Ratio Mathematica Volume 47, 2023

Some topological properties of revised fuzzy
cone metric spaces

A. Muraliraj *
R. Thangathamizh†

Abstract

In this paper, we introduced Revised fuzzy cone Metric space with its
topological properties. Likewise A necessary and sufficient condition
for a Revised fuzzy cone metric space to be precompact is given. We
additionally show that each distinct Revised fuzzy cone metric space
is second countable and that a subspace of a separable Revised fuzzy
cone metric space is separable.
Keywords: Revised fuzzy metric space; Revised fuzzy cone metric
space; separable; second countable.
2020 AMS subject classifications: 54A40, 54E35, 54E15, 54H25.1

*Assistant Professor, Department of Mathematics, Urumu Dhanalakshmi College Trichy, In-
dia; karguzali@gmail.com

†Assistant Professor, Department of Mathematics, K. Ramakrishnan College of Engineering,
Trichy, India; thamizh1418@gmail.com
1Received on March 19, 2022. Accepted on September 12, 2022. Published online on January 13,
2023. doi: 10.23755/rm.v39i0.734. ISSN: 1592-7415. eISSN: 2282-8214. ©The Authors.
This paper is published under the CC-BY licence agreement.

42



A. Muraliraj and R. Thangathamizh

1 Introduction
After Zadeh [1965] introduced the idea of fuzzy sets, several authors have

introduced and studied many notions of metric indistinctness [Huang and Zhang,
2007, Kramosil and Michalek, 1975, Ahmad and Mesiarová-Zemánková, 2007,
Navara, 2007, Muraliraj and Thangathamizh, 2021a,b,d, Oner and Tanay, 2015,
Grigorenko et al., 2020] and metric cone indistinctness. By modifying the idea of
metric indistinctness introduced by George and Veeramani [1994], Zadeh [1965]
studied the notion of fuzzy cone metric areas. Especially, they evidenced that
every fuzzy cone topological space generates a Hausdorf first-countable topology.
Here we tend to study additional topological properties of these areas whose fuzzy
metric version are usually found in George and Veeramani [1994], Ghareeb and
Al-Omeri [2018], Grabiec [1988], Gregori et al. [2011].

Sostak [2018] additionally represented the idea of “George–Veeramani Fuzzy
Metrics Revised”. Presently Olga Grigorenko, Juan Jose Minana, Alexander
Sostak, Muraliraj and Thangathamizh [2021c] have introduced “On t-conorm pri-
marily based Fuzzy (Pseudo) metrics”. Recently Muraliraj and Thangathamizh
[2021a,c,d] proved various fixed point theorems in revised fuzzy metric spaces.
Muraliraj and Thangathamizh [2021b] introduce the concept of Revised fuzzy
modular metric space. Moreover, we tend to prove that a Revised fuzzy cone
topological space is precompact if and providing each sequence in it’s a Cauchy
subsequence. Further, we tend to show that X1 × X2 may be a complete Revised
fuzzy cone topological space if and providing X1 and X2 are complete Revised
fuzzy cone metric areas. Finally it’s tried that each divisible Revised fuzzy cone
topological space is second calculable and a mathematical space of a separable
Revised fuzzy cone topological space is separable.

2 Preliminaries
Definition 2.1 (Gregori et al. [2011]). Let E be a real Banach space, θ the zero of
E and P a subset of E. Then P is called a cone if and only if

(i) P is closed, nonempty, and P ̸= {θ},
(ii) if ab ∈ R, ab ≥ 0 and xy ∈ P , then ax + by ∈ P ,

(iii) if both x ∈ P and −x ∈ P , then x = θ.

Given a cone P , a partial ordering ≾ on E with respect to P is defined by
x ≾ y if only if y − x ∈ P . The notation x ≺ y will stand for x ≾ y and x ̸= y,
while x ≪ y will stand for y − x ∈ int(P). Throughout this paper, we assume
that all the cones have nonempty interiors.

43



Some topological properties of revised fuzzy cone metric spaces

There are two kinds of cones: normal and non-normal ones. A cone P is
called normal if there exists a constant K ≥ 1 such that for all t, s ∈ E, θ ≾ t ≾ s
implies ∥t∥ ≤ K ∥s∥, and the least positive number K having this property is
called normal constant of P Gregori et al. [2011]. It is clear that K ≥ 1.

Definition 2.2 (Sostak [2018]). A binary operation ⊕ : [0, 1] × [0, 1] → [0, 1] is a
t-conorm if it satisfies the following conditions:

(i) ⊕ is associative andcommutative,
(ii) ⊕ is continuous,

(iii) a ⊕ 0 = a for all a ∈ [0, 1],
(iv) a ⊕ b ≤ c ⊕ d whenever a ≤ c and b ≤ d for all a, b, c, d ∈ [0, 1].

2.1 Examples Sostak and Öner [2020]
(i) Lukasievicz t-conorm: a ⊕ b = max{a, b};

(ii) Product t-conorm: a ⊕ b = a + b − ab;
(iii) Minimum t-conorm: a ⊕ b = min(a + b, 1).

Definition 2.3 (Oner and Tanay [2015]). A 3-tuple (U, M0, ∗) is called a fuzzy
cone metric space (FCM space) if C is a cone of E, U is an arbitrary set, (∗) is a
continuous t-norm, and M0 is a fuzzy set on U2 × int(P) satisfying the following
conditions:

(i) M0(λ1, λ2, t) > 0 and M0(λ1, λ2, t) = 1 ⇔ λ1 = λ2,
(ii) M0 (λ1, λ2, t) = M0(λ2, λ1, t)

(iii) M0 (λ1, λ2, t) ∗M0(λ2, λ3, s) ≤ M0(λ1, λ3, t + s),
(iv) M0 (λ1, λ2, ) : int(P) → [0, 1] is continuous ∀λ1, λ2, λ3 ∈ U and t, s ∈

int(p).

Definition 2.4 (Sostak [2018]). A Revised fuzzy metric space is an ordered triple
(X, µ, ⊕) such that X is a non empty set, ⊕ is a continuous t-conorm and µ is a
Revised fuzzy set on µ : X2 × R+ → [0, 1] satisfies the following conditions:

(RFM1) µ (x, y, t) < 1;

(RFM2) µ(x, y, t) = 0 if and only if x = y;

(RFM3) µ(x, y, t) = µ(y, x, t);

(RFM4) µ(x, z, t + s) ≤ µ(x, y, t) ⊕ µ(y, z, s);
(RFM5) µ(x, y, −) : (0, ∞) → [0, 1) is continuous ∀x, y, z ∈ X and t, s ∈ R+.

Then µ is called a Revised fuzzy metric on X.

44



A. Muraliraj and R. Thangathamizh

3 Main Results
Definition 3.1. A Revised fuzzy cone metric space is an 3-triple (X, µ, ⊕) such
that P is a cone of E, X is a non empty set, ⊕ is a continuous t-conorm and µ is a
Revised fuzzy set on X2 × int(P) satisfies the following conditions, ∀x, y, z ∈ X
and s, t ∈ int(P) (that is s ≫ θ, t ≫ θ),

(RFCM 1) µ (x, y, t) < 1 ,

(RFCM 2) µ(x, y, t) = 0 if and only if x = y,

(RFCM 3) µ(x, y, t) = µ(y, x, t),

(RFCM 4) µ(x, z, t + s) ≤ µ(x, y, t) ⊕ µ(y, z, s),
(RFCM 5) µ(x, y, −) : int(P) → [0, 1] is continuous.

Then µ is called a Revised fuzzy cone metric on X.

If (X, µ, ⊕) is a Revised fuzzy cone metric space, we will say that µ is a
Revised fuzzy cone metric on X.

Every revised fuzzy cone metric space (X, µ, ⊕) induces a Hausdorff first-
countable topology τfc on X which has as a base the family of sets of the form
{B(x, r, t) : x ∈ X; 0 < r < 1, t ≫ θ},
where {B(x, r, t) : y ∈ X; µ(x, y, t) < r} for every r with 0 < r < 1 and t ≫ θ.

A Revised fuzzy cone metric space (X, µ, ⊕) is called complete if every Cauchy
sequence in it is convergent, where a sequence {xn} is said to be a Cauchy se-
quence if for any ε ∈ (0, 1) and any t ≫ θ there exists a natural number n0 such
that µ (xn, xm, t) < ε for all n, m ≥ n0, and a sequence {xn} is said to converge
to x if for any t ≫ θ and any r ∈ (0, 1) there exists a natural number n0 such that
µ (xn, x, t) < r for all n ≥ n0. A sequence {xn} converges to x if and only if
µ (xn, x, t) → 0 for each t ≫ θ.

Definition 3.2. Let (X, µ, ⊕) be a Revised fuzzy cone metric space. For t ≫ θ,
the closed ball B[x, r, t] with center x and radius r ∈ (0, 1) is defined by

B[x, r, t] = {y ∈ X; µ(x, y, t) < r} .

Lemma 3.3. Every closed ball in a Revised fuzzy cone metric space (X, µ, ⊕) is
a closed set.

Proof. Let y ∈ B[x, r, t]. Since X is first countable, there exits a sequence {yn}
in B[x, r, t] converging to y. Therefore µ (yn, y, t) converges to 0 for all t ≫ θ.
For a given ϵ ≫ 0, we have, µ(x, y, t + ϵ) ≤ µ (x, yn, t) ⊕ µ (yn, y, ϵ)

Hence, µ(x, y, t + ϵ) ≤ µ (x, yn, t) ⊕ µ (yn, y, ϵ) ≤ 0 ⊕ 0 = 0.

45



Some topological properties of revised fuzzy cone metric spaces

(If µ (x, yn, t) is bounded, then the sequence yn has a subsequence, which we
again denote by yn, for which µ (x, yn, t) exists.) In particular for n ∈ N, take
ϵ = t

n
. Then, µ

(
x, y, t + t

n

)
< r. Hence, µ(xyt) ≤ µ

(
x, y, t + t

n

)
< r. Thus

y ∈ B[x, r, t]. Therefore B[x, r, t] is a closed set.
Definition 3.4. A Revised fuzzy cone metric space (X, µ, ⊕) is called precompact
if for each r, with 0 < r < 1, and each t ≫ θ, there is a finite subset A of X, such
that X =

⋃
a∈A B(a, r, t). In this case, we say that µ is a precompact Revised

fuzzy cone metric on X.

Lemma 3.5. A Revised fuzzy cone metric space is precompact if and only if every
sequence has a Cauchy subsequence.

Proof. Suppose that (X, µ, ⊕) is a precompact Revised fuzzy cone metric space.
Let xn be a sequence in X. For each m ∈ N there is a finite subset Am of
X such that X =

⋃
a∈A B

(
a, a

m
, t0
m∥t0∥

)
where t0, t ≫ θ is a constant. Hence,

for m = 1, there exists an a1 ∈ A1 and a subsequence x1(n) of xn such that
x1(n) ∈ B

(
a1, 1,

t0
m∥t0∥

)
for every n ∈ N. Similarly, there exist an a2 ∈ A2 and a

subsequence
{
x2(n)

}
of x1(n) such that x2(n) ∈ B

(
a2,

1
2
, t0
m∥t0∥

)
for every n ∈ N.

By continuing this process, we get that for m ∈ N, m > 1, there is an am ∈ Am
and a subsequence

{
xm(n)

}
of xm−1(n) such that xm(n) ∈ B

(
am,

1
m
, t0
m∥t0∥

)
for

every n ∈ N. Now, consider the subsequence xn(n) of xn. Given r with 0 < r < 1
and t ≫ θ there is an n0 ∈ N such that 1n0 ⊕

1
n0

< r and 2t0
n0∥t0∥

≪ t. Then, for
every km ≥ n0, we have

µ
(
xk(k), xm(m), t

)
≤ µ

(
xk(k), xm(m),

t0
n0 ∥t0∥

)
≤ µ

(
xk(k), an0,

t0
n0 ∥t0∥

)
⊕ µ

(
an0, xm(m),

t0
n0 ∥t0∥

)
≤

1

n0
⊕

1

n0
< r

Hence
(
xn(n)

)
is a Cauchy sequence in (X, µ, ⊕).

Conversely, suppose that (X, µ, ⊕) is a nonprecompact Revised fuzzy cone
metric space. Then there exist an r with 0 < r < 1 and t ≫ θ such that for
each finite subset A of X, we have X ̸=

⋃
a∈A B(a, r, t) fix x1 ∈ X. There

is x2 ∈ X − B(x1rt). Moreover, there is an x3 ∈ X −
2⋃

k=1

B(xk, r, t). By

continuing this process, we construct a sequence xn of distinct points in X such

that xn+1 /∈ X −
n⋃

k=1

B(xk, r, t) for every n ∈ N. Therefore xn has no Cauchy

subsequence. This completes the proof.

46



A. Muraliraj and R. Thangathamizh

Lemma 3.6. Let (X, µ, ⊕) be a Revised fuzzy cone metric space. If a Cauchy
sequence clusters around a point x ∈ X, then the sequence converges to x.
Proof. Let xn be a Cauchy sequence in (X, µ, ⊕) having a cluster point x ∈ X.
Then, there is a subsequence

{
xk(n)

}
of xn that converges to x with respect to τfc.

Thus, given r with 0 < r < 1 and t ≫ θ, there is an n0 ∈ N such that for each
n ≥ n0, µ

(
x, xk(n),

t
2

)
< s where s > 0 satisfies s ⊕ s < r. On the other hand,

there is n0 ≥ k(n0) such that for each nm ≥ n1, we have µ
(
xn, xm,

t
2

)
< s.

Therefore, for each n ≥ n1, we have
µ (x, xn, t) ≤ µ

(
x, xk(n),

t
2

)
⊕ µ

(
xk(n), x,

t
2

)
≤ s ⊕ s < r.

We conclude that the Cauchy sequence xn converges to x.

Proposition 3.7. Let (X1, µ1, ⊕) and (X2, µ2, ⊕) be Revised fuzzy cone metric
spaces. For (x1, x2) , (y1, y2) ∈ X1, X2, let µ ((x1, x2) , (y1, y2) , t) = µ1 (x1, y1, t)⊕
µ2 (x2, y2, t), Then µ is a Revised fuzzy cone metric on X1 × X2.
Proof. RFCM 1: Since µ1 (x1, y1, t) < 1 and µ2 (x2, y2, t) < 1, this implies that
µ1 (x1, y1, t) ⊕ µ2 (x2, y2, t) < 1. Therefore, µ ((x1, x2) , (y1, y2) , t) < 1.
RFCM 2: Suppose that for all t ≫ θ, (x1, y1, t) = (x2, y2, t). This implies that
x1 = y1 and x2 = y2 for all t ≫ θ. Hence, µ1 (x1, y1, t) = 0 and µ2 (x2, y2, t) = 0.

It follows that, µ ((x1, x2) , (y1, y2) , t) = 0.
Conversely, suppose that µ ((x1, x2) , (y1, y2) , t) = 0.

This implies that µ1 (x1, y1, t) ⊕ µ2 (x2, y2, t) = 0.
Since, 0 < µ1 (x1, y1, t) < 1 and 0 < µ2 (x2, y2, t) < 1. It follows that,

µ1 (x1, y1, t) = 0 and µ2 (x2, y2, t) = 0. Thus x1 = y1 and x2 = y2. Therefore
(x1, x2) = (y1, y2).
RFCM 3: To prove that µ ((x1, x2) , (y1, y2) , t) = µ ((y1, y2) , (x1, x2) , t) we
observe that µ1 (x1, y1, t) = µ1 (y1, x1, t) and µ2 (x2, y2, t) = µ2 (y2, x2, t). It
follows that for all (x1, x2) (y1, y2) ∈ X1 × X2 and t ≫ θ,

µ ((x1, x2) , (y1, y2) , t) = µ ((y1, y2) , (x1, x2) , t)

RFCM 4: Since (X1, µ1, ⊕) and (X2, µ2, ⊕) are Revised fuzzy cone metric spaces,
we have that, µ1 (x1, z1, t + s) ≤ µ1 (x1, y1, t) ⊕ µ1 (y1, z1, s) and
µ2 (x2, z2, t + s) ≤ µ2 (x2, y2, t) ⊕ µ2 (y2, z2, s) , for all

(x1, x2) (y1, y2) (z1, z2) ∈ X1 × X2 and t, s ≫ θ. Therefore,

µ ((x1, x2) , (z1, z2) , t + s)

= µ1 (x1, z1, t + s) ⊕ µ2 (x2, z2, t + s)
≤ µ1 (x1, y1, t) ⊕ µ1 (y1, z1, s) ⊕ µ2 (x2, y2, t) ⊕ µ2 (y2, z2, s)
≤ µ1 (x1, y1, t) ⊕ µ2 (x2, y2, t) ⊕ µ1 (y1, z1, s) ⊕ µ2 (y2, z2, s)
≤ µ1 (x1, y1, t) ⊕ µ2 (x2, y2, t) ⊕ µ1 (y1, z1, s) ⊕ µ2 (y2, z2, s)
≤ µ ((x1, x2) , (y1, y2) , t) ⊕ µ ((y1, y2) , (z1, z2) , t)

47



Some topological properties of revised fuzzy cone metric spaces

RFCM 5: Note that µ1 (x1, y1, t) and µ2 (x2, y2, t) are continuous with respect to
t and ⊕ is continuous too. It follows that,
µ ((x1, x2) , (y1, y2) , t) = µ1 (x1, y1, t) ⊕ µ2 (x2, y2, t) is also continuous.

Proposition 3.8. Let (X1, µ1, ⊕) and (X2, µ2, ⊕) be Revised fuzzy cone metric
spaces. We define, µ ((x1, x2) , (y1, y2) , t) = µ1 (x1, y1, t) ⊕ µ2 (x2, y2, t). Then µ
is a complete Revised fuzzy cone metric on X1 × X2 if and only if (X1, µ1, ⊕) and
(X2, µ2, ⊕) are complete.

Corollary 3.9. Every separable Revised fuzzy cone metric space is second count-
able.

Proof. Let (X, µ, ⊕) be the given separable Revised fuzzy cone metric space. Let
A = {an : n ∈ N} be a countable dense subset of X. Consider

B =

{(
aj,

1

k
,

t1
k ∥t1∥

)
: j, k ∈ N

}
where t1 ≫ θ is constant. Then B is countable. We claim that B is a base for the
family of all open sets in X. Let G be an open set in X. Let x ∈ G then there
exists r with 0 < r < 1 and t ≫ θ such that B(x, rt) ⊂ G.

Since r ∈ (0, 1), we can find an s ∈ (0, 1) such that s⊕s < r. Choose m ∈ N
such that 1

m
< s and t1

m∥t1∥
≪ t

2
. Since A is dense in X, there exists an aj ∈ A

such that aj ∈ B
(
x, 1

m
, t1
m∥t1∥

)
. Now if y ∈ B

(
aj,

1
m
, t1
m∥t1∥

)
, then

µ(x, y, t) ≤ µ
(
x, aj,

t

2

)
⊕ µ

(
y, aj,

t

2

)
≤ µ

(
x, aj,

t1
k ∥t1∥

)
⊕ µ

(
x, aj,

t1
k ∥t1∥

)
≤

1

m
⊕

1

m
≤ s ⊕ s < r < r.

Thus y ∈ B(x, r, t) and hence B is a basis.

Proposition 3.10. A subspace of a separable Revised fuzzy cone metric space is
separable.

Proof. Let X be a separable Revised fuzzy cone metric space and Y a subspace
of X. Let A = {xn : n ∈ N} be a countable dense subset of X. For arbitrary but
fixed nk ∈ N, if there are points x ∈ X such that µ

(
xn, x,

t1
k∥t1∥

)
< 1

k
where

t1 ≫ θ is constant, choose one of them and denote it by xnk .

48



A. Muraliraj and R. Thangathamizh

Let B = {xnk : n, k ∈ N} then B is countable. Now we claim that Y ⊂ B̄.
Let x ∈ Y . Given r with 0 < r < 1 and t ≫ θ we can find k ∈ N such that
1
k
⊕ 1

k
< r and t1

k∥t1∥
≪ t

2
.

Since A is dense in X, there exists an m ∈ N such that µ
(
xm, y,

t1
k∥t1∥

)
< 1

k
.

But by definition of B, there exists an xmk such that µ
(
xmk, xm,

t1
k∥t1∥

)
< 1

k
. Now

µ

(
xmk, y,

t1
k ∥t1∥

)
≤ µ

(
xmk, xm,

t

2

)
⊕ µ

(
xm, y,

t

2

)
≤ µ

(
xmk, xm,

t1
k ∥t1∥

)
⊕ µ

(
xm, y,

t1
k ∥t1∥

)
≤

1

k
⊕

1

k
< r.

Thus Y ⊂ B̄ and hence Y is separable.

Corollary 3.11. Let (X, µ, ⊕) be a Revised fuzzy cone metric space. Then (X, τfc)
is Hausdorff.

Corollary 3.12. Let (X, µ, ⊕) be a Revised fuzzy cone metric space. Define
τfc= A ⊂ X : x ∈ A if and only if there exist r ∈ (0, 1), and t ≫ θ such that
B(x, r, tt) ⊂ A, then τfc is a topology on X.

Corollary 3.13. In a Revised fuzzy cone metric space, every compact set is closed
and RFC-bounded.

4 Conclusion

In this paper we proved a necessary and sufficient condition for a revised fuzzy
cone metric space to be precompact. We also show that every separable revised
fuzzy cone metric space is second countable and that a subspace of a separable
revised fuzzy cone metric space is separable.

Acknowledgements

We are grateful to the Professor Fabrizio Maturo, Chief Editor of Ratio Math-
ematica and the reviewers for their interesting comments on this paper.

49



Some topological properties of revised fuzzy cone metric spaces

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