Some Topological Properties on The Cartesian Product of Fuzzy

b-Metric Space

Abstract: Abstract: The aim of this paper is to introduce the concept of the Cartesian product of two fuzzy

b-metric spaces with related properties. We have introduced the property that the Cartesian product of

two complete fuzzy b-metric spaces is a complete fuzzy b-metric space in terms of the product t-norm (.),

as well as the minimum t-norm (∧). The properties of convergent and Cauchy sequence are introduced

by using this concept.

Key-Words:Fuzzy b-metric space, Convergent sequence, Cauchy sequence, Cartesian product,Complete

Received: August 12, 2023. Revised: September 13, 2024. Accepted: October 12, 2024. Published: November 18, 2024.

1 Introduction

There are numerous ways to generalize the idea

of metric spaces, one of them is a b-metric space.

Now a days The theory of metric space has be-

come an emerging ﬁeld of research and one of

those methods is the fuzzy idea. With the study

in new horizon of research, Zadeh [5] introduced

the concept of fuzzy sets. Many eminent schol-

ars have introduced the applications of fuzzy set

theory to the terminology used in topology and

analysis. It is well recognized that scientists and

mathematicians can beneﬁt by using the concept

of fuzzy metric space as a generalization. Several

researchers have introduced fuzzy metric spaces

using various methods. In general it is not pos-

sible to measure the exact distance between any

two places precisely. Thus we conclude that

while measuring the same distance between two

places in different times, we will get the differ-

ent results. This situation can be handled by two

ways probabilistic and statistical approach. But

by using the probabilistic approach it uses the

idea of distribution function instead non-negative

real numbers. As the uncertainty in the distance

between two points is due to fuzziness instead of

randomness. Consequently, by using the contin-

uous t-norm, fuzzy metric space was deﬁned by

many researchers. Latter on it was Updated by

George and Veeramani [1].

In 2012, Sedghi and Shobe [11] found the new

idea with common ﬁxed point theorem in fuzzy

b-metric space. Hussain et al.[4] established

the concept of a fuzzy b-metric space and ob-

tained various ﬁxed point theorems for this type

of space through their publication of an article

in 2015. Although Hussain’s is formally cor-

rect, it is mathematically unjustiﬁed. Nadaban

[10] introduced some of the topological aspects

of a fuzzy b-metric space and presented the con-

cept of a fuzzy b-metric space and also explored

some of its topological properties. The contrac-

tion mapping in partial b-metric space was intro-

duced by Amar et al. [3] and has certain appli-

cations. On the other hand, the study on prod-

uct spaces in the probabilistic framework was ini-

tiated by Istratescu, Vaduva, Mohd. Raﬁ and

M.S.M Noorani subsequently by Egbert, Alsina

[1] and Schweizer [2]. Recently, Lafuerza-

Guillen [13] has studied ﬁnite products of prob-

abilistic normed spaces and proved some inter-

esting results.The concepts of product of prob-

abilistic metric (normed) spaces studied by Eg-

bert (Lafuerza -Guillen). Similarly, The Carte-

sian product of two fuzzy metric spaces was in-

troduced by Jehad R. Kider [8] by publishing his

paper in 2011. Later on in 2022, Mayada et al.[6]

developed the new properties in fuzzy b-metric

space by introducing two continuous t-norms on

the completeness of the characterization.

The objective of our research is to discuss the

different topological properties in fuzzy b-metric

by using convergent sequence, Cauchy sequence

and completeness to introduce the concept of

Cartesian product in fuzzy b-metric space. Also

other inherited signiﬁcant properties related to

fuzzy b-metric with Cartesian product space are

examined.

2 Preliminaries

In 1942, the operation of t-norm was intro-

duced by K. Menger and by using the concept

of continuity in these operations and in 1960, B.

Schweizer and Sklar introduced the operation of

t-norm and using the concept of continuity.

THANESHWAR BHANDARI1, K. B. MANANDHAR2, KANHAIYA JHA3

1Department of Mathematics, Butwal Multiple Campus

Tribhuvan University

NEPAL

2Department of Mathematics, School of Science

Kathmandu University

NEPAL

3Department of Mathematics, School of Science

Kathmandu University

NEPAL

International Journal of Computational and Applied Mathematics & Computer Science

DOI: 10.37394/232028.2024.4.10

Thaneshwar Bhandari,

K. B. Manandhar, Kanhaiya Jha

E-ISSN: 2769-2477

Volume 4, 2024

Deﬁnition 2.1 [2] A binary operation ∗: [0,1] ×

[0,1] →[0,1] is continuous triangular norm (t-

norm) if for all p, q, r, s ∈[0,1], the following

conditions are satisﬁed

(T1) p∗1 = p;

(T2) p∗q=q∗p;

(T3) If p≤rand q≤sthen p∗q≤r∗s;

(T4) p∗(q∗r)=(p∗q)∗r

The basic examples of continuous t-norm are:

Tp(p, q) = p.q (usual multiplication in[0,1]),

Tm(p, q) = min{p, q}and TL(p, q) = max(p+

q−1,0) (the Lukasiewicz t-norm).

Deﬁnition 2.2 [1] A triplet (U, F, ∗)is known

as a fuzzy metric space if Uis any set, ∗is a

continuous t-norm and Fis a fuzzy set deﬁned

on U×U×(0,∞)→[0,1] if it satisﬁes the

properties given as below, for all u, v, w ∈Uand

t, s > 0

(FM-1) F(u, v, t)>0,

(FM-2)F(u, v, t) = 1 ⇐⇒ u=v

(FM-3)F(u, v, t) = F(v, u, t),

(FM-4)F(u, v, t)∗F(v, w, s)≤F(u, v, t +s)

for all t, s > 0,

(FM-5)F(u, v, .) : (0,∞)→[0,1] is continuous

The degree of nearness between uand vwith re-

spect to t > 0is denoted by F(u, v, t).

Example 2.3 [1] Let Ube a non empty set and

dbe a metric deﬁned on Uand ∗is continuous

t-norm deﬁned as p∗q=p.q for all p, q ∈[0,1]

then the relation deﬁned by

F(u, v, t) = t

t+d(u, v)

is fuzzy metric in [0,1] and (U, F, ∗)is a fuzzy

metric space.

In 2016, Nadaban [10] explored the idea of fuzzy

b-metric space to generalize the notion of the

fuzzy metric spaces introduced by Kramosil and

Michalek.

Deﬁnition 2.4 [10] Let Ube a nonempty set, ∗is

a continuous t-norm and Fbe a fuzzy set. Then

a mapping F:U×U×R+→[0,1] is called a

fuzzy b-metric space. If there exists k≥1such

that the following properties are fulﬁlled

for all u, v, w ∈Uand t, s > 0

(Fb-i) F(u, v, t)>0;

(Fb-ii) F(u, v, t)=1for all t > 0if and only if

u=v;

(Fb-iii) F(u, v, t) = F(v, u, t);

(Fb-iv) F(u, v, t)∗F(v, w, s)≤F(u, w, k(t+s))

for all t, s > 0;

(Fb-v) F(u, v, .) : (0,∞)→[0,1] is left contin-

uous ;

(Fb-vi) lim

t→∞

F(u, v, t) = 1

The class of fuzzy b-metric space is larger than

the class of fuzzy metric space. In fuzzy b-metric

space if we put k= 1 then it becomes a fuzzy

metric space.

Deﬁnition 2.5 [1] Let (U, F, ∗)be a fuzzy b-

metric space. A sequence {un}in Uis said to

be convergent in Uif

lim

n→∞

F(un, u, t) = 1 for each t > 0.

where lim

n→∞

un=u

Deﬁnition 2.6 [1] A sequence {un}in Uis said

to be Cauchy sequence in Uif

lim

n→∞

F(un, um+n, t)=1where t > 0and m, n >

If every Cauchy sequence is convergent in a

fuzzy b-metric space the it is called the complete

fuzzy b-metric space.

An implementation of our ﬁndings in fuzzy b-

metric space is obtained from the constructive

discussion as below.

Example 2.7 [1] Let us deﬁne a mapping Ffrom

U×U×R+to the unit interval [0,1] as

F(u, v, t) = (t−d(u,v)

t+d(u,v)for t > d(u, v)

0for t≤d(u, v)

Then (U, F, ∗)is an fuzzy b-metric space, where

p∗q= min{p, q}dbe a metric on U.

By the above deﬁnition of fuzzy b-metric space,

the properties (i),(ii), as well (iii) are obviously

fulﬁlled. So, we will fulﬁll the property (iv)

where u, v, w ∈Uand t, s ∈(0,∞), the fol-

lowing cases follow:

a) If t≤d(u, v)or s≤d(v, w)or both, then

F(u, w, k(t+s)) ≥F(u, v, t)∗F(v, w, s)

is satisﬁed obviously.

(b) If t > d(u, v)and s>d(v, w), then

F(u, w, k(t+s))

≥k(t+s)−d(u, w)

k(t+s) + d(u, w)

≥k(t+s)−k[d(u, v) + d(v, w)]

k(t+s) + k[d(u, v) + d(v, w)]

≥(t+s)−d(u, v)−d(v, w)

(t+s) + d(u, v) + d(v, w)

≥min t−d(u, v)

t+d(u, v),s−d(v, w)

s+d(v, w)

=F(u, v, t)∗F(v, w, s).

It means that F(u, w, k(t+s)) ≥F(u, v, t)∗

F(v, w, s).

Now to fulﬁll the property (v), assume that {tn}

is a sequence in [0,∞)such that {tn}converges

International Journal of Computational and Applied Mathematics & Computer Science

DOI: 10.37394/232028.2024.4.10

Thaneshwar Bhandari,

K. B. Manandhar, Kanhaiya Jha

E-ISSN: 2769-2477

Volume 4, 2024

to t. Then, for all u, v ∈U,

lim

n→∞

F(u, v, tn) = lim

n→∞

tn−d(u, v)

tn+d(u, v)

=t−d(u, v)

t+d(u, v)

=F(u, v, t).

So F(u, v, tn)converges to F(u, v, t).

Hence F(u, v, ·) : [0,∞)→[0,1] is left-

continuous, and

lim

t→∞

F(u, v, t) = lim

t→∞

t−d(u, v)

t+d(u, v)= 1.

Hence the properties of fuzzy b-metric space are

satisﬁed. So (U, F, ∗)is a fuzzy b-metric space.

Now we will give the example of convergent se-

quence and Cauchy sequence in fuzzy b-metric

space deﬁned as:

Example 2.8 Let us deﬁne a mapping Ffrom

U×U×R+to the unit interval [0,1] as

F(u, v, t) = 





−

(u−v)2

t,if t > 0,

0,if t= 0

If U=R+and there exists k≥1. Then (U, F, ∗)

is a fuzzy b-metric space.

Example 2.9 Assume that {un}in Uwhere

{un}=1

nfor all n∈N, and the fuzzy b-

metric space is deﬁned as in the above example

(2.8), then clearly {un}is converges to 0, which

is shown as follow:

lim

n→∞

F(un,0, t) = lim

n→∞

−(un−0)2

= lim

n→∞

−(un)2

= lim

n→∞

−1

lim

n→∞

−1

n2t= 1.

Hence the sequence {un}is convergent in fuzzy

b-metric space.

Example 2.10 Consider a fuzzy b-metric space

(U, F, ∗)deﬁned as above example (2.8) and a

sequence {un}with

{un}=1

nfor all n∈N.

Now for all m∈N, we have

lim

n→∞

F(un, um+n, t) = lim

n→∞

−(un−um+n)2

= lim

n→∞

−(1

n−1

m+n)2

= 1

Hence {un}is a Cauchy sequence in fuzzy b-

metric space.

3 Cartesian Product of Two Fuzzy b-

Metric Spaces

Now, we use the concept of the Cartesian prod-

uct of two fuzzy b-metric spaces, then we prove

that the Cartesian product of two fuzzy b-metric

spaces is also fuzzy b-metric space.

Finally, we prove the completeness of the Carte-

sian product of two complete fuzzy b-metric

spaces together with the properties of limit point

and Cauchy sequence.

Deﬁnition 3.1 [9] Let (U1, F1,∗)and (U2, F2,∗)

be two fuzzy b-metric spaces. The Cartesian

product of (U1, F1,∗)and (U2, F2,∗)is the prod-

uct space (U1×U2, F, ∗)where (U1×U2)is the

Cartesian product of U1and U2and Fis a map-

ping from ((U1×U2)×(0,∞)) ×((U1×U2)×

(0,∞)) →[0,1] and

F((u1, u2),(v1, v2), t) = F1(u1, v1, t)·F2(u2, v2, t),

∀t > 0

where (u1, u2),(v1, v2)∈U1×U2and ∗is a con-

tinuous t-norm and F=F1·F2. Then the product

space (U1×U2, F, ∗)is known as the Cartesian

product of two fuzzy b-metric spaces.

Theorem 3.2 Let (U1, F1,∗)and (U2, F2,∗)be

any two fuzzy b-metric spaces, if there exists

areal number k≥1, where (u1, u2),(v1, v2)∈

U1×U2, such that

F((u1, u2),(v1, v2), t) = F1(u1, v1, t)·F2(u2, v2, t).

Then (U1×U2, F, ∗)is a fuzzy b-metric space.

Proof: In order to complete the proof of the theo-

rem, the following properties should be satisﬁed.

(i) Since F1(u1, v1, t)>0and F2(u2, v2, t)>

Which gives F1(u1, v1, t)∗F2(u2, v2, t)>0.

So, F((u1, u2),(v1, v2), t)>0.

(ii) Assume that (u1, u2) = (v1, v2).

Which gives u1=v1and u2=v2. Hence,

for all t > 0, we have F1(u1, v1, t) = 1 and

International Journal of Computational and Applied Mathematics & Computer Science

DOI: 10.37394/232028.2024.4.10

Thaneshwar Bhandari,

K. B. Manandhar, Kanhaiya Jha

E-ISSN: 2769-2477

Volume 4, 2024

F2(u2, v2, t) = 1.

It follows that F((u1, u2),(v1, v2), t)=1.

Conversely, let us assume that

F((u1, u2),(v1, v2), t) = 1.

Which gives F1(u1, v1, t)∗F2(u2, v2, t)=1.

Since 0< F1(u1, v1, t)≤1and 0<

F2(u2, v2, t)≤1,

it gives F1(u1, v1, t)=1and F2(u2, v2, t) =

Hence, u1=v1and u2=v2.

So (u1, u2)=(v1, v2).

(iii) To show F((u1, u2),(v1, v2), t)

=F((v1, v2),(u1, u2), t), we note that

F1(u1, v1, t) = F1(v1, u1, t), F2(u2, v2, t)

=F2(v2, u2, t).

It follows that for all (u1, u2),(v1, v2)∈

U1×U2and t > 0,

F((u1, u2),(v1, v2), t) = F((v1, v2),(u1, u2), t).

(iv) Since (U1, F1,∗)and (U2, F2,∗)are two

fuzzy b-metric spaces, we have

F1(u1, w1, k(t+s)≥F1(u1, v1, t)

∗F1(v1, w1, s),

F2(u2, w2, k(t+s))

≥F2(u2, v2, t)

∗F2(v2, w2, s)

for all (u1, u2),(v1, v2),(w1, w2)∈U1×U2

and t, s > 0.

F((u1, u2),(w1, w2), k(t+s))

=F1(u1, w1, k(t+s))

·F2(u2, w2, k(t+s))

≥[F1(u1, v1, t)

∗F1(v1, w1, s)].[F2(u2, v2, t)

∗F2(v2, w2, s)]

≥[F1(u1, v1, t)·F2(u2, v2, t)]

∗[F1(v1, w1, s)·F2(v2, w2, s)]

≥F((u1, u2),(v1, v2), t)

∗F((v1, v2),(w1, w2), s).

(v) Since F1(u1, v1, t)and F2(u2, v2, t)are non-

decreasing functions on R+.

So lim

t→∞

F1(u1, v1, t)=1and

lim

t→∞

F2(u2, v2, t) = 1.

Then lim

t→∞

F((u1, v1)(u2, v2), t)

= lim

t→∞

F1(u1, v1, t).lim

t→∞

F2(u2, v2, t)

= 1.1 = 1.

F((u1, u2),(v1, v2), t)

=F1(u1, v1, t)·F2(u2, v2, t)

is also continuous.

Hence (U1×U2, F, ∗)is a fuzzy b-metric

space.

Theorem 3.3 If {un}is a sequence in fuzzy b-

metric space (U1, F1,∗)converging to uin U1,

and {vn}is a sequence in the fuzzy b-metric

space (U2, F2,∗)converging to vin U2, then

{(un, vn)}is a sequence in the fuzzy b-metric

space (U1×U2, F )converging to (u, v)in U1×

U2, where F=F1·F2.

Proof: By since, (U1×U2, F, ∗)is a fuzzy b-

metric space. Now for each t > 0,

lim

n→∞

F((un, vn),(u, v), t)

=hlim

n→∞

F1(un, u, t)i

·hlim

n→∞

F2(vn, v, t)i

= 1.1 = 1.

Using the above theorem 3.2. Hence the pair of

sequence {(un, vn)}converges to (u, v).

Theorem 3.4 Let us assume that {un}be a

Cauchy sequence in a fuzzy b-metric space

(U1, F1,∗)and the sequence {vn}is Cauchy

in a fuzzy b-metric space (U2, F2,∗), where

∗is a continuous t-norm. Then {(un, vn)}

be the Cauchy sequence in the product space

(U1×U2, F, ∗).

Proof: Since (U1×U2, F, ∗)is a fuzzy b-metric

space. As our supposition {un}and {vn}are two

Cauchy sequences. So for each t > 0and m > 0,

lim

n→∞

F((un+m, vn+m),(un, vn), t)

= [ lim

n→∞

F1((un+mun, t)·lim

n→∞

F2((vn+mvn, t)]

Hence lim

n→∞

F1((un+mun, t) = 1 and

lim

n→∞

F2((vn+mvn, t) = 1

Thus {(un, vn)}is a Cauchy sequence in (U1×

U2, F, ∗).

Theorem 3.5 Let (U1, F1,∗)and (U2, F2,∗)are

any two fuzzy b-metric spaces. Then (U1×

U2, F, ∗)is complete if and only if (U1, F1,∗)

and (U2, F2,∗)are complete.

proof: Assume that (U1, F1,∗)and (U2, F2,∗)

are complete fuzzy b-metric spaces.

Suppose {(un, vn)}be a Cauchy sequence in

U1×U2.

that is for each t > 0and m > 0,

lim

n→∞

F((un+m, vn+m),(un, vn), t)

= [ lim

n→∞

F1((un+m, un, t)·lim

n→∞

F2((vn+m, vn, t)]

Hence

lim

n→∞

F1((un+m, un, t)=1and

International Journal of Computational and Applied Mathematics & Computer Science

DOI: 10.37394/232028.2024.4.10

Thaneshwar Bhandari,

K. B. Manandhar, Kanhaiya Jha

E-ISSN: 2769-2477

Volume 4, 2024

lim

n→∞

F2((vn+m, vn, t) = 1

Therefore {un}is a Cauchy sequence in

(U1, F1∗)and {vn}is a Cauchy sequence in

(U2, F2,∗).

But (U1, F1,∗)and (U2, F2,∗)are complete

fuzzy b-metric spaces, so there exists u∈U1

and v∈U2such that for each t > 0

lim

n→∞

F1(un, u, t)=1and lim

n→∞

F2(vn, v, t) = 1.

Hence {(un, vn)}converges to (u, v)in U1×U2.

Therefore (U1×U2, F, ∗)is a complete fuzzy b-

metric space.

Conversely, suppose that (U1×U2, M, ∗)is com-

plete.

We will show that (U1, F1,∗)and (U2, F2,∗)are

complete.

Let {un}and {vn}be Cauchy sequences in

(U1, F1,∗)and (U2, F2,∗), respectively.

Then F1(um+n, un, t)converges to 1and

F2(vm+n, vn, t)converges to 1for each t > 0

and m > 0. It follows that

lim

n→∞

F((un+m, vn+m),(un, vn), t)

= [ lim

n→∞

F1((un+m, un, t)·lim

n→∞

F2((vn+m, vn, t)]

converges to 1.

Thus {(un, vn)}is a Cauchy sequence in U1×U2.

Since (U1×U2, F, ∗)is complete, there exists

(u, v)∈U1×U2such that F((un, vn),(u, v), t)

converges to 1, as lim n→ ∞.

Clearly, F1(un, u, t)converges to 1and

F2(vn, v, t)converges to 1.

Hence, (U1, F1,∗)and (U2, F2,∗)are complete.

Theorem 3.6 Let ube a limit point of {un}in

a fuzzy b-metric space (U1, F1,∗)and u0be a

limit point of {u0

n}in a fuzzy b-metric space

(U2, F2,∗). Where ∗is a continuous t-norm, and

(u, u0)is the point of limit (un, u0

n)in the fuzzy

b-metric space (U1×U2, F, ∗).

Proof: The space (U1×U2, F, ∗)is fuzzy b-

metric space then by above Theorem (3.5).

Since ube a limit of (un), thus for all t > 0,

limn→∞ F1(un, u, t) = 1 and since u0be a limit

of (u0

n), thus limn→∞ F2(u0

n, u0, t) = 1. Now for

all t > 0,

lim

n→∞

F((un, u0

n),(u, u0), t)

= lim

n→∞

F1(un, u, t).

lim

n→∞

F2(u0

n, u0, t)

= 1.1 = 1.

Hence F((un, u0

n),(u, u0), t)=1, which gives

(u, u0)is the limit point of (un, u0

n).

4 Completeness of U1×U2with F=

F1∧F2

Here we introduce the fuzzy b- metric space in

terms of the minimum t-norm ∧. All the proper-

ties of fuzzy b-metric space are remained same,

only the triangle inequality is redeﬁned as:

F(u, v, k(s+t)) ≥F(u, w, s)∧F(w, v, t)for

each t, s > 0, where ∧is the minimum of

F(u, v, s)and F(w, v, t).

Then we will show that the Cartesian product of

two fuzzy b-metric spaces is a fuzzy b- metric

space and the properties of convergent sequence

and Cauchy sequence in terms of the ordered pair

in fuzzy b-metric space are established by using

the the minimum triangular norm ∧.

Theorem 4.1 If (U1, F1,∗)and (U2, F2,∗)are

fuzzy b-metric spaces. If there exists k≥1, then

(U1×U2, F, ∗)is a fuzzy metric space by deﬁning

F((u1, v1),(u2, v2), t) = F1(u1, u2, t)∧F2(v1, v2, t).

Proof: Let (u1, v1),(v2, y2),(u3, v3)∈U1×U2:

(i) Let t≥0, we have F1(u1, u2, t) = 0 and

F2(v1, v2, t) = 0. Hence,

F((u1, v1),(u2, v2), t)=0.

F1(u1, u2, t) = 1 for each t > 0⇐⇒ u1=

u2, and F2(v1, v2, t)=1for each t > 0⇐⇒

v1=v2.

Together with,

[F1(u1, u2, t)∧F2(v1, v2, t)] = 1

for t > 0⇐⇒ u1=u2and v1=v2. That is,

F((u1, v1),(u2, v2), t) = 1

⇐⇒ (u1, v1) = (u2, v2).

(iii) F1(u1, u2, t) = F1(u2, u1, t)and

F2(v1, v2, t) = F2(v2, v1, t)Now for each t > 0,

F((u1, v1),(u2, v2), t)=[F1(u1, u2, t)∧F2(v1, v2, t)]

= [F1(u2, u1, t)∧F2(v2, v1, t)]

=F((u2, v2),(u1, v1), t).

(iv) F1(u1, u2, k(s+t)) ≥[F1(u1, u3, s)∗

F1(u3, u2, t)] for each s, t > 0.

Also, F2(v1, v2, k(s+t)) ≥[F2(v1, v3, s)∗

F2(v3, v2, t)] for each s, t > 0. Now for each

t > 0,

F((u1, v1),(u2, v2), k(s+t))

= [F1(u1, u2, k(s+t)]

∧F2(v1, v2, k(s+t))

≥[F1(u1, u3, s)∗F1(u3, u2, t)

∧F2(v1, v3, s)∗F2(v3, v2, t)]

≥[F1(u1, u3, s)∧F2(v1, v3, s)] ∗[F1(u3, u2, t)∧

F2(v3, v2, t)]

≥[F((u1, v1),(u3, v3), s)∗F((u3, v3),(u2, v2), t)].

International Journal of Computational and Applied Mathematics & Computer Science

DOI: 10.37394/232028.2024.4.10

Thaneshwar Bhandari,

K. B. Manandhar, Kanhaiya Jha

E-ISSN: 2769-2477

Volume 4, 2024

(v) Since F1(u1, v1, t)and F2(u2, v2, t)are

non-decreasing functions on R+.

So lim

t→∞

F1(u1, v1, t)=1and lim

t→∞

F2(u2, v2, t) =

Now,

F((u1, v1),(u2, v2), t)=[F1(u1, u2, t)∧F2(v1, v2, t)]

≤[F1(u1, u2, t)∧F2(v1, v2, t)]

≤F((u1, v1),(u2, v2), t).

and

lim

t→∞

F((u1, v1),(u2, v2), t)

= [ lim

t→∞

F1(u1, u2, t)]∧

[ lim

t→∞

F2(v1, v2, t)] = 1.

Thus (U1×U2, F, ∗)is a fuzzy b-metric space.

Corollary 4.2: If {un}is a sequence in the

fuzzy b-metric space (U1, F1,∗)converging to

u∈U1, and {vn}is a sequence in the fuzzy b-

metric space (U2, F2,∗)converging to v∈U2,

then {(un, vn)}is a sequence in the fuzzy metric

space (U1×U2, F, ∗)converging to (u, v), where

F=F1∧F2.

Proof: By above Theorem 4.1, (U1×U2, F, ∗)is

a fuzzy b-metric space. Now for each t > 0,

lim

n→∞

F((un, vn),(u, v), t)

= [ lim

n→∞

F1(un, u, t)]

∧[ lim

n→∞

F2(vn, v, t)] = 1 ∧1 = 1.

Hence {(un, vn)}converges to (u, v).

Corollary 4.3: If {un}is a Cauchy sequence in

the fuzzy b-metric space (U1, F1,∗)and {vn}is

a Cauchy sequence in the fuzzy b-metric space

(U2, F2,∗), then {(un, vn)}is a Cauchy sequence

in the fuzzy b-metric space (U1×U2, F, ∗), where

F=F1∧F2.

Example 4.5 Let (U1, F, ∗)and (U2, F, ∗)be two

fuzzy b-metric spaces and assume (U1×U2, d)

be their product space, where

d(a, b) = max{U(u1, u2), U2(v1, v2)}

for each a= (u1, v1)and b= (u2, v2)in U1×U2.

Deﬁne p∧q= min(p, q)for all p, q ∈[0,1] and

assume

U1(a, b, t) = t

t+d(a, b).

Then (U1×U2, F, ∗)is a ∧-product of (U, dU1)

and (U2, dU2).

Proof: Since we have,

F(a, bt) = t

t+d(a, b)

t+ max{dU1(u1, u2), dU2(v1, v2)}

= min t

t+dU1(u1, u2),t

t+dU1(v1, v2)

=t

t+dU1(u1, u2)∧t

t+du2(v1, v2).

Thus, Md(a, b, t) = F dU1∧F dU2.

Hence, (U1×U2, F, ∗)is a ∧-product of (U, dU1)

and (U2, dU2).

5 Conclusion

By introducing the idea of Cartesian product of

two fuzzy b-metric spaces with suitable proper-

ties, we have proved that the Cartesian product

of two complete fuzzy b-metric spaces is again

a fuzzy b-metric space under the product t-norm

as well as the minimum t-norm. Also we have

proved the properties of convergence sequence

and Cauchy sequence in terms of the Cartesian

product in fuzzy b-metric space and we have pre-

sented some examples to verify the deﬁnition of

fuzzy b-metric space, convergent sequence and

Cauchy sequence as well. Further research can

be done by connecting the ﬁxed point theory in

fuzzy b-metric space in terms of the Cartesian

product. Further more, many topological proper-

ties can be connected with this topic.

Conﬂict of Interest

The authors have no conﬂicts of interest to de-

clare that are relevant to the content of this article.

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International Journal of Computational and Applied Mathematics & Computer Science

DOI: 10.37394/232028.2024.4.10

Thaneshwar Bhandari,

K. B. Manandhar, Kanhaiya Jha

E-ISSN: 2769-2477

Volume 4, 2024

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Policy)

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research, at all stages from the formulation of the

problem to the final findings and solution.

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Scientific Article or Scientific Article Itself

No funding was received for conducting this study.

Conflict of Interest

The authors have no conflicts of interest to declare

that are relevant to the content of this article.

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International Journal of Computational and Applied Mathematics & Computer Science

DOI: 10.37394/232028.2024.4.10

Thaneshwar Bhandari,

K. B. Manandhar, Kanhaiya Jha

E-ISSN: 2769-2477

100

Volume 4, 2024