Roundness, Completeness and Nested-Ball Fixed Points in Revised Fuzzy Metric Spaces

1. Introduction

Fuzzy metric spaces provide graded analogues of metric structures in which metric information is expressed by values in the unit interval. In Definition 7 of [1], Kramošil and Michálek introduced a fuzzy metric as a fuzzy relation whose value represents the degree of certainty that the distance between two points is smaller than a prescribed positive parameter. In Theorem 1 of [1], they related this construction to statistical metric spaces. George and Veeramani later introduced a modified formulation that induces a Hausdorff topology with a natural base of fuzzy open balls [2]. Fuzzy quasi-metric spaces extend this setting to asymmetric spaces and connect fuzzy topology with quasi-uniform structures, order theory and fixed-point theory [3,4].

Šostak introduced revised George–Veeramani fuzzy metrics by replacing the usual triangular norm with a triangular conorm and reversing the interpretation of the fuzzy value: zero represents coincidence, while larger values represent greater degrees of separation [5]. In Theorem 1 of [5], the George–Veeramani and revised George–Veeramani formulations were shown to be dual descriptions of the same underlying structure. This revised point of view was subsequently developed in the broader setting of t-conorm-based fuzzy pseudometrics by Grigorenko, Miñana, Šostak and Valero [6]. Revised fuzzy metrics induce first-countable Hausdorff topologies and support natural notions of convergence, Cauchy sequences and completeness.

A subtle issue concerns the relationship between open and closed balls. In an arbitrary metric space, the closure of an open ball is contained in the corresponding closed ball, but equality need not hold. Following Nathanson [7], a metric is called round when the closure of every open ball is the corresponding closed ball. Efe transferred this notion to George–Veeramani fuzzy metric spaces and studied several of its basic consequences [8]; further results concerning ordinary round metric spaces may be found in [9]. Under the duality $\mu =1-M$, Efe’s fuzzy open and closed balls coincide exactly with the corresponding revised fuzzy balls. Consequently, Efe’s roundness condition is equivalent to roundness in the revised formulation used here. In particular, the equality between the closure of an open revised fuzzy ball and the corresponding closed revised fuzzy ball, asserted in Corollary 2 of [5] and Corollary 3 of [6], does not hold without an additional hypothesis.

The first purpose of this paper is to develop this existing fuzzy roundness notion systematically in the revised t-conorm formulation and to determine its role in the published ball-closure assertions. In accordance with Efe’s terminology [8], we call a revised fuzzy metric space round when

$$\overline{B_{\mu }(x,r,t)}=B_{\mu }[x,r,t]$$

for every admissible centre x, radius r and time parameter t. We give a complete proof, in the revised setting, of the generally valid inclusion

$$\overline{B_{\mu }(x,r,t)}\subseteq B_{\mu }[x,r,t]$$

and establish the equivalent boundary-approximation formulation already implicit in Efe’s characterization. We then go beyond that earlier theory: more directly, we prove that nonroundness is equivalent to the attainment of a positive minimum by a radial map $y\mapsto \mu (x,y,t)$ on an open set not containing x, in the spirit of the metric characterization of Singh and Narang [9]. We also prove that a finite maximum product is round if and only if each factor is round. Finally, we show that every nontrivial revised fuzzy metric space admits a topologically equivalent truncated revised fuzzy metric that is not round. Thus roundness is a property of the chosen revised fuzzy metric rather than of its induced topology alone.

The second purpose is to investigate the relationship between completeness and decreasing families of sets. Cantor-type intersection principles are classical tools for characterizing completeness. In asymmetric fuzzy settings, several distinct completeness notions arise, including forward completeness and G-completeness; see Definition 3.2 and Definition 3.7 of [10]. In the present paper, we work in the symmetric revised fuzzy setting and characterize completeness through decreasing sequences of nonempty closed sets whose revised fuzzy diameters tend to zero. We then derive a corresponding nested-ball intersection principle. Cantor-type characterizations are already known for George–Veeramani fuzzy metrics and for variants based on strong completeness and p-completeness [2,12,13]. Our contribution here is the explicit separation-based formulation for revised fuzzy metrics and its use in the round-ball fixed-point construction. This viewpoint is also distinct from work that strengthens or transfers completeness in order to obtain fixed points, including p-completeness methods and metrization through an Archimedean t-norm [19,20].

The third purpose is to obtain a nested-ball fixed-point principle. Existing fixed-point results in fuzzy quasi-metric spaces frequently impose explicit algebraic contraction conditions. Examples include Theorem 3.3 and Theorem 3.9 of [10], as well as Theorem 4 and Theorem 5 of [11]. More recent contraction-based approaches include fuzzy F-contractions, extended $FZ$-simulation functions and nonlinear $\phi$-contractions [16,17,18]. Fixed-point theorems have also been proved directly in revised fuzzy metric spaces using Banach– and Edelstein-type assumptions and, subsequently, relation-theoretic $\mathcal{R}$-$\psi$-contractive conditions [14,15]. Our approach is instead local: a self-map is required to send each member of a decreasing sequence of open balls into the next ball. Roundness identifies the closures of these open balls with their corresponding closed balls, while completeness produces a unique point in their intersection. The resulting invariance of the closed balls then shows that this intersection point is fixed by the self-map.

1.1. Main Contributions and Comparison with Existing Results

The principal contributions, together with their relation to the existing literature, are as follows.

1.

Efe’s Definition 3.1 [8] introduced roundness for George–Veeramani fuzzy metrics. Under the duality $\mu =1-M$, the open and closed balls in that definition are precisely the revised fuzzy balls used here. Thus Definition 8 is an equivalent revised formulation, not a new underlying notion.

2.

Corollary 2 of [5] and Corollary 3 of [6] identify every closed revised fuzzy ball with the closure of the corresponding open ball. Example 2 verifies all the relevant axioms and shows that the reverse inclusion can fail even on a two-point space. Efe had proved the analogous inclusion in the George–Veeramani setting. Theorem 1 supplies a detailed proof in the revised setting and identifies the strongest conclusion valid in every revised fuzzy metric space:

$${\overline{B_{\mu }(x,r,t)}}^{{\tau }_{\mu }}\subseteq B_{\mu }[x,r,t].$$

3.

Proposition 8 gives the revised form of Efe’s boundary characterization and identifies the additional condition missing from the cited ball-closure assertions: every point of the level set $\mu (x,·,t)=r$ must be approachable by points satisfying $\mu (x,·,t)<r$. This condition is precisely the roundness property.

4.

Theorem 2 gives a direct criterion for the failure of roundness in terms of attainment of a positive minimum by a radial map on an open set not containing its centre. It is the revised fuzzy counterpart of the metric criterion of Singh and Narang ([9], Theorem 1).

5.

Efe obtained an obstruction from a decomposition into disjoint closed sets, one of which is compact, and a transfer theorem for a class of order-preserving surjections ([8], Theorems 3.5 and 3.7). Our isometry statement is consistent with that theory, while the dense- and open-subspace results address different permanence questions.

6.

Nathanson proved an equivalence for a particular countable weighted product of ordinary metrics ([7], Theorem 4). Theorem 4 establishes the corresponding equivalence for finite products equipped with the maximum of revised fuzzy metrics, while Theorem 5 gives a countable weighted-maximum construction and characterizes both its roundness and its completeness in terms of the factors. Theorem 6 further shows that every nontrivial revised fuzzy metric has a topologically equivalent nonround truncation.

7.

Cantor-type characterizations by nested closed sets of vanishing fuzzy diameter predate the present work in the George–Veeramani convention; strong and p-complete variants have also been obtained [2,12,13]. Theorem 7 gives the precise dual statement for revised fuzzy metrics and supplies the form needed here. The fixed-point results in Section 6 then apply this intersection principle to nested invariant balls. Unlike the algebraic contraction theorems in [10,11,14,15,16,17,18], these results express the shrinking behaviour through conditions on the designated family of balls. They also differ from metrization-based transfer theorems and from fixed-point results obtained by replacing ordinary completeness with a stronger completeness notion [19,20].

The paper is organized as follows. Section 2 recalls revised fuzzy metrics and their induced topologies. Section 3 formulates Efe’s roundness notion in revised language, corrects the general ball-closure statement and establishes the radial-minimum characterization. Section 4 examines isometries, subspaces, finite and countable weighted products, and topologically equivalent truncations. Section 5 establishes a Cantor-type characterization of completeness and a nested-ball intersection principle. Section 6 presents the fixed-point principles, Section 7 provides examples and applications, and Section 8 contains the conclusions.

2. Preliminaries

This section recalls the basic notions concerning revised fuzzy metric spaces, their induced topologies and convergence. We also introduce the notions of Cauchy sequence, completeness and revised fuzzy diameter that will be used in the subsequent sections. Throughout, we use the separation-based convention in which the value 0 represents coincidence and larger values represent greater degrees of separation.

We begin with the algebraic operation appearing in the revised fuzzy triangle inequality.

Definition 1.

A binary operation

$$\oplus :[0,1]\times [0,1]⟶[0,1]$$

is called a continuous triangular conorm , or a continuous t-conorm, if, for all $a,b,c,d\in [0,1]$, the following conditions hold:

1. 

$a\oplus b=b\oplus a$;

2. 

$(a\oplus b)\oplus c=a\oplus (b\oplus c)$;

3. 

$a\oplus 0=a$;

4. 

if $a\le c$ and $b\le d$, then

$$a\oplus b\le c\oplus d;$$

5. 

the operation ⊕ is continuous.

Standard examples are the maximum t-conorm

$$a{\oplus }_{max}b=max\{a,b\},$$

the probabilistic t-conorm

$$a{\oplus }_{P}b=a+b-ab,$$

and the Łukasiewicz t-conorm, also called the bounded sum,

$$a{\oplus }_{L}b=min\{a+b,1\}.$$

For every t-conorm ⊕ and every $a,b\in [0,1]$, monotonicity and the neutral-element condition give

$$max\{a,b\}\le a\oplus b.$$

The following definition is the revised analogue of the George–Veeramani fuzzy metric and is equivalent to Definition 1 of [5].

Definition 2.

Let X be a nonempty set and let ⊕ be a continuous t-conorm. A mapping

$$\mu :X\times X\times (0,\infty )⟶[0,1)$$

is a revised fuzzy metric on X if, for all $x,y,z\in X$ and $s,t>0$,

(RFM1) 

$\mu (x,y,t)=0$ if and only if $x=y$;

(RFM2) 

$\mu (x,y,t)=\mu (y,x,t)$;

(RFM3) 

$$\mu (x,z,t+s)\le \mu (x,y,t)\oplus \mu (y,z,s);$$

(RFM4) 

the map

$$\mu (x,y,·):(0,\infty )⟶[0,1)$$

is continuous.

The triple $(X,\mu ,\oplus )$ is called a revised fuzzy metric space.

In Definition 1 of [5], the range condition

$$\mu (x,y,t)<1$$

is displayed separately as axiom (1RGV). In Definition 2, this condition is incorporated into the codomain $[0,1)$. Thus (RFM1)–(RFM4) correspond, respectively, to axioms (2RGV)–(5RGV) of [5].

The George–Veeramani and revised George–Veeramani formulations are connected by an order-reversing involution on the unit interval. In particular, under the standard involution $a\mapsto 1-a$, a George–Veeramani fuzzy metric M determines a revised fuzzy metric

$$\mu (x,y,t)=1-M(x,y,t).$$

This is the duality established in Theorem 1 of [5].

The dependence of a revised fuzzy metric on the time parameter is monotone.

Proposition 1.

Let $(X,\mu ,\oplus )$ be a revised fuzzy metric space. For every $x,y\in X$, the mapping

$$t⟼\mu (x,y,t)$$

is nonincreasing.

Proof. 

Let $0<t<s$. Since $s=t+(s-t)$, the revised fuzzy triangle inequality gives

$$\begin{array}{cc}\hfill \mu (x,y,s)& \le \mu (x,y,t)\oplus \mu (y,y,s-t)\hfill \\ & =\mu (x,y,t)\oplus 0\hfill \\ & =\mu (x,y,t).\hfill \end{array}$$

Hence $t\mapsto \mu (x,y,t)$ is nonincreasing. □

This monotonicity also appears in Remark 1 of [5]. In the more general t-conorm-based fuzzy pseudometric setting, it appears as Proposition 2 of [6].

Definition 3.

Let $(X,\mu ,\oplus )$ be a revised fuzzy metric space. For $x\in X$, $r\in (0,1)$ and $t>0$, define

$$B_{\mu }(x,r,t)=\{y\in X:\mu (x,y,t)<r\}$$

and

$$B_{\mu }[x,r,t]=\{y\in X:\mu (x,y,t)\le r\}.$$

These sets are called, respectively, the open revised fuzzy ball and the closed revised fuzzy ball with centre x, radius r and time parameter t.

If M is a George–Veeramani fuzzy metric and $\mu =1-M$ is its revised dual, then, with the same parameters $x,r,t$,

$$B_{\mu }(x,r,t)=\{y:M(x,y,t)>1-r\}=B_M(x,r,t)$$

and

$$B_{\mu }[x,r,t]=\{y:M(x,y,t)\ge 1-r\}=B_M[x,r,t].$$

Thus the duality preserves the open and closed ball sets, as well as their closures in the common induced topology. In particular, the roundness notion of Efe ([8], Definition 3.1) is exactly the dual of Definition 8 below.

The open-ball construction is Definition 2 of [5]. The fact that every open revised fuzzy ball is open in the topology generated by these balls is Proposition 1 of [5]; compare Proposition 4 of [6].

Definition 4.

The topology induced by μ is

$$\begin{array}{cc}\hfill {\tau }_{\mu }=\{U\subseteq X:& \mathrm{for}\mathrm{every}x\in U,\mathrm{there}\mathrm{exist}\hfill \\ & r\in (0,1)\mathrm{and}t>0\mathrm{such}\mathrm{that}\hfill \\ & B_{\mu }(x,r,t)\subseteq U\}.\hfill \end{array}$$

Equivalently, the collection

$${\mathcal{B}}_{\mu }=\{B_{\mu }(x,r,t):x\in X,r\in (0,1),t>0\}$$

is a base for ${\tau }_{\mu }$. This is Corollary 1 of [5]. The induced topological space $(X,{\tau }_{\mu })$ is first countable and Hausdorff by Propositions 2 and 3 of [5]; see also Propositions 5 and 6 of [6]. Consequently, closure can be characterized sequentially.

Proposition 2.

Let $A\subseteq X$ and let $x\in X$. Then

$$x\in {\overline{A}}^{{\tau }_{\mu }}$$

if and only if there exists a sequence $\left(x_n\right)$ in A such that $x_n\to x$ in ${\tau }_{\mu }$.

Proof. 

Suppose first that a sequence $\left(x_n\right)$ in A converges to x. Every neighbourhood of x then contains all sufficiently large terms of the sequence. Hence every neighbourhood of x meets A, and therefore

$$x\in {\overline{A}}^{{\tau }_{\mu }}.$$

Conversely, suppose that $x\in {\overline{A}}^{{\tau }_{\mu }}$. Since $(X,{\tau }_{\mu })$ is first countable, let $\left(U_n\right)$ be a decreasing countable local base at x. For every $n\in \mathbb{N}$,

$$U_n\cap A\ne ⌀.$$

Choose $x_n\in U_n\cap A$. The local-base property then implies that $x_n\to x$ in ${\tau }_{\mu }$. □

Convergence in the induced topology admits the following direct characterization.

Proposition 3.

Let $\left(x_n\right)$ be a sequence in X and let $x\in X$. Then $x_n\to x$ in $(X,{\tau }_{\mu })$ if and only if

$$\underset{n\to \infty }{lim}\mu (x,x_n,t)=0$$

for every $t>0$.

Proof. 

Suppose first that $x_n\to x$ in ${\tau }_{\mu }$. Fix $t>0$ and $r\in (0,1)$. Since $B_{\mu }(x,r,t)$ is a neighbourhood of x, there exists $n_0\in \mathbb{N}$ such that

$$x_n\in B_{\mu }(x,r,t)$$

whenever $n\ge n_0$. Hence

$$\mu (x,x_n,t)<r$$

for every $n\ge n_0$. Since this holds for every $r\in (0,1)$, it follows that

$$\mu (x,x_n,t)⟶0.$$

Conversely, suppose that

$$\mu (x,x_n,t)⟶0$$

for every $t>0$. Let U be a neighbourhood of x. By Definition 4, there exist $r\in (0,1)$ and $t>0$ such that

$$B_{\mu }(x,r,t)\subseteq U.$$

There exists $n_0\in \mathbb{N}$ such that

$$\mu (x,x_n,t)<r$$

whenever $n\ge n_0$. Therefore $x_n\in U$ whenever $n\ge n_0$, and hence $x_n\to x$ in ${\tau }_{\mu }$. □

This is Theorem 2 of [5] in the revised fuzzy metric setting and Theorem 2 of [6] in the corresponding t-conorm-based setting.

The principal class of examples is obtained from ordinary metric spaces.

Example 1.

Let $(X,d)$ be a metric space and define

$${\mu }_d(x,y,t)={\displaystyle \frac{d(x,y)}{t+d(x,y)}}$$

for all $x,y\in X$ and $t>0$. Then

$$(X,{\mu }_d,{\oplus }_{max})$$

is a revised fuzzy metric space, where

$$a{\oplus }_{max}b=max\{a,b\}.$$

The identity, symmetry and continuity conditions follow immediately from the corresponding properties of d. To verify the revised fuzzy triangle inequality, put

$$a=d(x,y)\mathit{and}b=d(y,z).$$

Since $d(x,z)\le a+b$, it is enough to prove that

$${\displaystyle \frac{a+b}{t+s+a+b}}\le max\left\{{\displaystyle \frac{a}{t+a}},{\displaystyle \frac{b}{s+b}}\right\}.$$

Let

$$q=max\left\{{\displaystyle \frac{a}{t+a}},{\displaystyle \frac{b}{s+b}}\right\}.$$

Since $a,b\ge 0$ and $s,t>0$, we have $0\le q<1$. The inequalities

$${\displaystyle \frac{a}{t+a}}\le q\mathit{and}{\displaystyle \frac{b}{s+b}}\le q$$

give

$$a\le {\displaystyle \frac{qt}{1-q}}\mathit{and}b\le {\displaystyle \frac{qs}{1-q}}.$$

Consequently,

$$a+b\le {\displaystyle \frac{q(t+s)}{1-q}},$$

which is equivalent to

$${\displaystyle \frac{a+b}{t+s+a+b}}\le q.$$

Therefore

$${\mu }_d(x,z,t+s)\le {\mu }_d(x,y,t){\oplus }_{max}{\mu }_d(y,z,s).$$

For $r\in (0,1)$, one has

$$\begin{array}{cc}\hfill {\mu }_d(x,y,t)<r& \iff {\displaystyle \frac{d(x,y)}{t+d(x,y)}}<r\hfill \\ & \iff d(x,y)<{\displaystyle \frac{rt}{1-r}}.\hfill \end{array}$$

Consequently,

$$B_{{\mu }_d}(x,r,t)=B_d\left(x,{\displaystyle \frac{rt}{1-r}}\right).$$

Similarly,

$$B_{{\mu }_d}[x,r,t]=B_d\left[x,{\displaystyle \frac{rt}{1-r}}\right].$$

Thus the topology induced by ${\mu }_d$ coincides with the topology induced by d.

Example 1 is the metric-induced revised fuzzy metric given in Example 1 of [5]. Notice that this example is formulated with the maximum t-conorm.

We now introduce the notions of Cauchy sequence and completeness used throughout this paper.

Definition 5. 

A sequence $\left(x_n\right)$ in a revised fuzzy metric space $(X,\mu ,\oplus )$ is called a $\mu$-Cauchy sequence if, for every $r\in (0,1)$ and every $t>0$, there exists $n_0\in \mathbb{N}$ such that

$$\mu (x_m,x_n,t)<r$$

whenever $m,n\ge n_0$.

Equivalently, $\left(x_n\right)$ is $\mu$-Cauchy if and only if

$$\underset{m,n\to \infty }{lim}\mu (x_m,x_n,t)=0$$

for every $t>0$.

Definition 6. 

A revised fuzzy metric space $(X,\mu ,\oplus )$ is called complete if every μ-Cauchy sequence converges in $(X,{\tau }_{\mu })$.

By Proposition 3, completeness means that for every $\mu$-Cauchy sequence $\left(x_n\right)$, there exists $x\in X$ such that

$$\underset{n\to \infty }{lim}\mu (x,x_n,t)=0$$

for every $t>0$.

The metric-induced construction preserves Cauchy sequences and completeness.

Proposition 4. 

Let $(X,d)$ be a metric space, and let $(X,{\mu }_d,{\oplus }_{max})$ be the revised fuzzy metric space defined in Example 1. Then a sequence is d-Cauchy if and only if it is ${\mu }_d$-Cauchy. Consequently, $(X,d)$ is complete if and only if $(X,{\mu }_d,{\oplus }_{max})$ is complete.

Proof. 

Suppose first that $\left(x_n\right)$ is d-Cauchy. Let $r\in (0,1)$ and $t>0$. There exists $n_0\in \mathbb{N}$ such that

$$d(x_m,x_n)<{\displaystyle \frac{rt}{1-r}}$$

whenever $m,n\ge n_0$. By Example 1, this is equivalent to

$${\mu }_d(x_m,x_n,t)<r.$$

Thus $\left(x_n\right)$ is ${\mu }_d$-Cauchy.

Conversely, suppose that $\left(x_n\right)$ is ${\mu }_d$-Cauchy, and let $\epsilon >0$. Fix $t=1$ and put

$$r={\displaystyle \frac{\epsilon }{1+\epsilon }}\in (0,1).$$

There exists $n_0\in \mathbb{N}$ such that

$${\mu }_d(x_m,x_n,1)<r$$

whenever $m,n\ge n_0$. The ball formula in Example 1 gives

$$d(x_m,x_n)<{\displaystyle \frac{r}{1-r}}=\epsilon .$$

Hence $\left(x_n\right)$ is d-Cauchy.

Finally, the topologies induced by d and ${\mu }_d$ coincide by Example 1. The equivalence of Cauchy sequences therefore implies the equivalence of completeness. □

For the Cantor-type intersection results developed later, we require a time-dependent notion of diameter.

Definition 7. 

Let A be a nonempty subset of a revised fuzzy metric space $(X,\mu ,\oplus )$. For every $t>0$, the revised fuzzy diameter of A at time tis

$${diam}_{\mu }(A,t)=sup\{\mu (x,y,t):x,y\in A\}.$$

A sequence $\left(A_n\right)$ of nonempty subsets of X is said to have vanishing revised fuzzy diameter if

$$\underset{n\to \infty }{lim}{diam}_{\mu }(A_n,t)=0$$

for every $t>0$.

The revised fuzzy diameter has the following elementary properties.

Proposition 5. 

Let A and C be nonempty subsets of X. Then:

1. 

if $A\subseteq C$, then

$${diam}_{\mu }(A,t)\le {diam}_{\mu }(C,t)$$

for every $t>0$;

2. 

if $0<t<s$, then

$${diam}_{\mu }(A,s)\le {diam}_{\mu }(A,t);$$

3. 

${diam}_{\mu }(A,t)=0$ for some $t>0$ if and only if A is a singleton.

Proof. 

The first assertion follows because the supremum defining ${diam}_{\mu }(A,t)$ is taken over a subset of the collection of pairs used to define ${diam}_{\mu }(C,t)$.

For the second assertion, Proposition 1 gives

$$\mu (x,y,s)\le \mu (x,y,t)$$

for every $x,y\in A$. Taking suprema proves the required inequality.

If $A=\left\{x\right\}$, then

$${diam}_{\mu }(A,t)=\mu (x,x,t)=0$$

for every $t>0$. Conversely, if ${diam}_{\mu }(A,t)=0$ for some $t>0$, then

$$\mu (x,y,t)=0$$

for all $x,y\in A$. By (RFM1), $x=y$, and therefore A is a singleton. □

Proposition 6. 

Let $\left(A_n\right)$ be a decreasing sequence of nonempty subsets of X having vanishing revised fuzzy diameter. If $x_n\in A_n$ for every $n\in \mathbb{N}$, then $\left(x_n\right)$ is a μ-Cauchy sequence.

Proof. 

Fix $r\in (0,1)$ and $t>0$. Since

$${diam}_{\mu }(A_n,t)⟶0,$$

there exists $n_0\in \mathbb{N}$ such that

$${diam}_{\mu }(A_{n_0},t)<r.$$

If $m,n\ge n_0$, then the decreasing character of $\left(A_n\right)$ gives

$$x_m,x_n\in A_{n_0}.$$

Consequently,

$$\mu (x_m,x_n,t)\le {diam}_{\mu }(A_{n_0},t)<r.$$

Thus $\left(x_n\right)$ is a $\mu$-Cauchy sequence. □

Remark 1. 

Let $(X,{\mu }_d,{\oplus }_{max})$ be the revised fuzzy metric space associated with a metric space $(X,d)$ as in Example 1. Since

$$u⟼{\displaystyle \frac{u}{t+u}}$$

is increasing and continuous on $[0,\infty )$, every nonempty subset $A\subseteq X$ of finite metric diameter satisfies

$${diam}_{{\mu }_d}(A,t)={\displaystyle \frac{{diam}_d\left(A\right)}{t+{diam}_d\left(A\right)}}.$$

It follows that, for any sequence $\left(A_n\right)$ of nonempty subsets having finite metric diameter,

$${diam}_{{\mu }_d}(A_n,t)⟶0⟺{diam}_d\left(A_n\right)⟶0.$$

Thus vanishing revised fuzzy diameter recovers the usual vanishing-diameter condition in the metric-induced setting.

The definitions and elementary properties established in this section will first be used to analyze closures of revised fuzzy balls and will subsequently support the nested-set and fixed-point principles.

3. Roundness and Closures of Balls

We begin by showing that, although closed revised fuzzy balls are topologically closed, the closure of an open revised fuzzy ball need not coincide with the corresponding closed ball.

3.1. The Published Ball-Closure Assertions

For clarity, we first distinguish two statements in the literature. Efe [8] proved only the always-valid inclusion of the closure of an open George–Veeramani fuzzy ball in the corresponding closed ball and then used equality to define roundness. By contrast, the later revised-fuzzy statements below assert equality for every space, without a roundness hypothesis. We now state precisely the assertions being corrected. In the notation of Definition 1 of [5], Corollary 2 of that paper states that, for every RGV-fuzzy metric space, every $x\in X$, $r\in (0,1)$, and $t>0$,

$$B_{\mu }[x,r,t]={\overline{B_{\mu }(x,r,t)}}^{{\tau }_{\mu }}.$$

In the notation of [6], Corollary 3 states the same equality for a CB-fuzzy metric that is continuous in its third variable:

$$B_m[x,r,t]={\overline{B_m(x,r,t)}}^{{\tau }_m}.$$

The continuity assumed in these results is sufficient to prove that the closed ball is topologically closed, and consequently that

$${\overline{B_{\mu }(x,r,t)}}^{{\tau }_{\mu }}\subseteq B_{\mu }[x,r,t].$$

It does not imply the reverse inclusion. What is additionally required is that every boundary point y, satisfying $\mu (x,y,t)=r$, be a limit of points $y_n$ satisfying $\mu (x,y_n,t)<r$. As shown in Proposition 8, this boundary-accessibility condition is equivalent to roundness.

Example 2. 

Let $X=\{0,1\}$, let ⊕ be any continuous t-conorm, and define

$$\mu (x,y,t)=\left\{\begin{array}{cc}0,\hfill & x=y,\hfill \\ {\displaystyle \frac{1}{2}},\hfill & x\ne y,\hfill \end{array}\right.t>0.$$

Then $(X,\mu ,\oplus )$ is a revised fuzzy metric space.

Indeed, μ takes its values in $[0,1)$, and the identity, symmetry and continuity conditions are immediate. To verify the revised fuzzy triangle inequality, let $x,y,z\in X$ and $s,t>0$. If $x=z$, then

$$\mu (x,z,t+s)=0\le \mu (x,y,t)\oplus \mu (y,z,s).$$

If $x\ne z$, then y must coincide with either x or z. Hence one of the two terms on the right-hand side is 0 and the other is ${\displaystyle \frac{1}{2}}$. Consequently,

$$\mu (x,y,t)\oplus \mu (y,z,s)={\displaystyle \frac{1}{2}}=\mu (x,z,t+s).$$

Thus all the revised fuzzy metric axioms are satisfied.

We next verify that the same construction meets the hypotheses used in [6]. On the positive-time domain it satisfies axioms $\left(\mathrm{CB}1\right)$–$\left(\mathrm{CB}4\right)$: the diagonal values are 0, symmetry and the triangle inequality were verified above, and $\mu (x,y,·)$ is constant and therefore left-continuous. It also satisfies the stronger separation and range conditions $\left(\mathrm{CB}{1}_{\mathrm{gv}}^{\prime }\right)$ and $\left({\mathrm{CB}}_{\infty }^{\mathrm{gv}}\right)$ from Definition 7 of that paper. Hence it is a $C{B}_{gv}$-fuzzy metric and, by Remark 3 of [6], a CB-fuzzy metric on the positive-time domain. Moreover, it is continuous, not merely left-continuous, in the third variable.

If the optional value at $t=0$ in Definition 4 of [6] is retained, define

$$\tilde{\mu }(x,y,0)=1$$

for all $x,y\in X$, and let $\tilde{\mu }(x,y,t)=\mu (x,y,t)$ for $t>0$. Then axiom $\left(\mathrm{CB}0\right)$ also holds. As explained in Remark 1 of that paper, the topology and the ball statements use $t>0$, so this extension does not change any ball or topological calculation below.

For $r={\displaystyle \frac{1}{2}}$ and any $t>0$,

$$B_{\mu }\left(0,{\displaystyle \frac{1}{2}},t\right)=\left\{0\right\},B_{\mu }\left[0,{\displaystyle \frac{1}{2}},t\right]=X.$$

The topology induced by μ is discrete. In fact, if $0<\rho <{\displaystyle \frac{1}{2}}$, then

$$B_{\mu }(x,\rho ,t)=\left\{x\right\}$$

for every $x\in X$ and $t>0$. Therefore

$${\overline{B_{\mu }\left(0,{\displaystyle \frac{1}{2}},t\right)}}^{{\tau }_{\mu }}=\left\{0\right\}\ne X=B_{\mu }\left[0,{\displaystyle \frac{1}{2}},t\right].$$

Thus the closure of an open revised fuzzy ball need not equal the corresponding closed revised fuzzy ball.

The failure in Example 2 is exactly the failure of the reverse inclusions in (A) and (B). Indeed,

$$B_{\mu }\left[0,{\displaystyle \frac{1}{2}},t\right]=X⊈\left\{0\right\}={\overline{B_{\mu }\left(0,{\displaystyle \frac{1}{2}},t\right)}}^{{\tau }_{\mu }}.$$

Thus Example 2 satisfies all the hypotheses of Corollary 2 of [5] and, under the notation just verified, all the hypotheses of Corollary 3 of [6], but the asserted conclusion is false. The missing assumption is not another continuity condition: it is the boundary-accessibility, or roundness, condition described above.

The following one-sided inclusion is nevertheless always valid.

Theorem 1. 

Let $(X,\mu ,\oplus )$ be a revised fuzzy metric space. Then

$${\overline{B_{\mu }(x,r,t)}}^{{\tau }_{\mu }}\subseteq B_{\mu }[x,r,t]$$

for every $x\in X$, $r\in (0,1)$ and $t>0$.

Proof. 

Let

$$y\in {\overline{B_{\mu }(x,r,t)}}^{{\tau }_{\mu }}.$$

By Proposition 2, there exists a sequence $\left(y_n\right)$ in $B_{\mu }(x,r,t)$ such that $y_n\to y$ in ${\tau }_{\mu }$. Hence

$$\mu (x,y_n,t)<r$$

for every n.

Fix $s>0$. By the revised fuzzy triangle inequality and the monotonicity of the t-conorm,

$$\begin{array}{cc}\hfill \mu (x,y,t+s)& \le \mu (x,y_n,t)\oplus \mu (y_n,y,s)\hfill \\ & \le r\oplus \mu (y_n,y,s).\hfill \end{array}$$

By Proposition 3,

$$\mu (y_n,y,s)⟶0.$$

The continuity of ⊕ therefore gives

$$\mu (x,y,t+s)\le r\oplus 0=r.$$

Since this inequality holds for every $s>0$, continuity of $\mu (x,y,·)$ at t yields

$$\mu (x,y,t)=\underset{s\downarrow 0}{lim}\mu (x,y,t+s)\le r.$$

Thus

$$y\in B_{\mu }[x,r,t].$$

□

Remark 2. 

The second inequality in the proof of Theorem 1 is non-strict. Although $\mu (x,y_n,t)<r$, a t-conorm is only required to be nondecreasing, not strictly increasing. For example, the maximum t-conorm need not preserve strict inequalities.

The preceding inclusion alone does not prove that closed revised fuzzy balls are topologically closed. That fact requires a separate argument.

Proposition 7. 

Every closed revised fuzzy ball $B_{\mu }[x,r,t]$ is closed in $(X,{\tau }_{\mu })$.

Proof. 

Let $\left(y_n\right)$ be a sequence in $B_{\mu }[x,r,t]$ converging to $y\in X$. Thus

$$\mu (x,y_n,t)\le r$$

for every n. Fix $s>0$. The revised fuzzy triangle inequality gives

$$\begin{array}{cc}\hfill \mu (x,y,t+s)& \le \mu (x,y_n,t)\oplus \mu (y_n,y,s)\hfill \\ & \le r\oplus \mu (y_n,y,s).\hfill \end{array}$$

Since $y_n\to y$, Proposition 3 gives

$$\mu (y_n,y,s)⟶0.$$

Passing to the limit in n and using the continuity of ⊕, we obtain

$$\mu (x,y,t+s)\le r\oplus 0=r.$$

Finally, letting $s\downarrow 0$ and using the continuity of $\mu (x,y,·)$, we obtain

$$\mu (x,y,t)\le r.$$

Hence $y\in B_{\mu }[x,r,t]$.

We have shown that $B_{\mu }[x,r,t]$ is sequentially closed. Since $(X,{\tau }_{\mu })$ is first countable, sequential closedness is equivalent to closedness. □

Proposition 7 is Theorem 3 of [5] in the revised fuzzy metric setting and Theorem 3 of [6] in the t-conorm-based fuzzy pseudometric setting. These theorems establish that closed balls are closed; they do not, by themselves, imply that a closed ball is the closure of the corresponding open ball.

Efe ([8], Definition 3.1), building on Nathanson’s metric notion [7], defined a George–Veeramani fuzzy metric to be round when every closed fuzzy ball is the closure of the corresponding open fuzzy ball. In view of the ball identities following Definition 3, its exact revised formulation is the following.

Definition 8. 

A revised fuzzy metric space $(X,\mu ,\oplus )$ is called round if

$${\overline{B_{\mu }(x,r,t)}}^{{\tau }_{\mu }}=B_{\mu }[x,r,t]$$

for every $x\in X$, $r\in (0,1)$ and $t>0$.

Theorem 1 shows that roundness is precisely the requirement that every point satisfying $\mu (x,y,t)=r$ can be approximated by points lying in $B_{\mu }(x,r,t)$.

Proposition 8. 

Let $(X,\mu ,\oplus )$ be a revised fuzzy metric space. The following conditions are equivalent:

1. 

$(X,\mu ,\oplus )$ is round;

2. 

whenever $x,y\in X$, $t>0$ and

$$\mu (x,y,t)=r\in (0,1),$$

every ${\tau }_{\mu }$-neighbourhood of y meets $B_{\mu }(x,r,t)$;

3. 

whenever $x,y\in X$, $t>0$ and

$$\mu (x,y,t)=r\in (0,1),$$

there exists a sequence $\left(y_n\right)$ in X such that

$$y_n⟶y\mathit{in}{\tau }_{\mu }$$

and

$$\mu (x,y_n,t)<r$$

for every $n\in \mathbb{N}$.

Proof. 

Suppose first that $(X,\mu ,\oplus )$ is round and that $\mu (x,y,t)=r\in (0,1)$. Then

$$y\in B_{\mu }[x,r,t]={\overline{B_{\mu }(x,r,t)}}^{{\tau }_{\mu }}.$$

Therefore every neighbourhood of y meets $B_{\mu }(x,r,t)$. Thus (1) implies (2).

Suppose that (2) holds. Since $(X,{\tau }_{\mu })$ is first countable, Proposition 2 implies that

$$y\in {\overline{B_{\mu }(x,r,t)}}^{{\tau }_{\mu }}$$

if and only if there exists a sequence in $B_{\mu }(x,r,t)$ converging to y. Hence (2) implies (3).

Finally, assume (3). Let

$$y\in B_{\mu }[x,r,t].$$

If $\mu (x,y,t)<r$, then

$$y\in B_{\mu }(x,r,t)\subseteq {\overline{B_{\mu }(x,r,t)}}^{{\tau }_{\mu }}.$$

If $\mu (x,y,t)=r$, condition (3) gives a sequence $\left(y_n\right)$ in $B_{\mu }(x,r,t)$ converging to y. Therefore

$$y\in {\overline{B_{\mu }(x,r,t)}}^{{\tau }_{\mu }}.$$

Consequently,

$$B_{\mu }[x,r,t]\subseteq {\overline{B_{\mu }(x,r,t)}}^{{\tau }_{\mu }}.$$

The reverse inclusion follows from Theorem 1. Hence equality holds and the space is round. □

The following result gives an open-set characterization of the obstruction to roundness. It is the revised fuzzy analogue of the radial-minimum criterion for ordinary metrics obtained by Singh and Narang ([9], Theorem 1).

Theorem 2. 

Let $(X,\mu ,\oplus )$ be a revised fuzzy metric space. The following conditions are equivalent:

1. 

$(X,\mu ,\oplus )$ is not round;

2. 

there exist $x\in X$, $t>0$, a nonempty ${\tau }_{\mu }$-open set $U\subseteq X$ with $x\notin U$, and $y\in U$ such that

$$\mu (x,y,t)=\underset{z\in U}{min}\mu (x,z,t)>0.$$

Proof. 

Assume first that $(X,\mu ,\oplus )$ is not round. By Theorem 1, there exist $x,y\in X$, $t>0$ and $r\in (0,1)$ such that

$$\mu (x,y,t)=r\mathrm{and}y\notin {\overline{B_{\mu }(x,r,t)}}^{{\tau }_{\mu }}.$$

Set

$$U=X\setminus {\overline{B_{\mu }(x,r,t)}}^{{\tau }_{\mu }}.$$

Then U is open, $y\in U$, and $x\notin U$. If $z\in U$, then $z\notin B_{\mu }(x,r,t)$, so

$$\mu (x,z,t)\ge r=\mu (x,y,t).$$

Hence the radial map $z\mapsto \mu (x,z,t)$ attains the positive minimum r on U.

Conversely, suppose that (2) holds and put

$$r=\mu (x,y,t)>0.$$

The minimality of r gives

$$B_{\mu }(x,r,t)\cap U=⌀.$$

Since $X\setminus U$ is closed,

$${\overline{B_{\mu }(x,r,t)}}^{{\tau }_{\mu }}\subseteq X\setminus U.$$

Thus $y\notin {\overline{B_{\mu }(x,r,t)}}^{{\tau }_{\mu }}$, whereas $\mu (x,y,t)=r$ implies $y\in B_{\mu }[x,r,t]$. Therefore

$${\overline{B_{\mu }(x,r,t)}}^{{\tau }_{\mu }}\ne B_{\mu }[x,r,t],$$

and the space is not round. □

Corollary 1. 

A revised fuzzy metric space is round if and only if, for every $x\in X$, $t>0$, and nonempty open set U not containing x, the radial map $z\mapsto \mu (x,z,t)$ does not attain a positive minimum on U.

For revised fuzzy metrics induced by ordinary metrics, roundness is exactly the classical roundness property.

Theorem 3. 

Let $(X,d)$ be a metric space, and denote by ${\tau }_d$ the topology induced by d. Let $(X,{\mu }_d,{\oplus }_{max})$ be the standard revised fuzzy metric space of Example 1. Then $(X,{\mu }_d,{\oplus }_{max})$ is round if and only if d is a round metric on X in the sense of Nathanson [7].

Proof. 

Fix $x\in X$, $r\in (0,1)$ and $t>0$, and put

$$R={\displaystyle \frac{rt}{1-r}}.$$

By Example 1,

$$B_{{\mu }_d}(x,r,t)=B_d(x,R)$$

and

$$B_{{\mu }_d}[x,r,t]=B_d[x,R].$$

Moreover, ${\tau }_{{\mu }_d}={\tau }_d$. Therefore

$${\overline{B_{{\mu }_d}(x,r,t)}}^{{\tau }_{{\mu }_d}}=B_{{\mu }_d}[x,r,t]$$

if and only if

$${\overline{B_d(x,R)}}^{{\tau }_d}=B_d[x,R].$$

For every $R>0$, one can choose any $t>0$ and then set

$$r={\displaystyle \frac{R}{t+R}}\in (0,1),$$

so that

$$R={\displaystyle \frac{rt}{1-r}}.$$

Thus the revised fuzzy ball equality holds for every admissible $r,t$ if and only if the classical metric ball equality holds for every positive radius R. □

Corollary 2. 

Let $(X,d)$ be a geodesic metric space. Then its standard revised fuzzy metric space $(X,{\mu }_d,{\oplus }_{max})$ is round.

Proof. 

It is enough, by Theorem 3, to prove that d is round. Fix $x\in X$ and $R>0$, and let

$$y\in B_d[x,R].$$

If $d(x,y)<R$, then $y\in B_d(x,R)$. Suppose that $d(x,y)=R$. Let

$$\gamma :[0,R]⟶X$$

be a geodesic segment from x to y, parametrized by arc length, so that

$$\gamma \left(0\right)=x,\gamma \left(R\right)=y,$$

and

$$d\left(\gamma \right(u),\gamma (v\left)\right)=|u-v|$$

for all $u,v\in [0,R]$. Define

$$y_n=\gamma \left(R-{\displaystyle \frac{R}{n+1}}\right).$$

Then

$$d(x,y_n)=R-{\displaystyle \frac{R}{n+1}}<R,$$

so $y_n\in B_d(x,R)$, while

$$d(y_n,y)={\displaystyle \frac{R}{n+1}}⟶0.$$

Hence

$$y\in {\overline{B_d(x,R)}}^{{\tau }_d}.$$

Together with the general inclusion

$${\overline{B_d(x,R)}}^{{\tau }_d}\subseteq B_d[x,R],$$

this proves

$${\overline{B_d(x,R)}}^{{\tau }_d}=B_d[x,R].$$

Therefore d is round, and Theorem 3 completes the proof. □

Remark 3. 

Corollary 2 is consistent with the more general intermediate-point criterion given in Theorem 5 of [7]. That theorem assumes that there is an $\alpha \in (0,1)$ such that, for every $x,y\in X$, some $z\in X$ satisfies

$$\begin{array}{cc}\hfill d(z,x)& =(1-\alpha )d(x,y),\hfill \\ \hfill d(z,y)& =\alpha d(x,y).\hfill \end{array}$$

then d is round. Every geodesic metric space satisfies this condition: one may choose

$$z=\gamma \left((1-\alpha )d(x,y)\right)$$

on a geodesic segment γ joining x to y.

4. Permanence Properties

We investigate the behaviour of roundness under isometries, subspaces, finite products and countable weighted products. Roundness is invariant under revised fuzzy isometries and is inherited by dense and open subspaces, but not by arbitrary subspaces.

Definition 9. 

Let $(X,\mu ,\oplus )$ and $(Y,\nu ,\oplus )$ be revised fuzzy metric spaces. A bijection

$$T:X⟶Y$$

is called a revised fuzzy isometry if

$$\nu (Tx,Ty,t)=\mu (x,y,t)$$

for all $x,y\in X$ and $t>0$.

Proposition 9. 

Roundness is invariant under revised fuzzy isometries.

Proof. 

Let

$$T:(X,\mu ,\oplus )⟶(Y,\nu ,\oplus )$$

be a revised fuzzy isometry. For every $x\in X$, $r\in (0,1)$ and $t>0$, one has

$$T\left(B_{\mu }(x,r,t)\right)=B_{\nu }(Tx,r,t)$$

and

$$T\left(B_{\mu }[x,r,t]\right)=B_{\nu }[Tx,r,t].$$

Indeed,

$$\begin{array}{cc}\hfill y\in B_{\mu }(x,r,t)& \iff \mu (x,y,t)<r\hfill \\ & \iff \nu (Tx,Ty,t)<r\hfill \\ & \iff Ty\in B_{\nu }(Tx,r,t),\hfill \end{array}$$

and the closed-ball identity follows similarly.

Since open revised fuzzy balls form bases for the induced topologies, T is a homeomorphism. Suppose that $(X,\mu ,\oplus )$ is round. Then

$$\begin{array}{cc}\hfill {\overline{B_{\nu }(Tx,r,t)}}^{{\tau }_{\nu }}& ={\overline{T\left(B_{\mu }(x,r,t)\right)}}^{{\tau }_{\nu }}\hfill \\ & =T\left({\overline{B_{\mu }(x,r,t)}}^{{\tau }_{\mu }}\right)\hfill \\ & =T\left(B_{\mu }[x,r,t]\right)\hfill \\ & =B_{\nu }[Tx,r,t].\hfill \end{array}$$

Surjectivity of T shows that $(Y,\nu ,\oplus )$ is round. The converse follows by applying the same argument to $T^{-1}$. □

Roundness is not inherited by arbitrary subspaces.

Example 3. 

The real line $\mathbb{R}$, equipped with its usual metric

$$d(x,y)=|x-y|,$$

is geodesic and hence round. Consider the subspace

$$Y=\{0,1\}.$$

In the restricted metric,

$$B_Y(0,1)=\left\{0\right\},B_Y[0,1]=Y.$$

Since Y is discrete,

$${\overline{B_Y(0,1)}}^Y=\left\{0\right\}\ne Y=B_Y[0,1].$$

Thus the restricted metric on Y is not round. By Theorem 3, the corresponding standard revised fuzzy metric

$$\left(Y,{\mu }_{{d|}_{Y\times Y}},{\oplus }_{max}\right)$$

is not round. This also agrees, under duality, with Efe’s observation that a fuzzy metric space with an isolated point cannot carry a round fuzzy metric ([8], Conclusion 3.6).

Remark 4. 

The preceding example is consistent with the classical examples of Nathanson [7]. In particular,

$$(0,1)\cup (2,3)$$

is round with the metric inherited from $\mathbb{R}$, whereas

$$[0,1]\cup [2,3]$$

is not round. More generally, Theorem 1 of [7] states that if a metrizable space X is a disjoint union

$$X=A\cup K$$

of nonempty closed sets, with K compact, then no metric compatible with the topology of X is round. Thus roundness is not a hereditary topological property. Efe established the corresponding fuzzy-metric obstruction in Theorem 3.5 of [8].

Dense subspaces inherit roundness.

Proposition 10. 

Let $(X,\mu ,\oplus )$ be a round revised fuzzy metric space, and let Y be dense in $(X,{\tau }_{\mu })$. Then

$$\left({Y,\mu |}_{Y\times Y\times (0,\infty )},\oplus \right)$$

is round.

Proof. 

Write

$${\mu }_Y{=\mu |}_{Y\times Y\times (0,\infty )}.$$

For $x\in Y$, $r\in (0,1)$ and $t>0$,

$$B_{{\mu }_Y}(x,r,t)=B_{\mu }(x,r,t)\cap Y$$

and

$$B_{{\mu }_Y}[x,r,t]=B_{\mu }[x,r,t]\cap Y.$$

The first identity shows that ${\tau }_{{\mu }_Y}$ is exactly the subspace topology inherited from $(X,{\tau }_{\mu })$.

Let

$$y\in B_{{\mu }_Y}[x,r,t].$$

Roundness of X gives

$$y\in {\overline{B_{\mu }(x,r,t)}}^{{\tau }_{\mu }}.$$

Let V be a neighbourhood of y in Y. There exists a ${\tau }_{\mu }$-open neighbourhood U of y such that

$$V=U\cap Y.$$

Because y lies in the closure of $B_{\mu }(x,r,t)$, the set

$$U\cap B_{\mu }(x,r,t)$$

is nonempty. It is also ${\tau }_{\mu }$-open. Since Y is dense in $(X,{\tau }_{\mu })$,

$$U\cap B_{\mu }(x,r,t)\cap Y\ne ⌀.$$

Therefore

$$V\cap B_{{\mu }_Y}(x,r,t)\ne ⌀,$$

and hence

$$y\in {\overline{B_{{\mu }_Y}(x,r,t)}}^{{\tau }_{{\mu }_Y}}.$$

Thus

$$B_{{\mu }_Y}[x,r,t]\subseteq {\overline{B_{{\mu }_Y}(x,r,t)}}^{{\tau }_{{\mu }_Y}}.$$

The reverse inclusion follows from Theorem 1, applied to the restricted revised fuzzy metric. Hence the restricted space is round. □

Open subspaces also inherit roundness.

Proposition 11. 

Every open subspace of a round revised fuzzy metric space is round.

Proof. 

Let Y be open in $(X,{\tau }_{\mu })$, and write

$${\mu }_Y{=\mu |}_{Y\times Y\times (0,\infty )}.$$

As in the proof of Proposition 10, ${\tau }_{{\mu }_Y}$ is the subspace topology on Y. Fix $x\in Y$, $r\in (0,1)$ and $t>0$, and let

$$y\in B_{{\mu }_Y}[x,r,t].$$

Then $y\in B_{\mu }[x,r,t]$. Since X is round,

$$y\in {\overline{B_{\mu }(x,r,t)}}^{{\tau }_{\mu }}.$$

By Proposition 2, there exists a sequence

$$\left(y_n\right)\subseteq B_{\mu }(x,r,t)$$

converging to y in ${\tau }_{\mu }$.

Because Y is an open neighbourhood of y, the sequence is eventually contained in Y. Its resulting tail lies in

$$B_{\mu }(x,r,t)\cap Y=B_{{\mu }_Y}(x,r,t)$$

and converges to y in ${\tau }_{{\mu }_Y}$. Therefore

$$y\in {\overline{B_{{\mu }_Y}(x,r,t)}}^{{\tau }_{{\mu }_Y}}.$$

The reverse inclusion follows from Theorem 1. Hence

$${\overline{B_{{\mu }_Y}(x,r,t)}}^{{\tau }_{{\mu }_Y}}=B_{{\mu }_Y}[x,r,t],$$

so the restricted space is round. □

For finite maximum products, roundness is equivalent to roundness of every factor.

Theorem 4. 

For $i=1,\dots ,m$, let $(X_i,{\mu }_i,\oplus )$ be revised fuzzy metric spaces, where all factors use the same continuous t-conorm. On

$$X={\displaystyle \prod _{i=1}^m}{X}_i$$

define

$$\mu (x,y,t)=\underset{1\le i\le m}{max}{\mu }_i(x_i,y_i,t).$$

Then $(X,\mu ,\oplus )$ is a revised fuzzy metric space and ${\tau }_{\mu }$ is the finite product topology

$${\displaystyle \prod _{i=1}^m}{\tau }_{{\mu }_i}.$$

Moreover, $(X,\mu ,\oplus )$ is round if and only if every $(X_i,{\mu }_i,\oplus )$ is round.

Proof. 

Because the maximum is taken over finitely many values in $[0,1)$,

$$\mu (x,y,t)<1.$$

Moreover,

$$\mu (x,y,t)=0$$

if and only if

$${\mu }_i(x_i,y_i,t)=0$$

for every i, which is equivalent to $x=y$. Symmetry and continuity in the third variable follow from the corresponding properties of the finitely many functions ${\mu }_i$.

For the revised fuzzy triangle inequality, put

$$A=\underset{1\le i\le m}{max}{\mu }_i(x_i,y_i,t),B=\underset{1\le i\le m}{max}{\mu }_i(y_i,z_i,s).$$

For each $i=1,\dots ,m$, monotonicity of ⊕ gives

$$\begin{array}{cc}\hfill {\mu }_i(x_i,z_i,t+s)& \le {\mu }_i(x_i,y_i,t)\oplus {\mu }_i(y_i,z_i,s)\hfill \\ & \le A\oplus B.\hfill \end{array}$$

Taking the maximum over i, we obtain

$$\begin{array}{cc}\hfill \mu (x,z,t+s)& =\underset{1\le i\le m}{max}{\mu }_i(x_i,z_i,t+s)\hfill \\ & \le A\oplus B\hfill \\ & =\mu (x,y,t)\oplus \mu (y,z,s).\hfill \end{array}$$

Thus $(X,\mu ,\oplus )$ is a revised fuzzy metric space.

For every $x\in X$, $r\in (0,1)$ and $t>0$,

$$B_{\mu }(x,r,t)={\displaystyle \prod _{i=1}^m}{B}_{{\mu }_i}(x_i,r,t).$$

Therefore every $\mu$-ball is open in the finite product topology, so

$${\tau }_{\mu }\subseteq {\displaystyle \prod _{i=1}^m}{\tau }_{{\mu }_i}.$$

Conversely, let

$$U={\displaystyle \prod _{i=1}^m}{U}_i$$

be a basic product neighbourhood of x. For each i, choose $r_i\in (0,1)$ and $t_i>0$ such that

$$B_{{\mu }_i}(x_i,r_i,t_i)\subseteq U_i.$$

Set

$$r=\underset{1\le i\le m}{min}{r}_i,t=\underset{1\le i\le m}{min}{t}_i.$$

If $y\in B_{\mu }(x,r,t)$, then

$${\mu }_i(x_i,y_i,t)<r\le r_i$$

for every i. Since $t\le t_i$ and the functions ${\mu }_i(x_i,y_i,·)$ are nonincreasing,

$${\mu }_i(x_i,y_i,t_i)\le {\mu }_i(x_i,y_i,t)<r_i.$$

Hence

$$y_i\in B_{{\mu }_i}(x_i,r_i,t_i)\subseteq U_i$$

for every i, and therefore

$$B_{\mu }(x,r,t)\subseteq U.$$

It follows that

$${\tau }_{\mu }={\displaystyle \prod _{i=1}^m}{\tau }_{{\mu }_i}.$$

Similarly,

$$B_{\mu }[x,r,t]={\displaystyle \prod _{i=1}^m}{B}_{{\mu }_i}[x_i,r,t].$$

Assume first that every factor is round. Then, since the closure of a Cartesian product is the Cartesian product of the closures in the finite product topology,

$$\begin{array}{cc}\hfill {\overline{B_{\mu }(x,r,t)}}^{{\tau }_{\mu }}& ={\overline{{\prod }_{i=1}^m{B}_{{\mu }_i}(x_i,r,t)}}^{{\prod }_{i=1}^m{\tau }_{{\mu }_i}}\hfill \\ & ={\displaystyle \prod _{i=1}^m}{\overline{B_{{\mu }_i}(x_i,r,t)}}^{{\tau }_{{\mu }_i}}\hfill \\ & ={\displaystyle \prod _{i=1}^m}{B}_{{\mu }_i}[x_i,r,t]\hfill \\ & =B_{\mu }[x,r,t].\hfill \end{array}$$

Thus $(X,\mu ,\oplus )$ is round.

Conversely, suppose that $(X,\mu ,\oplus )$ is round. Fix $i\in \{1,\dots ,m\}$, choose $a_j\in X_j$ for $j\ne i$, and let $x_i,y_i\in X_i$, $t>0$, and $r\in (0,1)$ satisfy

$${\mu }_i(x_i,y_i,t)=r.$$

Define $x,y\in X$ by taking their i-th coordinates to be $x_i,y_i$, respectively, and setting $x_j=y_j=a_j$ for $j\ne i$. Then

$$\mu (x,y,t)=r.$$

By Proposition 8, there exists a sequence $\left(y^{\left(n\right)}\right)$ converging to y in the product topology such that

$$\mu (x,y^{\left(n\right)},t)<r$$

for every n. Hence $y_i^{\left(n\right)}\to y_i$ and

$${\mu }_i(x_i,y_i^{\left(n\right)},t)\le \mu (x,y^{\left(n\right)},t)<r.$$

Another application of Proposition 8 shows that $(X_i,{\mu }_i,\oplus )$ is round. Since i was arbitrary, every factor is round. □

The finite maximum construction admits a countable weighted extension. The decay of the weights is essential: it makes the supremum continuous in the time variable and ensures that a ball of any fixed positive radius restricts only finitely many coordinates.

Theorem 5. 

For each $i\in \mathbb{N}$, let $(X_i,{\mu }_i,{\oplus }_{max})$ be a revised fuzzy metric space, and let ${\left(a_i\right)}_{i\in \mathbb{N}}\subseteq (0,1]$ satisfy

$$\underset{i\to \infty }{lim}{a}_i=0.$$

On

$$X={\displaystyle \prod _{i=1}^{\infty }}{X}_i$$

define

$$\mu (x,y,t)=\underset{i\in \mathbb{N}}{sup}{a}_i{\mu }_i(x_i,y_i,t).$$

Then $(X,\mu ,{\oplus }_{max})$ is a revised fuzzy metric space and ${\tau }_{\mu }$ is the product topology

$${\displaystyle \prod _{i=1}^{\infty }}{\tau }_{{\mu }_i}.$$

Moreover:

1. 

$(X,\mu ,{\oplus }_{max})$ is round if and only if every $(X_i,{\mu }_i,{\oplus }_{max})$ is round;

2. 

$(X,\mu ,{\oplus }_{max})$ is complete if and only if every $(X_i,{\mu }_i,{\oplus }_{max})$ is complete.

Proof. 

We first verify that $\mu$ takes its values in $[0,1)$. Choose $N\in \mathbb{N}$ such that

$$\underset{i>N}{sup}{a}_i<{\displaystyle \frac{1}{2}}.$$

For fixed $x,y\in X$ and $t>0$,

$$\underset{1\le i\le N}{max}{a}_i{\mu }_i(x_i,y_i,t)<1,$$

while

$$\underset{i>N}{sup}{a}_i{\mu }_i(x_i,y_i,t)\le \underset{i>N}{sup}{a}_i<{\displaystyle \frac{1}{2}}.$$

Hence $\mu (x,y,t)<1$. Furthermore, $\mu (x,y,t)=0$ if and only if ${\mu }_i(x_i,y_i,t)=0$ for every i, which is equivalent to $x=y$. Symmetry follows coordinatewise.

For the triangle inequality, let $x,y,z\in X$ and $s,t>0$. For every $i\in \mathbb{N}$,

$$\begin{array}{cc}\hfill a_i{\mu }_i(x_i,z_i,t+s)& \le a_{i}max\{{\mu }_i(x_i,y_i,t),{\mu }_i(y_i,z_i,s)\}\hfill \\ & =max\{a_i{\mu }_i(x_i,y_i,t),a_i{\mu }_i(y_i,z_i,s)\}\hfill \\ & \le max\left\{\mu \right(x,y,t),\mu (y,z,s\left)\right\}.\hfill \end{array}$$

Taking the supremum over i gives

$$\mu (x,z,t+s)\le \mu (x,y,t){\oplus }_{max}\mu (y,z,s).$$

It remains to prove continuity in the third variable. For $N\in \mathbb{N}$, put

$$g_N\left(t\right)=\underset{1\le i\le N}{max}{a}_i{\mu }_i(x_i,y_i,t).$$

Each $g_N$ is continuous and

$$0\le \mu (x,y,t)-g_N\left(t\right)\le \underset{i>N}{sup}{a}_i$$

for every $t>0$. Thus $\left(g_N\right)$ converges uniformly on $(0,\infty )$ to $\mu (x,y,·)$, which is therefore continuous. This proves that $(X,\mu ,{\oplus }_{max})$ is a revised fuzzy metric space.

We next identify its topology. Fix $r\in (0,1)$, and set

$$I_r=\{i\in \mathbb{N}:a_i>r\}.$$

The set $I_r$ is finite. Directly from the definition,

$$B_{\mu }(x,r,t)=\prod _{i\in I_r}{B}_{{\mu }_i}\left(x_i,{\displaystyle \frac{r}{a_i}},t\right)\times \prod _{i\notin I_r}{X}_i.$$

Thus every $\mu$-ball is open in the product topology.

Conversely, let U be a basic product neighbourhood of x, so that there is a finite set $F\subseteq \mathbb{N}$ and open sets $U_i\in {\tau }_{{\mu }_i}$ such that

$$U=\prod _{i\in F}{U}_i\times \prod _{i\notin F}{X}_i\mathrm{and}{x}_i\in U_i.$$

If $F=⌀$, then $U=X$ and there is nothing to prove. Assume therefore that $F\ne ⌀$. For every $i\in F$, choose $r_i\in (0,1)$ and $t_i>0$ with

$$B_{{\mu }_i}(x_i,r_i,t_i)\subseteq U_i.$$

Choose $t={min}_{i\in F}{t}_i$ and

$$0<r<\underset{i\in F}{min}{a}_i{r}_i.$$

If $y\in B_{\mu }(x,r,t)$, then, for every $i\in F$,

$${\mu }_i(x_i,y_i,t)<{\displaystyle \frac{r}{a_i}}<r_i.$$

Since $t\le t_i$ and ${\mu }_i(x_i,y_i,·)$ is nonincreasing,

$${\mu }_i(x_i,y_i,t_i)\le {\mu }_i(x_i,y_i,t)<r_i.$$

Therefore $B_{\mu }(x,r,t)\subseteq U$, and ${\tau }_{\mu }$ is precisely the product topology.

For the roundness assertion, observe similarly that

$$B_{\mu }[x,r,t]=\prod _{i\in I_r}{B}_{{\mu }_i}\left[x_i,{\displaystyle \frac{r}{a_i}},t\right]\times \prod _{i\notin I_r}{X}_i.$$

If every factor is round, then, because $I_r$ is finite and ${\tau }_{\mu }$ is the product topology,

$$\begin{array}{cc}\hfill {\overline{B_{\mu }(x,r,t)}}^{{\tau }_{\mu }}& =\prod _{i\in I_r}{\overline{B_{{\mu }_i}(x_i,r/a_i,t)}}^{{\tau }_{{\mu }_i}}\times \prod _{i\notin I_r}{X}_i\hfill \\ & =\prod _{i\in I_r}{B}_{{\mu }_i}\left[x_i,{\displaystyle \frac{r}{a_i}},t\right]\times \prod _{i\notin I_r}{X}_i\hfill \\ & =B_{\mu }[x,r,t].\hfill \end{array}$$

Hence the product is round.

Conversely, suppose that the product is round. Fix $i\in \mathbb{N}$, choose base points $b_j\in X_j$ for $j\ne i$, and suppose that

$${\mu }_i(x_i,y_i,t)=\rho \in (0,1).$$

Define $x,y\in X$ by using $x_i,y_i$ in the i-th coordinate and putting $x_j=y_j=b_j$ for $j\ne i$. Then

$$\mu (x,y,t)=a_i\rho .$$

Roundness and Proposition 8 give a sequence $\left(y^{\left(n\right)}\right)$ converging to y such that

$$\mu (x,y^{\left(n\right)},t)<a_i\rho$$

for every n. Product convergence gives $y_i^{\left(n\right)}\to y_i$, and

$${\mu }_i(x_i,y_i^{\left(n\right)},t)<\rho .$$

Proposition 8 now shows that the i-th factor is round. Since i was arbitrary, every factor is round.

Finally, suppose that every factor is complete and let $\left(x^{\left(n\right)}\right)$ be a $\mu$-Cauchy sequence. Fix $i\in \mathbb{N}$, $q\in (0,1)$ and $t>0$. Applying the $\mu$-Cauchy condition with radius $a_{i}q$ shows that

$${\mu }_i(x_i^{\left(n\right)},x_i^{\left(m\right)},t)<q$$

for all sufficiently large $m,n$. Hence $\left(x_i^{\left(n\right)}\right)$ is ${\mu }_i$-Cauchy and converges to some $x_i\in X_i$. Thus $x^{\left(n\right)}\to x=\left(x_i\right)$ in the product topology, which equals ${\tau }_{\mu }$. Therefore the product is complete.

For the converse, assume that the product is complete. Fix $i\in \mathbb{N}$, choose points $b_j\in X_j$ for $j\ne i$, and let $\left(u_n\right)$ be a ${\mu }_i$-Cauchy sequence in $X_i$. Embed it in X by letting the i-th coordinate of $x^{\left(n\right)}$ be $u_n$ and all other coordinates be the corresponding $b_j$. Then

$$\mu (x^{\left(n\right)},x^{\left(m\right)},t)=a_i{\mu }_i(u_n,u_m,t),$$

so $\left(x^{\left(n\right)}\right)$ is $\mu$-Cauchy: indeed, for a prescribed $r\in (0,1)$, the required inequality is automatic if $r\ge a_i$, whereas if $r<a_i$, it follows from the ${\mu }_i$-Cauchy condition with radius $r/a_i$. The sequence therefore converges in X, and convergence in the product topology implies that $\left(u_n\right)$ converges in $(X_i,{\tau }_{{\mu }_i})$. Hence every factor is complete. □

Remark 5. 

Theorem 4 is related to the classical product result in Theorem 4 of [7]. Nathanson considered a countable family of metric spaces, with all but finitely many factors having finite diameter, and proved that a specified weighted product metric is round if and only if every factor metric is round. The corollary following his Theorem 4 states that a countable product of round metrizable spaces is round metrizable.

Theorem 4 concerns finite products equipped with the unweighted maximum of the factor revised fuzzy metrics. In contrast, Theorem 5 uses weights converging to zero to obtain a revised fuzzy metric inducing the countable product topology. It gives factorwise equivalences for both roundness and completeness.

Theorem 6. 

Let $(X,\mu ,\oplus )$ be a revised fuzzy metric space with at least two points. Then there exists a revised fuzzy metric ν on X, using the same t-conorm ⊕, such that

$${\tau }_{\nu }={\tau }_{\mu }$$

and $(X,\nu ,\oplus )$ is not round.

Proof. 

Choose distinct points $x,y\in X$ and $t_0>0$. By (RFM1),

$$\mu (x,y,t_0)>0.$$

Fix

$$c\in \left(0,\mu (x,y,t_0)\right)$$

and define

$$\nu (u,v,t)=min\{c,\mu (u,v,t\left)\right\}.$$

The separation, symmetry and continuity axioms are immediate. To verify the triangle inequality, write $a=\mu (u,v,t)$ and $b=\mu (v,w,s)$. If $a<c$ and $b<c$, then

$$min\{c,a\oplus b\}\le a\oplus b=min\{c,a\}\oplus min\{c,b\}.$$

If $a\ge c$ or $b\ge c$, monotonicity of ⊕ and its neutral element 0 give

$$min\{c,a\oplus b\}\le c\le min\{c,a\}\oplus min\{c,b\}.$$

Combining this inequality with the triangle inequality for $\mu$ shows that $\nu$ is a revised fuzzy metric.

For every $r\in (0,c]$,

$$B_{\nu }(u,r,t)=B_{\mu }(u,r,t).$$

Balls with radii at most c form a neighbourhood base for both induced topologies: inside a $\mu$-ball of radius r, one may take a ball of radius $min\{r,c\}/2$. Hence ${\tau }_{\nu }={\tau }_{\mu }$.

Finally,

$$B_{\nu }[x,c,t_0]=X,$$

whereas

$$B_{\nu }(x,c,t_0)=B_{\mu }(x,c,t_0).$$

By Theorem 1, applied to $\mu$,

$${\overline{B_{\mu }(x,c,t_0)}}^{{\tau }_{\mu }}\subseteq B_{\mu }[x,c,t_0].$$

Since $\mu (x,y,t_0)>c$, the point y does not belong to the set on the right. Using ${\tau }_{\nu }={\tau }_{\mu }$, we conclude that

$${\overline{B_{\nu }(x,c,t_0)}}^{{\tau }_{\nu }}\ne B_{\nu }[x,c,t_0].$$

Thus $(X,\nu ,\oplus )$ is not round. □

Corollary 3. 

Let X have at least two points. Every topology on X induced by a round revised fuzzy metric is also induced by a nonround revised fuzzy metric. Consequently, roundness is not a topological invariant of revised fuzzy metrics.

Proof. 

Apply Theorem 6 to the given round revised fuzzy metric. □

Remark 6. 

Roundness does not imply compactness, connectedness or geodesicity. The real line is round but not compact. The open disconnected subspace

$$(-2,-1)\cup (1,2)$$

of $\mathbb{R}$ is round by Proposition 11, although it is disconnected and is not geodesic as a whole. Thus geodesicity is a useful sufficient condition for roundness, but it is not necessary.

Nathanson also showed that the unit circle with its usual Euclidean subspace metric is round but admits an equivalent metric that is not round [7]. Consequently, roundness is a property of the selected metric or revised fuzzy metric and is not determined by the underlying topology alone.

5. Nested Balls and Completeness

We now connect completeness with decreasing families of closed sets and nested revised fuzzy balls. The notions of $\mu$-Cauchy sequence, completeness and vanishing revised fuzzy diameter are those introduced in Definitions 5, 6 and 7, respectively.

After reversing the interpretation of the fuzzy value, our symmetric Cauchy condition is analogous to the forward Cauchy condition in Definition 3.2 of [10]. It should be distinguished from the forward G-Cauchy condition introduced in Definition 3.7 of that paper.

The following result is the separation-based dual of the Cantor intersection characterization for George–Veeramani fuzzy metrics [2]; compare also the strong-completeness and p-completeness variants in [12,13]. We include the proof because the revised diameter convention and the subsequent nested-ball application require the statement in this precise form.

Theorem 7. 

A revised fuzzy metric space $(X,\mu ,\oplus )$ is complete if and only if every decreasing sequence

$$F_1\supseteq F_2\supseteq F_3\supseteq \dots$$

of nonempty ${\tau }_{\mu }$-closed subsets of X satisfying

$$\underset{n\to \infty }{lim}{diam}_{\mu }(F_n,t)=0$$

for every $t>0$ has nonempty intersection.

Whenever these conditions hold,

$${\displaystyle \bigcap _{n=1}^{\infty }}{F}_n$$

consists of exactly one point.

Proof. 

Suppose first that $(X,\mu ,\oplus )$ is complete. For every $n\in \mathbb{N}$, choose

$$x_n\in F_n.$$

By Proposition 6, the sequence $\left(x_n\right)$ is $\mu$-Cauchy. Completeness therefore gives a point $x\in X$ such that

$$x_n⟶x$$

in ${\tau }_{\mu }$.

Fix $k\in \mathbb{N}$. Since the sequence $\left(F_n\right)$ is decreasing,

$$x_n\in F_n\subseteq F_k$$

whenever $n\ge k$. Thus the tail ${\left(x_n\right)}_{n\ge k}$ lies in the closed set $F_k$. Consequently,

$$x\in F_k.$$

Since k was arbitrary,

$$x\in {\displaystyle \bigcap _{k=1}^{\infty }}{F}_k.$$

Hence the intersection is nonempty.

To prove uniqueness, suppose that

$$x,y\in {\displaystyle \bigcap _{n=1}^{\infty }}{F}_n.$$

Then, for every $n\in \mathbb{N}$ and every $t>0$,

$$\mu (x,y,t)\le {diam}_{\mu }(F_n,t).$$

Letting $n\to \infty$, we obtain

$$\mu (x,y,t)=0.$$

Axiom (RFM1) now gives $x=y$.

Conversely, suppose that every decreasing sequence of nonempty closed sets with vanishing revised fuzzy diameter has nonempty intersection. Let $\left(x_n\right)$ be a $\mu$-Cauchy sequence. For each $n\in \mathbb{N}$, define

$$A_n=\{x_m:m\ge n\}$$

and

$$F_n={\overline{A_n}}^{{\tau }_{\mu }}.$$

Each $F_n$ is nonempty and closed, and

$$F_{n+1}\subseteq F_n$$

for every n.

We show that $\left(F_n\right)$ has vanishing revised fuzzy diameter. Fix $t>0$ and $r\in (0,1)$. Choose $\rho \in (0,r)$. Since ⊕ is continuous and

$$0\oplus 0\oplus 0=0,$$

there exists $q\in (0,1)$ such that

$$q\oplus q\oplus q<\rho .$$

Indeed, the iterated map

$$\Phi (a,b,c)=(a\oplus b)\oplus c$$

is continuous at $(0,0,0)$ and satisfies $\Phi (0,0,0)=0$. Hence ${\Phi }^{-1}\left([0,\rho )\right)$ is a neighbourhood of $(0,0,0)$, so it contains a cube ${[0,q)}^3$ for some $q>0$.

Because $\left(x_n\right)$ is $\mu$-Cauchy, there exists $n_0\in \mathbb{N}$ such that

$$\mu (x_m,x_{\ell },t/3)<q$$

whenever $m,\ell \ge n_0$.

Let $n\ge n_0$ and let $u,v\in F_n$. By Proposition 2, there exist sequences $\left(x_{m_j}\right)$ and $\left(x_{{\ell }_j}\right)$ in $A_n$ such that

$$x_{m_j}⟶u,x_{{\ell }_j}⟶v$$

in ${\tau }_{\mu }$. Since $m_j,{\ell }_j\ge n\ge n_0$,

$$\mu (x_{m_j},x_{{\ell }_j},t/3)<q$$

for every j.

By Proposition 3,

$$\mu (u,x_{m_j},t/3)⟶0$$

and

$$\mu (x_{{\ell }_j},v,t/3)⟶0.$$

Therefore, for all sufficiently large j,

$$\mu (u,x_{m_j},t/3)<q$$

and

$$\mu (x_{{\ell }_j},v,t/3)<q.$$

The three occurrences of $t/3$ arise because the comparison from u to v uses the three successive links

$$u⟶x_{m_j}⟶x_{{\ell }_j}⟶v.$$

Their time parameters add to t, allowing two applications of (RFM3).

Applying the revised fuzzy triangle inequality twice and using the monotonicity of ⊕, we obtain

$$\begin{array}{cc}\hfill \mu (u,v,t)& \le \mu (u,x_{m_j},t/3)\oplus \mu (x_{m_j},x_{{\ell }_j},t/3)\hfill \\ & \oplus \mu (x_{{\ell }_j},v,t/3)\hfill \\ & \le q\oplus q\oplus q<\rho .\hfill \end{array}$$

Thus

$$\mu (u,v,t)<\rho$$

for all $u,v\in F_n$, and hence

$${diam}_{\mu }(F_n,t)\le \rho <r$$

whenever $n\ge n_0$. Since $r\in (0,1)$ was arbitrary,

$$\underset{n\to \infty }{lim}{diam}_{\mu }(F_n,t)=0.$$

As $t>0$ was arbitrary, the sequence $\left(F_n\right)$ has vanishing revised fuzzy diameter.

By the assumed intersection property, there exists

$$x\in {\displaystyle \bigcap _{n=1}^{\infty }}{F}_n.$$

We show that $x_n\to x$. Fix $r\in (0,1)$ and $t>0$. Choose $n_1\in \mathbb{N}$ such that

$${diam}_{\mu }(F_{n_1},t)<r.$$

For every $n\ge n_1$,

$$x_n\in A_{n_1}\subseteq F_{n_1},$$

while $x\in F_{n_1}$. Therefore

$$\mu (x,x_n,t)\le {diam}_{\mu }(F_{n_1},t)<r.$$

By Proposition 3, $x_n\to x$ in ${\tau }_{\mu }$. Hence $(X,\mu ,\oplus )$ is complete. □

We now specialize the Cantor characterization to decreasing sequences of closed revised fuzzy balls.

Definition 10. 

A sequence of closed revised fuzzy balls

$$C_n=B_{\mu }[x_n,r_n,t_n],n\in \mathbb{N},$$

is called a vanishing nested-ball sequence if

$$C_{n+1}\subseteq C_n$$

for every $n\in \mathbb{N}$ and

$$\underset{n\to \infty }{lim}{diam}_{\mu }(C_n,t)=0$$

for every $t>0$.

Corollary 4. 

Let $(X,\mu ,\oplus )$ be a complete revised fuzzy metric space. Every vanishing nested-ball sequence $\left(C_n\right)$ satisfies

$${\displaystyle \bigcap _{n=1}^{\infty }}{C}_n=\left\{p\right\}$$

for a unique point $p\in X$.

Proof. 

By Proposition 7, every $C_n$ is a closed subset of X. Moreover, every $C_n$ is nonempty because it contains its centre $x_n$. The conclusion therefore follows directly from Theorem 7. □

The preceding principle only concerns nested sequences whose revised fuzzy diameters vanish. A stronger intersection property can be imposed without any diameter restriction.

Definition 11. 

A revised fuzzy metric space $(X,\mu ,\oplus )$ is called sequentially spherically complete if every decreasing sequence

$$C_1\supseteq C_2\supseteq C_3\supseteq \dots$$

of nonempty closed revised fuzzy balls has nonempty intersection.

The adjective “sequentially” emphasizes that Definition 11 concerns decreasing sequences rather than arbitrary totally ordered families of balls.

Proposition 12. 

Let $(X,\mu ,\oplus )$ be sequentially spherically complete. Then every vanishing nested-ball sequence $\left(C_n\right)$ has an intersection consisting of exactly one point.

Proof. 

Sequential spherical completeness gives

$${\displaystyle \bigcap _{n=1}^{\infty }}{C}_n\ne ⌀.$$

Suppose that

$$x,y\in {\displaystyle \bigcap _{n=1}^{\infty }}{C}_n.$$

For every $n\in \mathbb{N}$ and $t>0$,

$$\mu (x,y,t)\le {diam}_{\mu }(C_n,t).$$

Letting $n\to \infty$, we obtain

$$\mu (x,y,t)=0.$$

Thus $x=y$ by (RFM1). □

Remark 7. 

Theorem 7 concerns symmetric sequential completeness. In a fuzzy quasi-metric space, the forward and backward Cauchy conditions need not coincide, and convergence may be considered relative to the topology induced by the original or conjugate fuzzy quasi-metric. Definition 3.2 of [10] uses forward Cauchy sequences and convergence with respect to the conjugate topology in its associated completeness condition. Definition 3.7 of [10] introduces the weaker forward G-Cauchy condition and the corresponding notion of G-completeness. The symmetric revised fuzzy setting considered here avoids this directional ambiguity.

Remark 8. 

Roundness is not required for the abstract closed-set formulation in Theorem 7. Nor is it needed in Corollary 4 once the closed balls are given directly. Its role is to provide the identification

$$B_{\mu }[x,r,t]={\overline{B_{\mu }(x,r,t)}}^{{\tau }_{\mu }}.$$

This identification becomes essential when hypotheses are formulated in terms of open revised fuzzy balls and conclusions must be transferred to their closed analogues. It is precisely this role that roundness plays in the fixed-point principles of Section 6.

6. Fixed Points from Nested Round Balls

We now apply the nested-ball intersection principle to self-maps. The main idea is that roundness transfers inclusions involving open balls to their closures, while completeness and vanishing revised fuzzy diameter produce a unique intersection point.

Theorem 8. 

Let $(X,\mu ,\oplus )$ be a complete round revised fuzzy metric space, and let

$$f:X⟶X$$

be continuous. Suppose that there exists a sequence of open revised fuzzy balls

$$U_n=B_{\mu }(x_n,r_n,t_n),n\in \mathbb{N},$$

with corresponding closed revised fuzzy balls

$$C_n=B_{\mu }[x_n,r_n,t_n],n\in \mathbb{N},$$

such that:

1. 

$$U_{n+1}\subseteq U_n$$

for every $n\in \mathbb{N}$;

2. 

$$f\left(U_n\right)\subseteq U_{n+1}$$

for every $n\in \mathbb{N}$;

3. 

$$\underset{n\to \infty }{lim}{diam}_{\mu }(C_n,t)=0$$

for every $t>0$.

Then

$${\displaystyle \bigcap _{n=1}^{\infty }}{C}_n=\left\{p\right\}$$

for some $p\in X$, and

$$f\left(p\right)=p.$$

Moreover, p is the unique fixed point of f contained in $C_1$.

Proof. 

Since $(X,\mu ,\oplus )$ is round,

$$C_n={\overline{U_n}}^{{\tau }_{\mu }}$$

for every n. The inclusion $U_{n+1}\subseteq U_n$ therefore gives

$$\begin{array}{cc}\hfill C_{n+1}& ={\overline{U_{n+1}}}^{{\tau }_{\mu }}\hfill \\ & \subseteq {\overline{U_n}}^{{\tau }_{\mu }}\hfill \\ & =C_n.\hfill \end{array}$$

Thus $\left(C_n\right)$ is a decreasing sequence of nonempty closed revised fuzzy balls.

Continuity of f gives

$$f\left({\overline{A}}^{{\tau }_{\mu }}\right)\subseteq {\overline{f\left(A\right)}}^{{\tau }_{\mu }}$$

for every $A\subseteq X$. To justify this inclusion directly, let $x\in {\overline{A}}^{{\tau }_{\mu }}$ and let V be a neighbourhood of $f\left(x\right)$. By continuity, $f^{-1}\left(V\right)$ is a neighbourhood of x, and therefore it meets A. Hence V meets $f\left(A\right)$, proving that $f\left(x\right)\in {\overline{f\left(A\right)}}^{{\tau }_{\mu }}$. Applying this observation to $A=U_n$, we obtain

$$\begin{array}{cc}\hfill f\left(C_n\right)& =f\left({\overline{U_n}}^{{\tau }_{\mu }}\right)\hfill \\ & \subseteq {\overline{f\left(U_n\right)}}^{{\tau }_{\mu }}\hfill \\ & \subseteq {\overline{U_{n+1}}}^{{\tau }_{\mu }}\hfill \\ & =C_{n+1}.\hfill \end{array}$$

By the vanishing-diameter hypothesis, $\left(C_n\right)$ is a vanishing nested-ball sequence. Corollary 4 therefore gives a unique point $p\in X$ such that

$${\displaystyle \bigcap _{n=1}^{\infty }}{C}_n=\left\{p\right\}.$$

For every $n\in \mathbb{N}$, one has $p\in C_n$, and hence

$$f\left(p\right)\in f\left(C_n\right)\subseteq C_{n+1}.$$

It follows that

$$f\left(p\right)\in {\displaystyle \bigcap _{n=1}^{\infty }}{C}_n=\left\{p\right\},$$

so $f\left(p\right)=p$.

Finally, let $q\in C_1$ be any fixed point of f. Since

$$q=f\left(q\right)\in f\left(C_1\right)\subseteq C_2,$$

we have $q\in C_2$. Inductively, if $q\in C_n$, then

$$q=f\left(q\right)\in f\left(C_n\right)\subseteq C_{n+1}.$$

Thus

$$q\in {\displaystyle \bigcap _{n=1}^{\infty }}{C}_n=\left\{p\right\},$$

and therefore $q=p$. Hence p is the unique fixed point of f contained in $C_1$. □

The following formulation starts directly with closed revised fuzzy balls. It does not require roundness or continuity because the required closed-ball invariance is assumed explicitly.

Corollary 5. 

Let $(X,\mu ,\oplus )$ be a complete revised fuzzy metric space, let

$$f:X⟶X$$

be any self-map, and let $\left(C_n\right)$ be a vanishing nested-ball sequence such that

$$f\left(C_n\right)\subseteq C_{n+1}$$

for every $n\in \mathbb{N}$. Then

$${\displaystyle \bigcap _{n=1}^{\infty }}{C}_n=\left\{p\right\}$$

for some $p\in X$, and

$$f\left(p\right)=p.$$

Moreover, p is the unique fixed point of f contained in $C_1$.

Proof. 

By Corollary 4, there exists a unique point $p\in X$ such that

$${\displaystyle \bigcap _{n=1}^{\infty }}{C}_n=\left\{p\right\}.$$

Since $p\in C_n$ for every n,

$$f\left(p\right)\in f\left(C_n\right)\subseteq C_{n+1}$$

for every n. Therefore

$$f\left(p\right)\in {\displaystyle \bigcap _{n=1}^{\infty }}{C}_n=\left\{p\right\},$$

and hence $f\left(p\right)=p$.

Let $q\in C_1$ be another fixed point. Then

$$q=f\left(q\right)\in f\left(C_1\right)\subseteq C_2.$$

Repeating the argument inductively gives $q\in C_n$ for every n. Consequently,

$$q\in {\displaystyle \bigcap _{n=1}^{\infty }}{C}_n=\left\{p\right\},$$

so $q=p$. □

The closed-ball formulation yields an orbital convergence principle.

Corollary 6. 

Let $(X,\mu ,\oplus )$ be a complete revised fuzzy metric space, let

$$f:X⟶X$$

be a self-map, and let $x_0\in X$. Suppose that there exists a vanishing nested-ball sequence $\left(C_n\right)$ such that

$$f^n\left(x_0\right)\in C_n$$

and

$$f\left(C_n\right)\subseteq C_{n+1}$$

for every $n\in \mathbb{N}$. Then there exists a unique point

$$p\in {\displaystyle \bigcap _{n=1}^{\infty }}{C}_n,$$

the orbit $\left(f^n\left(x_0\right)\right)$ converges to p, and

$$f\left(p\right)=p.$$

Furthermore, p is the unique fixed point of f contained in $C_1$.

Proof. 

The existence and uniqueness of the intersection point, together with the identity $f\left(p\right)=p$, follow from Corollary 5.

It remains to prove convergence of the orbit. Fix $r\in (0,1)$ and $t>0$. Since the revised fuzzy diameters of the $C_n$’s vanish, there exists $n_0\in \mathbb{N}$ such that

$${diam}_{\mu }(C_{n_0},t)<r.$$

For every $n\ge n_0$, nestedness gives

$$f^n\left(x_0\right)\in C_n\subseteq C_{n_0}.$$

Also,

$$p\in C_{n_0}.$$

Therefore

$$\mu (f^n\left(x_0\right),p,t)\le {diam}_{\mu }(C_{n_0},t)<r$$

whenever $n\ge n_0$. By Proposition 3,

$$f^n\left(x_0\right)⟶p$$

in ${\tau }_{\mu }$. □

The full continuity of the self-map in Theorem 8 can be weakened to a condition imposed only on the designated family of open balls.

Proposition 13. 

In Theorem 8, continuity of f may be replaced by the requirement

$$f\left({\overline{U_n}}^{{\tau }_{\mu }}\right)\subseteq {\overline{f\left(U_n\right)}}^{{\tau }_{\mu }}$$

for every $n\in \mathbb{N}$.

Proof. 

Under this assumption, roundness and the open-ball invariance condition give

$$\begin{array}{cc}\hfill f\left(C_n\right)& =f\left({\overline{U_n}}^{{\tau }_{\mu }}\right)\hfill \\ & \subseteq {\overline{f\left(U_n\right)}}^{{\tau }_{\mu }}\hfill \\ & \subseteq {\overline{U_{n+1}}}^{{\tau }_{\mu }}\hfill \\ & =C_{n+1}.\hfill \end{array}$$

The remainder of the proof is identical to that of Theorem 8. □

Remark 9. 

Theorem 8 differs structurally from the contraction principles previously obtained in fuzzy quasi-metric spaces. Theorem 3.3 of [10] assumes an orbital inequality and derives convergence under forward completeness, whereas Theorem 3.9 of that paper uses forward G-completeness. Theorem 4 of [11] combines a preordered completeness condition with a Matkowski-type contractive implication, and Theorem 5 specializes the argument to the specialization order.

Other recent directions remain algebraic or transfer-based. Huang et al. [16] formulate a fuzzy F-contraction, Moussaoui et al. [17] use extended $FZ$-simulation functions in an M-complete fuzzy metric space, and Saheli et al. [18] employ one- and five-dimensional comparison functions to define nonlinear φ-contractions. Miñana, Šostak and Valero [19] instead transfer Matkowski- and Meir–Keeler-type results through an associated complete classical metric, while Gregori et al. [20] study how stronger p-completeness permits a weakening of structural assumptions on the fuzzy metric.

In the revised fuzzy setting itself, Muraliraj and Thangathamizh [14] proved fixed-point results under Banach– and Edelstein-type hypotheses. Their later relation-theoretic work [15] uses $\mathcal{R}$-ψ-contractive mappings in non-Archimedean revised fuzzy metric spaces. These results likewise impose pointwise algebraic conditions on the self-map and the revised fuzzy metric.

Theorem 8 does not claim to generalize those results. It employs a different approach: a decreasing family of balls in a round revised fuzzy metric space, with each ball mapped into its successor and with revised fuzzy diameters tending to zero. The contractive behaviour is expressed geometrically through the nested-ball structure rather than through a global algebraic inequality.

7. Examples and Applications

We illustrate the nested-ball principles with nonlinear, globally noncontractive and metric-induced examples.

Example 4. 

Let $X=\mathbb{R}$ with the Euclidean metric and the associated standard revised fuzzy metric

$$\mu (x,y,t)={\displaystyle \frac{|x-y|}{t+|x-y|}},x,y\in \mathbb{R},t>0,$$

taken with the maximum t-conorm ${\oplus }_{max}$. Since $(\mathbb{R},|·\left|\right)$ is complete and geodesic, Proposition 4 and Corollary 2 show that this revised fuzzy metric space is complete and round.

Define

$$f\left(x\right)={\displaystyle \frac{x}{1+\left|x\right|}}.$$

This map is nonlinear and is not a Banach contraction on $\mathbb{R}$. Indeed, if a constant $q<1$ satisfied

$$\left|f\right(x)-f(y\left)\right|\le q|x-y|$$

for all $x,y\in \mathbb{R}$, then taking $y=0$ and $x\ne 0$ would give

$${\displaystyle \frac{\left|f\right(x)-f(0\left)\right|}{\left|x\right|}}={\displaystyle \frac{1}{1+\left|x\right|}}\le q.$$

Letting $x\to 0$ yields $1\le q$, a contradiction.

Fix $R_0>0$ and define recursively

$$\begin{array}{cc}\hfill R_{n+1}& ={\displaystyle \frac{R_n}{1+R_n}},\hfill \\ \hfill R_n& ={\displaystyle \frac{R_0}{1+n{R}_0}}\downarrow 0.\hfill \end{array}$$

Set

$$\begin{array}{cc}\hfill r_n& ={\displaystyle \frac{R_n}{1+R_n}},\hfill \\ \hfill U_n& =B_{\mu }(0,r_n,1),\hfill \\ \hfill C_n& =B_{\mu }[0,r_n,1].\hfill \end{array}$$

By Example 1,

$$U_n=(-R_n,R_n),C_n=[-R_n,R_n].$$

Since the function $s\mapsto s/(1+s)$ is increasing on $[0,\infty )$,

$$f\left(U_n\right)=U_{n+1},f\left(C_n\right)=C_{n+1}.$$

For every $t>0$, the largest Euclidean distance between two points of $C_n$ is $2R_n$. Consequently,

$${diam}_{\mu }(C_n,t)={\displaystyle \frac{2R_n}{t+2R_n}}⟶0.$$

After shifting the indices, the hypotheses of Theorem 8 are therefore satisfied, and

$${\displaystyle \bigcap _{n=0}^{\infty }}{C}_n=\left\{0\right\}.$$

Thus 0 is the unique fixed point of f contained in $C_0$. In this example, it is also the unique fixed point of f on $\mathbb{R}$, since

$${\displaystyle \frac{x}{1+\left|x\right|}}=x$$

implies $x=0$. Hence the nested-ball principle applies even though no global Banach contraction constant exists.

Example 5. 

Let $X=[0,1]$ with the Euclidean metric and its associated standard revised fuzzy metric

$$\mu (x,y,t)={\displaystyle \frac{|x-y|}{t+|x-y|}},$$

again taken with ${\oplus }_{max}$. Since $[0,1]$ is a complete geodesic metric space, Proposition 4 and Corollary 2 show that the corresponding revised fuzzy metric space is complete and round.

Define

$$f\left(x\right)=x^2.$$

Fix $a\in (1/2,1)$, and set

$$a_n=a^{2^n},n\ge 0.$$

Then

$$a_{n+1}=a_n^2\mathrm{and}{a}_n⟶0.$$

The restriction of f to the initial invariant interval $[0,a]$ is not a Banach contraction. Indeed, for $0<\epsilon <2a-1$,

$${\displaystyle \frac{\left|f\right(a)-f(a-\epsilon \left)\right|}{|a-(a-\epsilon \left)\right|}}=2a-\epsilon >1.$$

Put

$$r_n={\displaystyle \frac{a_n}{1+a_n}}$$

and define

$$U_n=B_{\mu }(0,r_n,1)\mathit{and}{C}_n=B_{\mu }[0,r_n,1].$$

Because the balls are taken in the subspace $X=[0,1]$,

$$U_n=[0,a_n)\mathit{and}{C}_n=[0,a_n].$$

Moreover,

$$U_{n+1}\subseteq U_n,C_{n+1}\subseteq C_n,$$

and

$$f\left(U_n\right)=U_{n+1},f\left(C_n\right)=C_{n+1}.$$

For every $t>0$, the largest Euclidean distance between two points of $C_n$ is $a_n$. Hence

$${diam}_{\mu }(C_n,t)={\displaystyle \frac{a_n}{t+a_n}}⟶0.$$

After shifting the indices, Theorem 8 gives

$${\displaystyle \bigcap _{n=0}^{\infty }}{C}_n=\left\{0\right\},$$

and 0 is a fixed point of f.

Notice, however, that $f\left(1\right)=1$. Thus 0 is not the unique fixed point of f on the whole space X; it is the unique fixed point contained in the initial ball $C_0=[0,a]$, and hence the unique fixed point detected by this nested family. This illustrates the local nature of the nested-ball method.

The following criterion is useful when the centres of the balls are chosen along an orbit.

Proposition 14. 

Let $(X,\mu ,\oplus )$ be a complete revised fuzzy metric space, let $f:X\to X$, and let $x_0\in X$. Suppose that there exist $r_n\in (0,1)$ and $t_n>0$ such that the closed revised fuzzy balls

$$C_n=B_{\mu }[f^n\left(x_0\right),r_n,t_n],n\ge 0,$$

satisfy

$$C_{n+1}\subseteq C_n\mathit{and}f\left(C_n\right)\subseteq C_{n+1}$$

for every $n\ge 0$. Suppose also that

$$\underset{n\to \infty }{lim}{diam}_{\mu }(C_n,t)=0$$

for every $t>0$. Then there exists a unique point

$$p\in {\displaystyle \bigcap _{n=0}^{\infty }}{C}_n.$$

Furthermore,

$$f\left(p\right)=p\mathrm{and}{f}^n\left(x_0\right)⟶p.$$

The point p is the only fixed point of f contained in $C_0$.

Proof. 

The sequence ${\left(C_n\right)}_{n\ge 0}$ is decreasing, consists of nonempty closed revised fuzzy balls and has vanishing revised fuzzy diameter. Applying Corollary 4 after shifting the indices gives a unique point

$$p\in {\displaystyle \bigcap _{n=0}^{\infty }}{C}_n.$$

Since

$$f\left(C_n\right)\subseteq C_{n+1}$$

for every $n\ge 0$, we have

$$f\left(p\right)\in C_{n+1}$$

for every $n\ge 0$. Therefore

$$f\left(p\right)\in {\displaystyle \bigcap _{n=0}^{\infty }}{C}_n=\left\{p\right\},$$

and hence $f\left(p\right)=p$.

Because $f^n\left(x_0\right)$ is the centre of $C_n$,

$$f^n\left(x_0\right)\in C_n$$

for every $n\ge 0$. Given $r\in (0,1)$ and $t>0$, choose $n_0\ge 0$ such that

$${diam}_{\mu }(C_{n_0},t)<r.$$

If $n\ge n_0$, then

$$f^n\left(x_0\right),p\in C_n\subseteq C_{n_0}.$$

Consequently,

$$\mu (f^n\left(x_0\right),p,t)\le {diam}_{\mu }(C_{n_0},t)<r.$$

By Proposition 3,

$$f^n\left(x_0\right)⟶p$$

in ${\tau }_{\mu }$.

Finally, if $q\in C_0$ and $f\left(q\right)=q$, then

$$q=f\left(q\right)\in f\left(C_0\right)\subseteq C_1.$$

Inductively, if $q\in C_n$, then

$$q=f\left(q\right)\in f\left(C_n\right)\subseteq C_{n+1}.$$

Thus $q\in C_n$ for every $n\ge 0$, and hence

$$q\in {\displaystyle \bigcap _{n=0}^{\infty }}{C}_n=\left\{p\right\}.$$

Therefore $q=p$. □

Corollary 7. 

Let $(X,d)$ be a complete metric space, and let

$${\mu }_d(x,y,t)={\displaystyle \frac{d(x,y)}{t+d(x,y)}}$$

be its standard revised fuzzy metric, taken with ${\oplus }_{max}$. Suppose that $f:X\to X$ and that there exists a decreasing sequence of nonempty closed metric balls

$$D_n=B_d[x_n,R_n],R_n>0,$$

such that

$$f\left(D_n\right)\subseteq D_{n+1}\mathrm{and}{R}_n⟶0.$$

Then there exists a unique point

$$p\in {\displaystyle \bigcap _{n=1}^{\infty }}{D}_n,$$

and $f\left(p\right)=p$. Moreover, p is the only fixed point of f contained in $D_1$.

Proof. 

Fix any sequence $\left(t_n\right)$ of positive real numbers and define

$$r_n={\displaystyle \frac{R_n}{t_n+R_n}}.$$

Then $r_n\in (0,1)$, and Example 1 gives

$$D_n=B_{{\mu }_d}[x_n,r_n,t_n].$$

For every fixed $t>0$ and every $u,v\in D_n$, the metric triangle inequality yields

$$d(u,v)\le d(u,x_n)+d(x_n,v)\le 2R_n.$$

Since the function

$$s⟼{\displaystyle \frac{s}{t+s}}$$

is increasing on $[0,\infty )$, it follows that

$${\mu }_d(u,v,t)\le {\displaystyle \frac{2R_n}{t+2R_n}}.$$

Therefore

$${diam}_{{\mu }_d}(D_n,t)\le {\displaystyle \frac{2R_n}{t+2R_n}}⟶0.$$

By Proposition 4, $(X,{\mu }_d,{\oplus }_{max})$ is complete because $(X,d)$ is complete. Consequently, Corollary 5 gives a unique point

$$p\in {\displaystyle \bigcap _{n=1}^{\infty }}{D}_n$$

satisfying $f\left(p\right)=p$.

If $q\in D_1$ is another fixed point, then

$$q=f\left(q\right)\in f\left(D_1\right)\subseteq D_2.$$

Inductively, the inclusions $f\left(D_n\right)\subseteq D_{n+1}$ imply that $q\in D_n$ for every n. Hence

$$q\in {\displaystyle \bigcap _{n=1}^{\infty }}{D}_n=\left\{p\right\},$$

so $q=p$. □

Remark 10. 

Corollary 7 is a revised fuzzy formulation of the classical nested-ball argument for complete metric spaces. Its purpose is not to provide a new version of the metric Cantor intersection theorem, but to demonstrate that the revised fuzzy setting recovers the expected metric conclusion.

Roundness is not required in this corollary because the hypotheses are stated directly in terms of closed balls. If the assumptions are instead formulated using open balls and their closures, roundness supplies the identification

$${\overline{B_{{\mu }_d}(x,r,t)}}^{{\tau }_{{\mu }_d}}=B_{{\mu }_d}[x,r,t].$$

This distinction reflects the role played by round metrics in the sense of Nathanson [7].

8. Conclusions

We formulated Efe’s George–Veeramani fuzzy roundness [8] in the equivalent revised t-conorm language through the ball-closure identity

$${\overline{B_{\mu }(x,r,t)}}^{{\tau }_{\mu }}=B_{\mu }[x,r,t].$$

A two-point example showed that this identity is not automatic. In particular, the ball-closure equalities asserted in Corollary 2 of [5] and Corollary 3 of [6] can fail. Under the defining axioms of a revised fuzzy metric, the generally valid relation is

$${\overline{B_{\mu }(x,r,t)}}^{{\tau }_{\mu }}\subseteq B_{\mu }[x,r,t].$$

Roundness is therefore the additional property that ensures the reverse inclusion. The contribution here is not the underlying definition itself, but its explicit revised formulation, its use in correcting the later ball-closure assertions, and the new structural and fixed-point results that follow.

We characterized nonroundness by the existence of a radial map that attains a positive minimum on an open set not containing its centre. We also established invariance under revised fuzzy isometries, examined how roundness behaves on subspaces, and proved factorwise characterizations for finite maximum products and countable weighted maximum products. For the countable construction, completeness is likewise equivalent to completeness of every factor. The truncation theorem shows that every nontrivial revised fuzzy metric has a topologically equivalent nonround truncation. Thus roundness is not determined by the induced topology. For the standard construction

$${\mu }_d(x,y,t)={\displaystyle \frac{d(x,y)}{t+d(x,y)}},$$

taken with the maximum t-conorm, we showed that revised fuzzy roundness is equivalent to classical metric roundness. We also established, in the separation-based revised convention, the Cantor-type characterization of completeness through decreasing sequences of nonempty closed sets with vanishing revised fuzzy diameter. This is the dual form of the corresponding George–Veeramani characterization and provides the intersection tool needed for the fixed-point argument.

The main fixed-point consequence combines completeness with roundness. Instead of imposing a global algebraic contraction inequality, it requires a decreasing sequence of open balls such that each ball is mapped into its successor and the corresponding closed balls have vanishing revised fuzzy diameter. Roundness transfers the open-ball conditions to the closed balls, while completeness produces a unique intersection point fixed by the self-map. The nonlinear example shows that the principle applies even when the map is not a Banach contraction on the relevant invariant domain. The result therefore complements both fuzzy quasi-metric contraction principles [10,11] and the existing Banach–, Edelstein– and relation-theoretic contraction results formulated directly in revised fuzzy metric spaces [14,15].

Further work may investigate preservation of roundness under completion, other infinite-product constructions and inverse limits, as well as extensions to multivalued, ordered and asymmetric revised fuzzy metric structures.