Answering an open question in fuzzy metric spaces

TL;DR

This paper proves that in stationary fuzzy metric spaces, the metric function and its slices are uniformly continuous, resolving an open problem and enhancing understanding of fuzzy metric space properties.

Contribution

It provides a positive answer to an open question by establishing the uniform continuity of the fuzzy metric function and its slices in stationary fuzzy metric spaces.

Findings

The function $M_y(x)$ is $ ext{R}$-uniformly continuous for all $y$ in $X$.

The fuzzy metric function $M$ itself is $ ext{R}$-uniformly continuous.

The paper resolves Problem 32 from GMM2012 affirmatively.

Abstract

This paper answers affirmatively Problem 32 posted in \cite{GMM2012}, proving that, for every stationary fuzzy metric space $(X, M, *)$, the function $M_y(x):=M(x,y)$ defined therein is $\mathbb{R}$-uniformly continuous for all $y\in X$, and furthermore proves that the function $M$ is $\mathbb{R}$-uniformly continuous.

Equations

$$M_d(x,y,t)=\frac{t}{t+d(x,y)},\text{ for }(x,y,t)\in X\times X\times (0,+\infty ).$$

$$M(y,x_{n_k})\ast M(x_{n_k},z_{n_k})-M(y,x_{n_k})\le M(y,z_{n_k})-M(y,x_{n_k})=M_y(z_{n_k})-M_y(x_{n_k})\le -{\epsilon }_0.$$

$$0=\xi \ast 1-\xi =\underset{k\to +\infty }{\lim }(M(y,x_{n_k})\ast M(x_{n_k},z_{n_k})-M(y,x_{n_k}))\le -{\epsilon }_0,$$

$${\epsilon }_0\le M_y(z_{n_k})-M_y(x_{n_k})=M(y,z_{n_k})-M(y,x_{n_k})\le M(y,z_{n_k})-M(y,z_{n_k})\ast M(z_{n_k},x_{n_k}).$$

$${\epsilon }_0\le \underset{k\to +\infty }{\lim }(M(y,z_{n_k})-M(y,z_{n_k})\ast M(z_{n_k},x_{n_k}))=\eta -\eta \ast 1=0,$$

$$\hat{M}((x_1,x_2),(y_1,y_2),t)=\min \left\{M(x_1,y_1),M(x_2,y_2)\right\},\text{ for }((x_1,x_2),(y_1,y_2),t)\in X^2\times X^2\times (0,+\infty ).$$

$$M:(X^2,\hat{M},\ast )$$

$$(x,y)$$

$$M(x_{n_k},z_{n_k})\ast M(z_{n_k}^{\prime },z_{n_k})\ast M(x_{n_k}^{\prime },x_{n_k})-M(x_{n_k},z_{n_k})\le M(x_{n_k}^{\prime },z_{n_k}^{\prime })-M(x_{n_k},z_{n_k})\le -{\epsilon }_0.$$

$$0=\xi \ast 1\ast 1-\xi =\underset{k\to +\infty }{\lim }(M(x_{n_k},z_{n_k})\ast M(z_{n_k}^{\prime },z_{n_k})\ast M(x_{n_k}^{\prime },x_{n_k})-M(x_{n_k},z_{n_k}))\le -{\epsilon }_0,$$

$${\epsilon }_0\le M(x_{n_k}^{\prime },z_{n_k}^{\prime })-M(x_{n_k},z_{n_k})\le M(x_{n_k}^{\prime },z_{n_k}^{\prime })-M(x_{n_k}^{\prime },z_{n_k}^{\prime })\ast M(z_{n_k}^{\prime },z_{n_k})\ast M(x_{n_k}^{\prime },x_{n_k}).$$

$${\epsilon }_0\le \underset{k\to +\infty }{\lim }(M(x_{n_k}^{\prime },z_{n_k}^{\prime })-M(x_{n_k}^{\prime },z_{n_k}^{\prime })\ast M(z_{n_k}^{\prime },z_{n_k})\ast M(x_{n_k}^{\prime },x_{n_k}))=\eta -\eta \ast 1\ast 1=0,$$

Full text

This work was supported by the National Natural Science Foundation of China (No. 11601449), the Science and Technology Innovation Team of Education Department of Sichuan for Dynamical System and its Applications (No. 18TD0013), and the Youth Science and Technology Innovation Team of Southwest Petroleum University for Nonlinear Systems (No. 2017CXTD02).

During the last three decades, many research works were devoted to the notion of fuzzy metric spaces in different ways [1, 2, 8]. To obtain a Hausdorff topology of a fuzzy metric space, George and Veeramani [3] modified the concept of fuzzy metric space initiated by Kramosil and Michalek [9] and introduced the fuzzy metric space in a new sense as follows.

• (GV1)

$M(x,y,t)>0$;

• (GV2)

$M(x,y,t)=1$ if and only if $x=y$;

• (GV3)

$M(x,y,t)=M(y,x,t)$;

• (GV4)

$M(x,y,t)\ast M(y,z,s)\leq M(x,z,t+s)$;

• (GV5)

$M(x,y,\_):(0,+\infty)\longrightarrow(0,1]$ is continuous.

$(M,\ast)$, or simply $M$, is usually called a fuzzy metric on $X$, and each value $M(x,y,t)$ can be understood as the nearness degree between $x$ and $y$ with respect to $t$.

According to Grabiec [4], the real function $M(x,y,\_)$ of Axiom (GV5) is nondecreasing for all $x,y\in X$. George and Veeramani [3] proved that every fuzzy metric space $(X,M,\ast)$ generates a topology $\mathscr{T}_{M}$ on $X$, which has as a base the family of open sets of the form $\{B_{M}(x,\varepsilon,t):x\in X,0<\varepsilon<1,t>0\}$, where $B_{M}(x,\varepsilon,t)=\{y\in X:M(x,y,t)>1-\varepsilon\}$.

Let $(X,d)$ be a metric space and $M_{d}$ be a function on $X\times X\times(0,+\infty)$ defined by

[TABLE]

It can be verified that $(X,M_{d},\cdot)$ is a fuzzy metric space, and $M_{d}$ is called the standard fuzzy metric induced by $d$. This shows that every ordinary metric induces a fuzzy metric in the sense of George and Veeramani.

Recently, Gregori et al. [5] studied the fuzzy metric defined by $M(x,y,t)=\frac{\min\{x,y\}+t}{\max\{x,y\}+t}$ and other fuzzy metrics related to it. In particular, they [5] proposed the following question for the real function $M(x,y)$ on stationary fuzzy metric spaces.

The following theorem proves that the function $M_{y}$ is $\mathbb{R}$-uniformly continuous for stationary fuzzy metric spaces, giving a positive answer to Question 4. Denote $\mathbb{N}=\left\{1,2,3,\ldots\right\}$.

Consider the following two cases:

• Case 1.

If $\xi>\eta$, without loss of generality, assume that $M_{y}(x_{n_{k}})>M_{y}(z_{n_{k}})$ for all $k\in\mathbb{N}$. This, together with $M(y,x_{n_{k}})\ast M(x_{n_{k}},z_{n_{k}})\leq M(y,z_{n_{k}})$, implies that

[TABLE]

Then,

[TABLE]

which is a contradiction.

• Case 2.

If $\xi<\eta$, without loss of generality, assume that $M_{y}(x_{n_{k}})<M_{y}(z_{n_{k}})$ for all $k\in\mathbb{N}$. This, together with $M(y,z_{n_{k}})\ast M(z_{n_{k}},x_{n_{k}})\leq M(y,x_{n_{k}})$, implies that

[TABLE]

Then,

[TABLE]

which is a contradiction.

Therefore, $M_{y}$ is $\mathbb{R}$-uniformly continuous. ∎

Next, let $(X,M,\ast)$ be a stationary fuzzy metric space and $\mathscr{T}_{M}$ be the topology generated by the fuzzy metric $M$. Define $\hat{M}:X^{2}\times X^{2}\times(0,+\infty)\longrightarrow[0,1]$ by

[TABLE]

From [10, Proposition 4.1, Theorem 4.4], it follows that

1. (1)

$(\hat{M},\ast)$ is a fuzzy metric on $X^{2}$, i.e., $(X^{2},\hat{M},\ast)$ is a fuzzy metric space; 2. (2)

the topology $\mathscr{T}_{\hat{M}}$ generated by $\hat{M}$ is the product topology for $X^{2}$.

Similarly to Theorem 5, the following theorem proves the $\mathbb{R}$-uniform continuity of $M$ on $(X^{2},\hat{M},\ast)$.

[TABLE]

is $\mathbb{R}$-uniformly continuous.

Consider the following two cases:

• Case 1.

If $\xi>\eta$, without loss of generality, assume that $M(x_{n_{k}},z_{n_{k}})>M(x^{\prime}_{n_{k}},z^{\prime}_{n_{k}})$ for all $k\in\mathbb{N}$. This, together with $M(x_{n_{k}},z_{n_{k}})\ast M(z^{\prime}_{n_{k}},z_{n_{k}})\ast M(x^{\prime}_{n_{k}},x_{n_{k}})\leq M(x^{\prime}_{n_{k}},z^{\prime}_{n_{k}})$, implies that

[TABLE]

Then,

[TABLE]

which is a contradiction.

• Case 2.

If $\xi<\eta$, without loss of generality, assume that $M(x_{n_{k}},z_{n_{k}})<M(x^{\prime}_{n_{k}},z^{\prime}_{n_{k}})$ for all $k\in\mathbb{N}$. This, together with $M(x^{\prime}_{n_{k}},z^{\prime}_{n_{k}})\ast M(z^{\prime}_{n_{k}},z_{n_{k}})\ast M(x^{\prime}_{n_{k}},x_{n_{k}})\leq M(x_{n_{k}},z_{n_{k}})$, implies that

[TABLE]

Then,

[TABLE]

which is a contradiction.

Therefore, $M$ is $\mathbb{R}$-uniformly continuous. ∎