFMS is defined as $(A,{\mathscr {B}},\varDelta ,\nabla )$. A set $[D_k]_{k\in {\mathbb {N}}}$ is said to have bipolar fuzzy diameter zero if there exists $k_0\in {\mathbb {N}}$ for each $\delta \in (0,1)$ and each $\gamma >0$ so that $\mu _{A}^{r}(u,v,\gamma )>1-\varepsilon$ and $\mu _{A}^{\ell }(u,v,\gamma )<\varepsilon -1$ for all $u,v\in D_{k_0}$.

Remark 6

If and only if D is a singleton set, a nonempty subset D of a bipolar fuzzy metric space A has bipolar fuzzy diameter zero.

Theorem 9

If and only if every nested sequence $[D_k]_{k\in {\mathbb {N}}}$ of nonempty closed sets with bipolar fuzzy diameter zero has nonempty intersection, a bipolar fuzzy metric space $(A,{\mathscr {B}},\varDelta ,\nabla )$ is complete.

Proof

Assume that the specified condition exists first. $(A,{\mathscr {B}},\varDelta ,\nabla )$, we imply, is complete. In A, consider $(u_k)$ to be a Cauchy sequence. The set $N_k=\{u_n:n\ge k\}$ and $D_k=\overline{N_k}$ then $[D_k]$ is said to have a bipolar fuzzy diameter zero. For taken $\varepsilon \in (0,1)$ and $\gamma >0$, we select $\delta \in (0,1)$ as a result $(1-\delta )\varDelta (1-\delta )\varDelta (1-\delta )>1-\varepsilon$ and $(\delta -1)\nabla (\delta -1)\nabla (\delta -1)<\varepsilon -1$. Because $(u_k)$ is a Cauchy sequence, there will be $k_0\in {\mathbb {N}}$ such that $\mu _{A}^{r}(u_k,u_{\ell },\frac{\gamma }{3})>1-\varepsilon$ and $\mu _{A}^{\ell }(u_k,u_{\ell },\frac{\gamma }{3})<\varepsilon -1$ are valid for all $\ell ,k\ge k_0$. As a result, $\mu _{A}^{r}(u,v,\frac{\gamma }{3})>1-\varepsilon$ and $\mu _{A}^{\ell }(u,v,\frac{\gamma }{3})>\varepsilon -1$ for all $u,v\in N_{k_0}$. Let $u,v\in D_{k_0}$. Then in $N_{k_0}$, there are sequences $(u_k^{'})$ and $(v_k^{'})$ so that $u_k^{'}\rightarrow u$ and $v_k^{'}\rightarrow v$. Hence $u_k^{'}\in \phi (u,\delta ,\frac{\gamma }{3})$ and $v_k^{'}\in \phi (v,\delta ,\frac{\gamma }{3})$ for an enough large k. Now we have $\mu _{A}^{r}(u,v,\gamma )\ge \mu _{A}^{r}(u,u_k^{'},\frac{\gamma }{3})\varDelta \mu _{A}^{r}(u_k^{'},v_k^{'},\frac{\gamma }{3})\varDelta \mu _{A}^{r}(v_k^{'},v,\frac{\gamma }{3})>(1-\delta )\varDelta (1-\delta )\varDelta (1-\delta )>1-\varepsilon$ and $\mu _{A}^{\ell }(u,v,\gamma )\le \mu _{A}^{\ell }(u,u_k^{'},\frac{\gamma }{3})\nabla \mu _{A}^{\ell }(u_k^{'},v_k^{'},\frac{\gamma }{3})\nabla \mu _{A}^{\ell }(v_k^{'},v,\frac{\gamma }{3})<(\delta -1)\nabla (\delta -1)\nabla (\delta -1)<\varepsilon -1$. After that, $\mu _{A}^{r}(u,v,\gamma )>1-\varepsilon$ and $\mu _{A}^{\ell }(u,v,\gamma )<\varepsilon -1$ for all $u,v\in D_{k_0}$. Eventually, $[D_k]$ has bipolar fuzzy diameter zero and as a result of hypothesis the variable $\cap _{k\in {\mathbb {N}}}D_k$ is nonempty.

Take $u\in \cap _{k\in {\mathbb {N}}}D_k$. We show that $u_k\rightarrow u$. Then for $\delta \in (0,1)$ and $\gamma >0$, there exists $k_1\in {\mathbb {N}}$ such that $\mu _{A}^{r}(u_k,u,\gamma )>1-\delta$ and $\mu _{A}^{\ell }(u_k,u,\gamma )<\delta -1$ for all $k\ge k_1$. Therefore, for each $\gamma >0$ $\mu _{A}^{r}(u_k,u,\gamma )\rightarrow 1$ and $\mu _{A}^{\ell }(u_k,u,\gamma )\rightarrow -1$ as $k\rightarrow \infty$ and hence $u_k\rightarrow u$. Therefore, $(A,{\mathscr {B}},\varDelta ,\nabla )$ is complete.

In the opposite case, assume $(A,{\mathscr {B}},\varDelta ,\nabla )$ is complete and $[D_k]_{n\in {\mathbb {N}}}$ is a nested sequence of nonempty closed sets with bipolar fuzzy diameter zero. Get a point $u_k\in D_k$ for each $k\in {\mathbb {N}}$. A Cauchy sequence is defined as $(u_k)$. Because of the fact that $[D_k]$ has bipolar fuzzy diameter zero, for $\gamma >0$ and $\delta \in (0,1)$, there is $k_0\in {\mathbb {N}}$ so that $\mu _{A}^{r}(u,v,\gamma )>1-\delta$ and $\mu _{A}^{\ell }(u,v,\gamma )<\delta -1$ for all $u,v\in D_{k_0}$. Since, $[D_k]$ is nested sequence $\mu _{A}^{r}(u_k,u_{\ell },\gamma )>1-\delta$ and $\mu _{A}^{\ell }(u_k,u_{\ell },\gamma )<\delta -1$ for all $\ell ,k\ge k_0$. As a consequence, the sequence $(u_k)$ is a Cauchy sequence. Since $(A,{\mathscr {B}},\varDelta ,\nabla )$ is complete, $u_k\rightarrow u$ for some $u\in A$. As a result, for every k, $u\in \overline{D}_k=D_k$ and $u\in \cap _{k\in {\mathbb {N}}}D_k$. The proof is now finished. $\square$

Remark 7

The element $u\in \cap _{k\in {\mathbb {N}}}D_k$ is singular. If two elements are present as $u,v\in \cap _{k\in {\mathbb {N}}}D_k$, since $[D_k]$ has bipolar fuzzy diameter zero, for each fixed $\gamma >0$, $\mu _{A}^{r}(u,v,\gamma )>1-\frac{1}{k}$ and $\mu _{A}^{\ell }(u,v,\gamma )<\frac{1}{k}-1$ for each $k\in {\mathbb {N}}$. This denotes $\mu _{A}^{r}(u,v,\gamma )=1$ and $\mu _{A}^{\ell }(u,v,\gamma )=-1$ and consequently $u=-v$.

It’s worth noting that the regular bipolar fuzzy metric and the corresponding metric produce the same topologies. But here’s what we’ve got:

Corollary 1

Any nested sequence $[D_k]_{n\in {\mathbb {N}}}$ of nonempty closed sets with diameter tending to zero has nonempty intersection if and only if a metric space (X, d) is complete.

Theorem 10

Every separable bipolar fuzzy metric space is second countable.

Proof

The provided separable bipolar fuzzy metric space is $(A,{\mathscr {B}},\varDelta ,\nabla )$. A countable dense subset of A will be $X=\{u_k:k\in {\mathbb {N}}\}$. Consider the case of the family ${\mathscr {K}}=\{\phi (u_j,\frac{1}{n},\frac{1}{n}):j,n\in {\mathbb {N}}\}$. Then ${\mathscr {K}}$ can be counted. We conclude that ${\mathscr {K}}$ is a base for the family of all open sets in A. Take O be any open set in A and $u\in O$. Then there is $\gamma >0$ and $\delta \in (0,1)$ such that $\phi (u,\delta ,\gamma )\subset O$. Because of the fact that $\delta \in (0,1)$, $\varepsilon \in (0,1)$ can be selected such that $(1-\varepsilon )\varDelta (1-\varepsilon )>1-\delta$ and $(\varepsilon -1)\nabla (\varepsilon -1)<\delta -1$. Take $t\in {\mathbb {N}}$ such that $\frac{1}{t}<\min \{\varepsilon ,\frac{\gamma }{2}\}$. Since X is dense in A, there exists $u_j\in X$ such that $u_j\in \phi (u,\frac{1}{t},\frac{1}{t})$. Now, if $v\in \phi (u_j,\frac{1}{t},\frac{1}{t})$, then $\mu _{A}^{r}(u,v,\gamma )\ge \mu _{A}^{r}(u,u_j,\frac{\gamma }{2})\varDelta \mu _{A}^{r}(v,u_j,\frac{\gamma }{2})\ge \mu _{A}^{r}(u,u_j,\frac{1}{t})\varDelta \mu _{A}^{r}(v,u_j,\frac{1}{t})\ge (1-\frac{1}{t})\varDelta (1-\frac{1}{t})\ge (1-\varepsilon )\varDelta (1-\varepsilon )>1-\delta$ and $\mu _{A}^{\ell }(u,v,\gamma )\le \mu _{A}^{\ell }(u,u_j,\frac{\gamma }{2})\nabla \mu _{A}^{\ell }(v,u_j,\frac{\gamma }{2})\le \mu _{A}^{\ell }(u,u_j,\frac{1}{t}) \nabla \mu _{A}^{\ell }(v,u_j,\frac{1}{t})\le (\frac{1}{t}-1)\nabla (\frac{1}{t}-1)\le (\varepsilon -1)\nabla (\varepsilon -1)<\delta -1$. Finally, $v\in \phi (u,\delta ,\gamma )\subset O$ and as a result ${\mathscr {K}}$ is a base. $\square$

Remark 8

Since second countability is an inherited property and second countability means separability, we get: A separable bipolar fuzzy metric space has separable subspaces.

Definition 10

Let A be a set that is not empty and $(B,{\mathscr {B}},\varDelta ,\nabla )$ be an bipolar fuzzy metric space. If we take $\gamma >0$ and $\delta \in (0,1)$, then a sequence $(g_k)$ of functions from A to B converges uniformly to a function g from A to B and $k_0\in {\mathbb {N}}$ arises so that $\mu _{A}^{r}(g_k(u),g(u),\gamma )>1-\delta$ and $\mu _{A}^{\ell }(g_k(u),g(u),\gamma )>\delta -1$ for all $k\ge k_0$ and $u\in A$.

Theorem 11

(Uniform limit theorem) Let $g_k:A\rightarrow B$ denote a sequence of continuous functions connecting a topological space A to a bipolar fuzzy metric space$(B,{\mathscr {B}},\varDelta ,\nabla )$. Then, g is continuous, if $(g_k)$ converges uniformly to $g:A\rightarrow B$.

Proof

Assume Z is an open set of B and allow $u_0\in g^{-1}(Z)$. We want to explore a W neighborhood of $u_0$ where $g(W)\subset Z$. Because Z is open, $\gamma >0$ and $\delta \in (0,1)$ exist, allowing $\phi (g(u_0),\delta ,\gamma )\subset Z$. Because $\delta \in (0,1)$, we select a $\varepsilon \in (0,1)$ so that $(1-\varepsilon )\varDelta (1-\varepsilon )\varDelta (1-\varepsilon )>1-\delta$ and $(\varepsilon -1)\nabla (\varepsilon -1)\nabla (\varepsilon -1)<\delta -1$. Because $(g_k)$ converges to g uniformly, in the case of $\gamma >0$ and $\varepsilon \in (0,1)$, $k_0\in {\mathbb {N}}$ exists in a form that $\mu _{A}^{r}(g_k(u),g(u),\frac{\gamma }{3})>1-\varepsilon$ and $\mu _{A}^{\ell }(g_k(u),g(u),\frac{\gamma }{3})<\varepsilon -1$ for all $k\ge k_0$ and $u\in A$. While $g_k$ is continuous for all $k\in {\mathbb {N}}$ there is a neighborhood W of $u_0$ in which $g_k(W)\subset \phi (g_k(u_0),\varepsilon ,\frac{\gamma }{3})$. Consequently, $\mu _{A}^{r}(g_k(u),g_k(u_0),\frac{\gamma }{3})>1-\varepsilon$ and $\mu _{A}^{\ell }(g_k(u),g_k(u_0),\frac{\gamma }{3})<\varepsilon -1$ for all $u\in W$. Also, $\mu _{A}^{r}(g(u),g(u_0),\gamma )\ge \mu _{A}^{r}(g(u),g_k(u),\frac{\gamma }{3})\varDelta \mu _{A}^{r}(g_k(u),g_k(u_0),\frac{\gamma }{3})\varDelta \mu _{A}^{r}(g_k(u_0),g(u_0),\frac{\gamma }{3})\ge (1-\varepsilon )\varDelta (1-\varepsilon )\varDelta (1-\varepsilon )>1-\delta$ and $\mu _{A}^{\ell }(g(u),g(u_0),\gamma )\le \mu _{A}^{\ell }(g(u),g_k(u),\frac{\gamma }{3})\nabla \mu _{A}^{\ell }(g_k(u),g_k(u_0),\frac{\gamma }{3})$ $\nabla \mu _{A}^{\ell }(g_k(u_0),g(u_0),\frac{\gamma }{3}) \le (\varepsilon -1)\nabla (\varepsilon -1)\nabla (\varepsilon -1)>\delta -1$. Thus $g(u)\in \phi (g(u_0),\delta ,\gamma )\subset Z$ for all $u\in W$. From here, $g(W)\subset Z$ and so g is continuous. $\square$

5 Application to vaccine selection in COVID-19

The COVID-19 pandemic has emerged as a serious public health emergency and prompted a process to be addressed. As of March 11, 2020, the World Health Organization declared this event as a public health emergency in accordance with the International Health Regulations. It is important to choose the most appropriate vaccine for the groups to be administered COVID-19 vaccine by evaluating the risks of exposure to the disease, the risks of spreading and transmitting the disease, and the negative impact of the disease on the functioning of social life. Until now, serious side effects have not been encountered in both clinical trials conducted for COVID-19 vaccines and current vaccination practices. Post-vaccination side effects are often mild.

These; mild side effects such as fatigue, headache, fever, chills, muscle pain, vomiting, diarrhea, pain, redness, swelling in the area where the vaccine was applied.

No vaccine can be approved or put on the market without conducting phase studies and without transparently sharing the results of these studies with the relevant authorities. The organization conducting vaccine studies must comply with international quality standards during the development and production process. These standards are Good Laboratory Practices (GLP), Good Clinical Practices (GCP) and Good Manufacturing Practices (GMP). In addition, during the phase studies, when a situation that is thought to be related to the vaccine and significantly affects human health is detected, the studies are stopped. Studies are carried out only if it is ensured that the problem is not related to the vaccine, and if vaccination is performed, it is continued from where it left off. However, it is still important to choose the most suitable vaccine for the person.

Several different types of vaccines are being developed for COVID-19. All of these vaccines are designed to teach the body’s immune system to safely introduce and destroy the virus that causes COVID-19. We evaluate these vaccines in five categories.

• 1. Vaccines that do not cause disease but contain inactivated virus (Inactivated vaccines) that produce an immune response. $(L_1)$

• 2. Vaccines containing attenuated virus that do not cause disease but produce an immune response (Live attenuated vaccines). $(L_2)$

• 3. Protein-based vaccines that use protein fragments that mimic the structure of the COVID-19 virus to safely induce an immune response.$(L_3)$

• 4. Viral vector vaccines using non-pathogenic viruses that carry RNA particles of the COVID-19 virus to create a safe immune response. $(L_4)$

• 5. M-RNA and DNA vaccines, a state-of-the-art approach uses genetically engineered RNA and DNA fragments to produce proteins that induce a safe immune response on its own. $(L_5)$

In this section, we have included an application that shows which of these 5 vaccine types is more suitable. Appropriate measurements of similarity were used in this application that allow us to make a comparison between selected bipolar fuzzy sets.

Now, let’s start by giving a definition of similarity measure on BFS.

Definition 11

(Abdullah et al. 2014). Let U be the universe of discourse and E be the set of parameters and $B\subseteq E$. Let $\psi :B\rightarrow BF^{U}$ be a mapping, then a bipolar fuzzy soft set (BFSS) $(\psi , B)$ or $\psi _B$ is defined by

$$\begin{aligned} (\psi , B)= & {} \Bigg \{\bigg (e, \{ \ell , \mu _{A}^{r}(\ell ),\mu _{A}^{\ell }(\ell )\} \bigg ) : e\in B, \ell \in U \Bigg \}. \end{aligned}$$

If $U= \{\ell _1, \ldots , \ell _m\}$, $B= \{e_1, \ldots , e_n\}$, then BFSS $\psi _B$ in the tabular form is expressed as in the following:

Definition 12

Let $U=\{\ell _i:i=1,2,\ldots ,m\}$ be a classical set and $P=\{P_j:j=1,2,\ldots ,n\}$ is the sum of attributes. Then

$$\begin{aligned} \psi _1=(\psi _1,B)= & {} \begin{pmatrix} (\mu _{11}^r,\mu _{11}^{\ell })_{\psi _1} &{} (\mu _{12}^r,\mu _{12}^{\ell })_{\psi _1} &{} \cdot \cdot \cdot &{}(\mu _{1n}^r,\mu _{1n}^{\ell })_{\psi _1}\\ (\mu _{21}^r,\mu _{21}^{\ell })_{ \psi _1} &{} (\mu _{22}^r,\mu _{22}^{\ell })_{\psi _1} &{} \cdot \cdot \cdot &{} (\mu _{2n}^r,\mu _{2n}^{\ell })_{\psi _1}\\ \vdots &{} \vdots &{} \ddots &{} \vdots \\ (\mu _{m1}^r,\mu _{m1}^{\ell })_{\psi _1} &{} (\mu _{m2}^r,\mu _{m2}^{\ell })_{\psi _1} &{} \cdot \cdot \cdot &{} (\mu _{mn}^r,\mu _{mn}^{\ell })_{\psi _1}\\ \end{pmatrix} \end{aligned}$$

and

$$\begin{aligned} \psi _2=(\psi _2,B)= & {} \begin{pmatrix} (\mu _{11}^r,\mu _{11}^{\ell })_{\psi _2} &{} (\mu _{12}^r,\mu _{12}^{\ell })_{\psi _2} &{} \cdot \cdot \cdot &{}(\mu _{1n}^+,\mu _{1n}^{\ell })_{\psi _2}\\ (\mu _{21}^r,\mu _{21}^{\ell })_{\psi _2} &{} (\mu _{22}^r,\mu _{22}^{\ell })_{\psi _2} &{} \cdot \cdot \cdot &{} (\mu _{2n}^r,\mu _{2n}^{\ell })_{\psi _2}\\ \vdots &{} \vdots &{} \ddots &{} \vdots \\ (\mu _{m1}^r,\mu _{m1}^{\ell })_{\psi _2} &{} (\mu _{m2}^r,\mu _{m2}^{\ell })_{\psi _2} &{} \cdot \cdot \cdot &{} (\mu _{mn}^r,\mu _{mn}^{\ell })_{\psi _2}\\ \end{pmatrix} \end{aligned}$$

are BFS-matrices $(\psi _1, P)$ and $(\psi _2, P)$, then similarity measure between $(\psi _1, P)$ and $(\psi _2, P)$ is given as below:

$$\begin{aligned} \text {Sim}(\psi _1,\psi _2)=\frac{\langle \psi _1,\psi _2\rangle }{\Vert \psi _1\Vert \psi _2}. \end{aligned}$$

Here

$$\begin{aligned} \langle \psi _1, \psi _2\rangle= & {} \varSigma _{i, j} (\mu ^{r}_{ij}, \mu ^{\ell }_{ij})_{\psi _1} \cdot (\mu ^{r}_{ij}, \mu ^{\ell }_{ij})_{\psi _2},\\ \Vert \psi _1\Vert= & {} \sqrt{\langle \psi _1,\psi _1\rangle }. \end{aligned}$$

Proposition 1

The similarity measure defined in Definition 12 satisfies the following situations:

• 1. $0 \le \mathrm{Sim}(\psi _1, \psi _2) \le 1$

• 2. $\mathrm{Sim}(\psi _1, \psi _2) = 1 \Leftrightarrow \psi _1 = \psi _2$

• 3. $\mathrm{Sim}(\psi _1, \psi _2) = \mathrm{Sim}(\psi _2, \psi _1)$.

Definition 13

Let $\psi _1$ and $\psi _2$ be two BFSSs. Suppose that the weights of attributes $p_j$ are $\vartheta _j\in [0, 1]$ for $j=1,2,\ldots ,n$. The weighted similarity measure $\mathrm{Sim}_W$ between $\psi _1$ and $\psi _2$ is defined by

$$\begin{aligned} \mathrm{Sim}_W(\psi _1, \psi _2) = \frac{<\psi _1, \psi _2>}{\Vert \psi _1\Vert \Vert \psi _2\Vert } \end{aligned}$$

where

$$\begin{aligned}<\psi _1, \psi _2>= & {} \frac{\varSigma _{i, j} \vartheta _j (\mu ^{+}_{ij}, \mu ^{-}_{ij})_{\psi _1} . (\mu ^{+}_{ij}, \mu ^{-}_{ij})_{\psi _2}}{\varSigma _j \vartheta _j},\\ \Vert \psi _1\Vert= & {} \sqrt{<\psi _1, \psi _1>}. \end{aligned}$$

Proposition 2

The similarity measure defined in Definition 13 provides the circumstances listed below:

• 1. $0 \le \mathrm{Sim}_{W}(\psi _1, \psi _2) \le 1$

• 2. $\mathrm{Sim}_{W}(\psi _1, \psi _2) = 1 \Leftrightarrow \psi _1 = \psi _2$

• 3. $\mathrm{Sim}_{W}(\psi _1, \psi _2) = \mathrm{Sim}_{W}(\psi _2, \psi _1)$.

Example 3

Let us take $\psi _1$ and $\psi _2$, BFS-matrices of $(\psi _1,p)$ and $(\psi _2,p)$ as given below:

$$\begin{aligned} \psi _1= & {} \begin{pmatrix} (0.513, -0.279) &{} (0.312, -0.700) &{} (0.624, -0.861)\\ (0.342, -0.489) &{} (0.317, -0.817) &{} (0.754, -0.426)\\ (0.329, -0.544) &{} (0.604, -0.152) &{} (0.214, -0.191)\\ \end{pmatrix},\\ \psi _2= & {} \begin{pmatrix} (0.641, -0.040) &{} (0.542, -0.842) &{} (0.384, -0.216)\\ (0.045, -0.343) &{} (0.941, -0.525) &{} (0.782, -0.013)\\ (0.253, -0.517) &{} (0.317, -0.544) &{} (0.262, -0.415)\\ \end{pmatrix}. \end{aligned}$$

Then similarity measure between $(\psi _1,p)$ and $(\psi _2,p)$ will be determined as follows:

$$\begin{aligned} \langle \psi _1,\psi _2\rangle =3.7960\\ \Vert \psi _1\Vert =2.1909\\ \Vert \psi _2\Vert =2.1123\\ \because \mathrm{Sim}(\psi _1,\psi _2)=0.8203. \end{aligned}$$

Additionally, consider $\vartheta _1=0.12,\vartheta _1=0.67$ and $\vartheta _1=0.54$ as weights of $p_1, p_2$ and $p_3$. After, the subsequent computations can be made:

$$\begin{aligned} \langle \psi _1,\psi _2\rangle =1.9185\\ \Vert \psi _1\Vert =1.5352\\ \Vert \psi _2\Vert =1.5428\\ \because \mathrm{Sim}_{W}(\psi _1,\psi _2)=0.8100. \end{aligned}$$

Fig. 1

Diagram of Algorithm 1

Fig. 2

An example of decision making algorithm

The materials required of Algorithm 1 are illustrated in Figs. 1 and 2.

5.1 Numerical example

A government intends to purchase vaccines from corporations that have the vaccine due to the COVID-19 outbreak. The government selected two decision makers to determine the best type of vaccines from the five types of vaccines suitable for its conditions along with particularities $C=\{\alpha _1,\alpha _2,\alpha _3\}$. Here,

• $\alpha _1=\text {Sustainability}$

• $\alpha _2=\text {Efficiency}$

• $\alpha _3=\text {Cost}$

Take into account the collection of variables

$$\begin{aligned} E=\{e_1=\text {unimpressed},e_2=\text {impressed},e_3=\text {extremely impressed}\}, \end{aligned}$$

which are linguistic identifiers that define the degree of satisfaction with decision-making when evaluating a vaccine. Assume that $(\psi ,E)$ is the regular BFSS, which is considered as a model BFSS by the decision maker. A positive grade is given if the effectiveness of the type of vaccine is more appropriate than the country’s expectation standard. A negative grade is also granted if the suitability of the type of vaccine is less effective based on certain parameters. The government determines two decision makers from the management committee to assign the best form of vaccine to administer to the population. Conclude that $(\psi _i,E)$ is the BFSS model of the vaccine kind, with $L_i, i= 1,2,3,4,5$ being assigned by each decision manager. Then, by considering the maximum of the positive values and the minimum of the negative values, we arrive at a consensus BFS decision. To determine the SM between these BFSSs, we will compare the vaccines’ characteristics with the analysis in Table 3. BFS decision matrices are shown in Tables 1 and  2. Assume that the threshold value is 0.8. When we class with this value to the determined value of SM in Table 3, it would seem that the BFSS model $(\psi ,E)$ is very similar to $(\psi _1,E)$. Also, it means that, it is unlike any of the other sets. We acknowledge that the form of vaccine $L_1$ is the best option for satisfying certain requirements. Since $L_1$ is the most similar to model, the government should select it as their country’s vaccine provider.

In Table 4, the SMs between BFS-model and vaccine form are listed as below:

To compute the weighted SMs, we use the weights $w_1= 0.119$, $w_2=0.452$, and $w_3=0.817$, which correspond to $e_1$, $e_2$, and $e_3$, respectively. By this way we calculate the weighted SMs. Table 5 becomes

$$\begin{aligned} L_1 \succ L_4 \succ L_5 \succ L_2 \succ L_3 \end{aligned}$$

when ranking vaccine types based on weighted similarity measure.

Thus, as shown in the Tables 4, 5, it can be easily said that $L_1$ is the most suitable and preferable vaccine option.

Table 1 BFS decision matrix 1

Table 2 BFS decision matrix 2

Table 3 Unanimous BFS decision matrix

Table 4 SM between the BFSS method and the vaccine type

Table 4 becomes

$$\begin{aligned} L_1 \succ L_5 \succ L_4 \succ L_3 \succ L_1 \end{aligned}$$

when ranking vaccine types based on similarity measure.

Table 5 shows the weighted SM between the BFSS model and the vaccine type as given in the follows:

Table 5 Weighted SM between the BFSS method and the vaccine type

Fig. 3

Diagram of $\mathrm{Sim}$ and $\mathrm{Sim}_W$ of type of vaccines provider

The ranking of these two measures for each type of vaccines provider could be seen in Fig. 3.

6 Conclusion

The aim of this research is to define bipolar metric spaces and investigate some of their properties. The characteristic features of BFMSs have been developed, including open set, $T_2$-space, compactness, completeness and nowhere dense. Following that, Baire Category and Uniform Convergence Theorems for bipolar metric spaces are presented. Afterwards, an application on determining the appropriate type of vaccine in the treatment process of COVID-19 is given using similarity measure between bipolar fuzzy metric spaces. For vaccine selection and supply chain management, an advanced multi-attribute decision-making (MADM) algorithm is being developed using similarity measure between bipolar fuzzy metric spaces. The validity of the proposed MADM method for selecting the best proper form of vaccine is also established.