Some Remarks on Fuzzy sb-Metric Spaces

Abstract

Fuzzy strong b-metrics called here by fuzzy sb-metrics, were introduced recently as a fuzzy version of strong b-metrics. It was shown that open balls in fuzzy $sb$-metric spaces are open in the induced topology (as different from the case of fuzzy b-metric spaces) and thanks to this fact fuzzy $sb$-metrics have many useful properties common with fuzzy metric spaces which generally may fail to be in the case of fuzzy b-metric spaces. In the present paper, we go further in the research of fuzzy $sb$-metric spaces. It is shown that the class of fuzzy $sb$-metric spaces lies strictly between the classes of fuzzy metric and fuzzy b-metric spaces. We prove that the topology induced by a fuzzy $sb$-metric is metrizable. A characterization of completeness in terms of diameter zero sets in these structures is given. We investigate products and coproducts in the naturally defined category of fuzzy $sb$-metric spaces.

1. Introduction

The class of b-metric spaces, one of the generalizations of metric spaces, has been introduced by different authors under different names (b-metric by Czerwik [1], quasimetric by Bakhtin [2] and by Heinhonen [3], $NE{M}_r$ (nonlinear elastic matching) by Fagin et al. [4], metric type by Khamsi et al. [5]. To make our exposition homogeneous in the sequel we use the term b-metric in all these cases. The definition of a b-metric is obtained by replacing the triangular inequality in the definition of a metric space with the inequality $d(x,z)\le k·\left(d\right(x,y)+d(y,z\left)\right),\forall x,y,z\in X$ for some $k\ge 1,$ so called “relaxed triangle inequality” in [4]. The topology induced by a b-metric has some “unpleasant” features. For instance, open balls may be not open [6,7], closed balls may be not closed [7] and a b-metric as a mapping may be not continuous in the induced topology [8]. To remedy these defects, Kirk and Shahzad [9] introduced the notion of an $sb$-metric by using the inequality $d(x,z)\le d(x,y)+k·d(y,z),\forall x,y,z\in X$ for some $k\ge 1$. Therefore, a metric is an $sb$-metric and an $sb$-metric is a b-metric. They also investigated the fixed point theory for mappings of these structures and complained about the absence of nontrivial examples of such spaces.

Recently, in [10], the authors of this paper presented a series of examples of $sb$-metric spaces that fail to be metric. They also considered these metric type spaces in the context when the ordinary sum operation is replaced by an extended t-conorm that is an operation satisfying certain conditions.

In the papers [11,12], a fuzzy version of an $sb$-metric was introduced and the basic properties of fuzzy strong b-metric spaces (renamed here as fuzzy $sb$-metric spaces) were studied. Besides, this notion can be viewed as a generalization of fuzzy metric spaces in the sense of George and Veeramani [13] since its definition is obtained by replacing the fuzzy triangularity axiom in the definition of a fuzzy metric with $M(x,y,t)\ast M(y,z,s)\le M(x,z,t+k·s)$ for some $k\ge 1$. Thus the class of fuzzy $sb$-metric spaces lies between fuzzy metric spaces and fuzzy b-metric spaces [14,15,16]. As expected, fuzzy $sb$-metric spaces have useful properties in common with metric and fuzzy metric spaces such as openness of open balls whereas it is not true in general for b-metric and fuzzy b-metric spaces.

As one can expect $sb$-metric and fuzzy $sb$-metric spaces have important properties sharing with metric and fuzzy metric spaces but that are not valid for general b-metric and fuzzy b-metric spaces. Specifically, open balls in fuzzy $sb$-metric spaces are open in the topology induced by an $sb$-metric, but may fail, as our examples in [14,16] show, to be open in the topology of fuzzy b-metric spaces. Based on this fact we prove that fuzzy $sb$-metric is continuous, that fuzzy $sb$-metric spaces are metrizable and other important properties common with the properties of metric and fuzzy metric spaces but not sharing with b-metric spaces.

The problem of definition of a concept of a fuzzy metric has a very long and diverse theory. As the first works of definitions of fuzzy metric, in an essentially different way, were given by Deng [17], Kaleva and Seikkala [18] and Kramosil and Michalek [19]. In our work we follow Kramosil-Michalek approach to fuzzy metric in the form modified by George-Veeramani [13]. Therefore, in order to designate the place of our research, we give a very brief survey of some work done but different authors in the direction of generalization of George-Veeramani fuzzy metrics.

In [20], Park defined the notion of intuitionistic fuzzy metric spaces by using the idea of intuitionistic fuzzy sets and proved some known results such as Baire’s theorem and the Uniform limit theorem. In [21], the authors proposed a method for constructing a Hausdorff fuzzy metric on the set of the nonempty compact subsets for a given fuzzy metric space and discussed several important properties as completeness, completion and precompactness. By using complete latices, Adibi et al. [22], introduced the notion of L-fuzzy metric space and studied common fixed point and coincidence point theorems on these structures. Sedghi et al. [23] introduced the concept of partial fuzzy metric as a fuzzy analogy of partial metric spaces and gave the topological structure and proved some fixed point results. In [24], a fuzzy version of cone metric spaces was introduced by Öner et al., and some topological properties and Banach’s fixed point theorem were given. By employing a control function, Sezen [25] defined controlled fuzzy metric spaces and established some fixed point results.

The aim of the present paper is to go further in the research of fuzzy $sb$-metric spaces. The structure of the paper is as follows. We recall some terminology related to fuzzy $sb$-metrics, present some examples and prove some properties of fuzzy $sb$-metric spaces in Section 2. Section 3 is devoted to the study of (complete) metrizability of a (complete) fuzzy $sb$-metric space. Here we also prove the metrizabillity of $sb$-metric spaces by using fuzzy $sb$-metrics. Diameter zero sets in fuzzy $sb$-metrics and related properties such as completeness are given in Section 4. Moreover, by using fuzzy $sb$-metrics we indirectly characterized the completeness of an $sb$-metric space in terms of diameter zero sets. Products and coproducts in the category of fuzzy $sb$-metric spaces are studied in Section 5. In the last section we sketch some directions where the research in the field of fuzzy $sb$-metrics can be continued and its results could find some applications.

2. Fuzzy $sb$-Metric Spaces: Basic Concepts and Properties

In the following, we collect the terminology concerning fuzzy metric and its generalizations that can be found in the literature (see e.g., [11,12,13,14,15,16,26,27,28]).

Definition 1.

Let X be a nonempty set. A mapping $M:X\times X\times (0,\infty )\to [0,1]$ is called a fuzzy symmetric if it satisfies the following properties for all $x,y,z\in X$ and all $t>0$:

($f{m}_1$)

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

($f{m}_2$)

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

($f{m}_3$)

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

($f{m}_4$)

$M(x,y,.):(0,\infty )\to [0,1]$ is continuous.

The corresponding tuple $(X,M,\ast )$ is called a fuzzy symmetric space.

Definition 2.

Given a continuous t-norm $\ast :\left[0,1\right]\times \left[0,1\right]⟶\left[0,1\right]$, a fuzzy symmetric $M:X\times X\times (0,\infty )\to [0,1]$ is called

($f{m}_5$)

a fuzzy metric if

$M(x,y,t)\ast M(y,z,s)\le M(x,z,t+s)\forall x,y,z\in X,\forall s,t>0$

and the corresponding tuple $(X,M,\ast )$ is called a fuzzy metric space;

($fb{m}_5$)

a fuzzy b-metric (or fuzzy metric type) if there exists $k\ge 1$ such that

$M(x,y,t)\ast M(y,z,s)\le M(x,z,k·(t+s\left)\right)\forall x,y,z\in X,\forall s,t>0$

and the corresponding tuple $(X,M,\ast ,k)$ is called a fuzzy b-metric space (or fuzzy metric type space);

($fsb{m}_5$)

a fuzzy $sb$-metric if there exists $k\ge 1$ such that

$M(x,y,t)\ast M(y,z,s)\le M(x,z,t+k·s)\forall x,y,z\in X,\forall s,t>0$

and the corresponding tuple $(X,M,\ast ,k)$ is called a fuzzy $sb$-metric space.

Remark 1.

We prefer to call the notion of fuzzy strong b-metric as “fuzzy $sb$-metric” since the term a fuzzy strong b-metric may lead to a misunderstanding when it comes into a collision with the concepts of a strong fuzzy metric [29] that has a quite different meaning.

Remark 2.

It is clear that every fuzzy metric is also a fuzzy $sb$-metric. Furthermore, for a fuzzy $sb$-metric M, since $M\left(x,y,·\right):(0,\infty )⟶\left[0,1\right]$ is nondecreasing, we have

$$M(x,y,t)\ast M(y,z,s)\le M(x,z,t+k·s)\le M(x,z,k·(t+s\left)\right)\forall x,y,z\in X,\forall s,t>0$$

and it means that a fuzzy $sb$-metric is fuzzy b-metric.

For fuzzy metric-like structures, an open ball with center x, radius $r\in (0,1)$ and $t>0$ is defined as $B(x,r,t)=\{y\in X:M(x,y,t)>1-r\}$. The induced topology denoted ${\tau }_M$ is defined by open balls as follows:

$${\tau }_M=\{A\subset X:\forall a\in A\exists r\in (0,1),t>0 \mathrm{such} \mathrm{that} B(a,r,t)\subset A\}$$

Contrary to fuzzy b-metric spaces, in fuzzy $sb$-metric spaces open balls are open sets and form a base for the induced topology.

In the following we will need the following.

Remark 3.

Let a continuous t-norm * be given. Then for any $r_1>r_2$, we can find a $r_3$ such that $r_1\ast r_3\ge r_2$ and for any $r_4$ we can find a $r_5$ such that $r_5\ast r_5\ge r_4$ where $r_1,r_2,r_3,r_4,r_5\in (0,1)$. [13]

Definition 3.

Let X be a nonempty set and $k\ge 1$. A mapping $d:X\times X\to [0,\infty )$ is called an $sb$-metric if it satisfies the following properties for all $x,y,z\in X$ [9]

($m_1$)

$d(x,y)=0$ if and only if $x=y$,

($m_2$)

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

($s{b}_3$)

$d(x,z)\le d(x,y)+k·d(y,z)$.

The corresponding tuple $(X,d,k)$ is called an $sb$-metric space.

As in the case of fuzzy metrics and fuzzy b-metrics the construction of the so called “standard” fuzzy $sb$-metric plays an important role. Namely, given an $sb$-metric d on a set X we define a mapping $M_d:X\times X\times (0,\infty )\to [0,1]$ by setting $M_d\left(x,y,t\right)=\frac{t}{t+d\left(x,y\right)}$. In [11], see Example 2.2 and Proposition 2.15, it was proved that $M_d$ is a fuzzy $sb$-metric for the case of the product t-norm and the topologies induced on the set X by $sb$-metric d and the fuzzy $sb$-metric $M_d$ coincide. Here we extend this result for the case of the minimum t-norm ∧. Similar results for fuzzy metric and fuzzy b-metric spaces can be found in [16,30], respectively.

Proposition 1.

Let $\left(X,d,k\right)$ be an $sb$-metric space. Then $\left(X,M_d,\wedge ,k\right)$ is a fuzzy $sb$-metric space where $M_d\left(x,y,t\right)=\frac{t}{t+d\left(x,t\right)}$ and ${\tau }_d={\tau }_{M_d.}$

Proof. 

We prove only $\left(fsb{m}_5\right)$ since the other conditions are straightforward. To do this, first we need to prove the following inequality for $t,s>0$.

$$\frac{d\left(x,z\right)}{t+k·s}\le max\left\{\frac{d\left(x,y\right)}{t},\frac{d\left(y,z\right)}{s}\right\}$$

(1)

Consider three cases:

Case 1: $d\left(x,z\right)\le d\left(x,y\right)$.

Case 2: $d\left(x,z\right)\le k·d\left(y,z\right)$.

Case 3: $d\left(x,z\right)>d\left(x,y\right)$ and $d\left(x,z\right)>k·d\left(y,z\right)$.

Case 1 and Case 2 are obvious. Suppose Case 3 is satisfied. Then $d\left(x,z\right)\le d\left(x,y\right)+k·d\left(y,z\right)$. Without loss of generality we may assume that $d\left(x,z\right)=d\left(x,y\right)+k·d\left(y,z\right)$.

Since $d\left(x,z\right)>d\left(x,y\right),$ there exist $\beta \in \left(0,1\right)$ such that $d\left(x,y\right)=\beta ·d\left(x,z\right)$. Hence, we have $d\left(y,z\right)=\frac{\left(1-\beta \right)·d\left(x,z\right)}{k}$. Therefore, Equation (1) become

$$\frac{d\left(x,z\right)}{t+k·s}\le max\left\{\frac{\beta ·d\left(x,z\right)}{t},\frac{\left(1-\beta \right)·d\left(x,z\right)}{k·s}\right\}$$

and we should prove simply

$$\frac{1}{t+k·s}\le max\left\{\frac{\beta }{t},\frac{1-\beta }{k·s}\right\}.$$

To do this, consider the functions $f\left(\beta \right)=\frac{t}{\beta }$ and $g\left(\beta \right)=\frac{k·s}{1-\beta }$. Since f is decreasing and g is increasing, the largest value of $min\left\{\frac{t}{\beta },\frac{k·s}{1-\beta }\right\}$ is $t+k·s$ that is taken when $f\left(\beta \right)=g\left(\beta \right)$ where $\beta =\frac{t}{t+k·s}$. Then

$$\begin{array}{cc}\hfill t+k·s& =f\left(\frac{t}{t+k·s}\right)\hfill \\ \hfill & \ge min\left\{\frac{t}{\beta },\frac{k·s}{1-\beta }\right\}\hfill \end{array}$$

implies

$$\frac{1}{t+k·s}\le max\left\{\frac{\beta }{t},\frac{1-\beta }{k·s}\right\}.$$

If $d(x,z)<d(x,y)+k·d(y,z)$, then there exists $\alpha \in (0,1)$ such that

$$\frac{1}{\alpha }·d(x,z)=d(x,y)+k·d(y,z).$$

Further, $d\left(x,z\right)>d\left(x,y\right)$ and $d\left(x,z\right)>k·d\left(y,z\right)$ imply ${\displaystyle \frac{1}{\alpha }}·d(x,z)>d(x,y)$ and ${\displaystyle \frac{1}{\alpha }}·d(x,z)>k·d(y,z)$. Hence by above case, we have

$${\displaystyle \frac{1}{\alpha }}·\frac{d\left(x,z\right)}{t+k·s}\le max\left\{\frac{d\left(x,y\right)}{t},\frac{d\left(y,z\right)}{s}\right\}$$

which implies

$$\frac{d\left(x,z\right)}{t+k·s}\le max\left\{\frac{d\left(x,y\right)}{t},\frac{d\left(y,z\right)}{s}\right\}.$$

Now we are ready to prove $\left(fsb{m}_5\right)$. By Equation (1), we have

$$\begin{array}{cc}\hfill 1+\frac{d\left(x,z\right)}{t+k·s}& \le max\left\{1+\frac{d\left(x,y\right)}{t},1+\frac{d\left(y,z\right)}{s}\right\}⟹\hfill \\ \hfill \frac{t+k·s+d\left(x,z\right)}{t+k·s}& \le max\left\{\frac{t+d\left(x,y\right)}{t},\frac{s+d\left(y,z\right)}{s}\right\}⟹\hfill \\ \hfill \frac{t+k·s}{t+k·s+d\left(x,z\right)}& \ge min\left\{\frac{t}{t+d\left(x,y\right)},\frac{s}{s+d\left(y,z\right)}\right\}⟹\hfill \\ \hfill M\left(x,z,t+k·s\right)& \ge M\left(x,y,t\right)\wedge M\left(y,z,s\right).\hfill \end{array}$$

The proof of the equivalence of the induced topologies is the same as the case where $\ast =·$ (Proposition 2.15 in [11]) since the axiom $\left(fsb{m}_5\right)$ and t-norm do not effect it. □

Our next two examples present fuzzy $sb$-metric spaces which fail to be fuzzy metric spaces. Since on the other hand from the fact that open balls are open in fuzzy $sb$-metric, but need not be open in fuzzy b-metric, these examples will show also that the class of fuzzy $sb$-metric spaces lies strictly between the class of fuzzy metrics and fuzzy b-metrics.

Example 1.

Let $X=X_a\cup X_b\cup X_c$ where $X_i=\left\{i\right\}\times [0,1],i\in \{a,b,c\}$ and $d:X\times X\to [0,5]$ be the distance function defined as follows:

$$d(x,y)=d(y,x)=\left\{\begin{array}{cc}\mid \overline{x}-\overline{y}\mid & ,x,y\in X_i\hfill \\ 1& ,x\in X_a,y\in X_b\hfill \\ 2& ,x\in X_a,y\in X_c\hfill \\ 5& ,x\in X_b,y\in X_c\hfill \end{array}\right.$$

where $x=\left\{i\right\}\times \overline{x},\overline{x}\in [0,1]$. Then $(X,d,3)$ is an $sb$-metric space (see Example 7 in [10]). By the above example, $(X,M_d,\wedge ,3)$ is a fuzzy $sb$-metric space that fails to be a fuzzy metric space as seen below:

Let $x\in X_b,y\in X_a$ and $z\in X_c$. For $t=2$ and $s=3$, we have

$$\begin{array}{cc}\hfill M_d(x,z,t+s)& ={\displaystyle \frac{t+s}{t+s+d(x,z)}}={\displaystyle \frac{1}{2}}\hfill \\ \hfill & <{\displaystyle \frac{3}{5}}={\displaystyle \frac{2}{3}}\wedge {\displaystyle \frac{3}{5}}={\displaystyle \frac{t}{t+d(x,y)}}\wedge {\displaystyle \frac{s}{s+d(y,z)}}\hfill \\ \hfill & <M_d(x,y,t)\wedge M_d(y,z,s).\hfill \end{array}$$

Example 2.

Let X be the unit disk in ${\mathbb{R}}^2$ with center $(0,0)$ and $S^1$ the corresponding unit circle and let the distance function $d:X\times X\to \mathbb{R}$ be defined as follows: for $x=\left(x_1,x_2\right)$ and $y=\left(y_1,y_2\right)$

$$d\left(x,y\right)=d\left(y,x\right)=\left\{\begin{array}{cc}10& ,x,y\in S^{\prime }\hfill \\ d^{\prime }\left(x,y\right)& ,otherwise\hfill \end{array}\right.$$

where $d^{\prime }$ is the post office metric that is $d^{\prime }\left(x,y\right)=\sqrt{x_1^2+x_2^2}+\sqrt{y_1^2+y_2^2}$. Then $\left(X,d,9\right)$ is an $sb$-metric space (see Example 8 and Remark 7 in [10]). Hence $\left(X,M_d,·,9\right)$ is the standard fuzzy $sb$-metric space induced by d. Let $x,y\in S^{\prime }$ and z be the origin. Then for $t=s=1$ we have

$$\begin{array}{cc}\hfill M\left(x,z,t\right)·M\left(z,y,s\right)& =\frac{t}{t+d\left(x,z\right)}·\frac{s}{s+d\left(z,y\right)}\hfill \\ \hfill & =\frac{t}{t+d^{\prime }\left(x,z\right)}·\frac{s}{s+d^{\prime }\left(z,y\right)}\hfill \\ \hfill & =\frac{t}{t+1}·\frac{s}{s+1}\hfill \\ \hfill & =\frac{1}{4}>\frac{1}{6}=\frac{2}{2+10}=\frac{t+s}{t+s+d\left(x,y\right)}=M\left(x,y,t+s\right)\hfill \end{array}$$

and this means that $M_d$ fails to be a fuzzy metric.

A fuzzy metric space $(X,M,\ast )$ is said to be F-bounded if there exist $t>0$ and $r\in (0,1)$ such that $M(x,y,t)>1-r$ for all $x,y\in X$ [13]. We use the same definition for fuzzy $sb$-metrics. We shall obtain a F-bounded fuzzy $sb$-metric equivalent for a given one in the sense that they induce the same topologies. However, in order to construct this fuzzy $sb$-metric we need to recall the convergence of a sequence and its characterization.

Definition 4.

Let $(X,M,\ast ,k)$ be a fuzzy $sb$-metric space, $x\in X$ and $\left\{x_n\right\}$ be a sequence in X. Then [11]

(i)

$\left\{x_n\right\}$ is said to converge to x if for any $t>0$ and any $r\in \left(0,1\right)$ there exists a natural number $n_0$ such that $M\left(x_n,x,t\right)>1-r$ for all $n\ge n_0$. We denote this by ${lim}_{n\to \infty }{x}_n=x$ or $x_n\to x$ as $n\to \infty$.

(ii)

$\left\{x_n\right\}$ is said to be a Cauchy sequence if for any $r\in \left(0,1\right)$ and any $t>0$ there exists a natural number $n_0$ such that $M\left(x_n,x_m,t\right)>1-r$ for all $n,m\ge n_0$.

(iii)

$\left(X,M,\ast ,k\right)$ is said to be a complete fuzzy $sb$-metric space if every Cauchy sequence is convergent.

Theorem 1.

Let $(X,M,\ast ,k)$ be a fuzzy $sb$-metric space, $x\in X$ and $\left\{x_n\right\}$ be a sequence in X. $\left\{x_n\right\}$ converges to x if and only if $M\left(x_n,x,t\right)\to 1$ as $n\to \infty$, for each $t>0$. [11]

Proposition 2.

Let $(X,M,\ast ,k)$ be a fuzzy $sb$-metric space and $\lambda \in (0,1)$. Then $(X,N,\ast ,k)$ is an F-bounded fuzzy $sb$-metric space where $N(x,y,t)=max\left\{M\right(x,y,t),\lambda \}$ for $x,y\in X$ and $t>0$ and ${\tau }_N={\tau }_M$.

Proof. 

We only show the $\left(fsb{m}_5\right)$ since the other conditions are immediate. We distinguish two cases:

Case 1: $N(x,y,t)=\lambda$ or $N(y,z,s)=\lambda$. Assume that $N(x,y,t)=\lambda$. Then we have

$$\begin{array}{cc}\hfill N(x,z,t+k·s)& \ge \lambda =\lambda \ast 1\ge \lambda \ast N(y,z,s)\hfill \\ \hfill & \ge N(x,y,t)\ast N(y,z,s).\hfill \end{array}$$

Case 2: $N(x,y,t)=M(x,y,t)>\lambda$ and $N(y,z,s)=M(y,z,s)>\lambda$. Here we have

$$\begin{array}{cc}\hfill N(x,z,t+k·s)& \ge M(x,z,t+k·s)\ge M(x,y,t)\ast M(y,z,s)\hfill \\ \hfill & \ge N(x,y,t)\ast N(y,z,s).\hfill \end{array}$$

To establish the equivalence of the induced topologies, it is sufficient to show the equivalence of the convergence of sequences since induced topologies are first countable (Proposition 2.8. in [11]). Let $\left(x_n\right)$ be a sequence in X. For each $t>0$, we have

$$\begin{array}{cc}\hfill N(x_n,x,t)\to 1& ⟺max\{\lambda ,M(x_n,x,t)\}\to 1\hfill \\ \hfill & ⟺M(x_n,x,t)\to 1\hfill \end{array}$$

and it means that the induced topologies are the same. □

Proposition 3.

Let $(X,M_1,\ast ,k_1)$ and $(X,M_2,\ast ,k_2)$ be fuzzy $sb$-metric spaces and define

$$\begin{array}{cc}\hfill N_1(x,y,t)& =M_1(x,y,t)\ast M_2(x,y,t)\hfill \\ \hfill N_2(x,y,t)& =min\{M_1(x,y,t),M_2(x,y,t)\}.\hfill \end{array}$$

and $k=max\{k_1,k_2\}$. Then

(i) 

$(X,N_1,\ast ,k)$ is a fuzzy $sb$-metric space.

(ii) 

$(X,N_2,\ast ,k)$ is a fuzzy $sb$-metric space.

(iii) 

${\tau }_{N_1}={\tau }_{N_2}$.

Proof. 

We only show $\left(fsb{m}_5\right)$ for both cases.

(i)

$$\begin{array}{cc}\hfill N_1(x,z,t+k·s)& =M_1(x,z,t+k·s)\ast M_2(x,z,t+k·s)\hfill \\ \hfill & \ge M_1(x,z,t+k_1·s)\ast M_2(x,z,t+k_2·s)\hfill \\ \hfill & \ge M_1(x,y,t)\ast M_1(y,z,s)\ast M_2(x,y,t)\ast M_2(y,z,s)\hfill \\ \hfill & =M_1(x,y,t)\ast M_2(x,y,t)\ast M_1(y,z,s)\ast M_2(y,z,s)\hfill \\ \hfill & =N_1(x,y,t)\ast N_1(y,z,s).\hfill \end{array}$$

(ii)

$$\begin{array}{cc}\hfill N_2(x,z,t+k·s)& =min\{M_1(x,z,t+k·s),M_2(x,z,t+k·s)\}\hfill \\ \hfill & \ge min\{M_1(x,z,t+k_1·s),M_2(x,z,t+k_2·s)\}\hfill \\ \hfill & \ge min\{M_1(x,y,t)\ast M_1(y,z,s),M_2(x,y,t)\ast M_2(y,z,s)\}\hfill \\ \hfill & \ge min\{M_1(x,y,t)\ast M_2(x,y,t)\}\ast min\{M_1(y,z,s)\ast M_2(y,z,s)\}\hfill \\ \hfill & =N_2(x,y,t)\ast N_2(y,z,s).\hfill \end{array}$$

(iii) Let $\left(x_n\right)$ be a sequence in X. Then for all $t>0$, we have

$$\begin{array}{cc}\hfill N_1(x_n,x,t)\to 1& ⟺M_1(x_n,x,t)\ast M_2(x_n,x,t)\to 1\hfill \\ \hfill & ⟺M_1(x_n,x,t)\to 1\mathrm{and}{M}_2(x_n,x,t)\to 1\hfill \\ \hfill & ⟺min\{M_1(x_n,x,t),M_2(x_n,x,t)\}\to 1\hfill \\ \hfill & ⟺N_2(x_n,x,t)\to 1\hfill \end{array}$$

and it means that the induced topologies are the same. □

Example 3.

Let X be a set and * be a continuous t-norm. By Example 7 in [31] and Lemma 3.1 in [30],

$$M(x,y,t)=\{\begin{array}{cc}1& ,x=y\hfill \\ \lambda & ,x\ne y\hfill \end{array}$$

is a discrete fuzzy metric on X where $\lambda \in (0,1)$. Then $(X,M,\ast ,k)$ can be considered as a fuzzy $sb$-metric for any $k\ge 1$. Hence, by Example 3, $(X,N_i,\ast ,k),i=1,2,$ are also discrete fuzzy $sb$-metric spaces where

$$\begin{array}{cc}\hfill N_1(x,y,t)& =\{\begin{array}{cc}1& ,x=y\hfill \\ M_1(x,y,t)\ast \lambda & ,x\ne y\hfill \end{array}\hfill \\ \hfill N_2(x,y,t)& =\{\begin{array}{cc}1& ,x=y\hfill \\ min\{M_1(x,y,t),\lambda \}& ,x\ne y\hfill \end{array}\hfill \end{array}$$

and $(X,M_1,\ast ,k)$ is an arbitrary fuzzy $sb$-metric space.

Theorem 2.

Let $\left(X,M,\ast ,k\right)$ be a fuzzy $sb$-metric space. Then $M\left(-,-,t\right):X\times X\to \left[0,1\right]$ is continuous.

Proof. 

Since $\left(X,{\tau }_M\right)$ is first countable, it is also sequential. Moreover, product topology on $X\times X$ is also first countable and sequential. So, it is enough to show that $M\left(-,-,t\right)$ is sequentially continuous. Let two convergent sequences $\left(x_n\right)$ and $\left(y_n\right)$ in X be given such that $x_n\to x$ and $y_n\to y$ for some $x,y\in X$. Hence ${lim}_{n\to \infty }M\left(x_n,x,t\right)\to 1$ and ${lim}_{n\to \infty }M\left(y_n,y,t\right)\to 1$ for any $t>0$. For a given $\epsilon >0$ and $t>0$,

$$\begin{array}{cc}\hfill M\left(x,y,t\right)\ast M\left(x_n,x,\frac{\epsilon }{2k}\right)& \le M\left(x_n,y,t+\frac{\epsilon }{2}\right)\hfill \\ \hfill M\left(x,y,t\right)\ast M\left(x_n,x,\frac{\epsilon }{2k}\right)\ast M\left(y_n,y,\frac{\epsilon }{2k}\right)& \le M\left(x_n,y,t+\frac{\epsilon }{2}\right)\ast M\left(y_n,y,\frac{\epsilon }{2k}\right)\hfill \\ \hfill M\left(x,y,t\right)\ast M\left(x_n,x,\frac{\epsilon }{2k}\right)\ast M\left(y_n,y,\frac{\epsilon }{2k}\right)& \le M\left(x_n,y_n,t+\epsilon \right).\hfill \end{array}$$

As $n\to \infty$, we have

$$\begin{array}{cc}\hfill \underset{n\to \infty }{lim}\left[M\left(x,y,t\right)\ast M\left(x_n,x,\frac{\epsilon }{2k}\right)\ast M\left(y_n,y,\frac{\epsilon }{2k}\right)\right]& \le \underset{n\to \infty }{lim}M\left(x_n,y_n,t+\epsilon \right)\hfill \\ \hfill \underset{n\to \infty }{lim}M\left(x,y,t\right)\ast \underset{n\to \infty }{lim}M\left(x_n,x,\frac{\epsilon }{2k}\right)\ast \underset{n\to \infty }{lim}M\left(y_n,y,\frac{\epsilon }{2k}\right)& \le \underset{n\to \infty }{lim}M\left(x_n,y_n,t+\epsilon \right)\hfill \\ \hfill \underset{n\to \infty }{lim}M\left(x,y,t\right)\ast 1\ast 1& \le \underset{n\to \infty }{lim}M\left(x_n,y_n,t+\epsilon \right)\hfill \\ \hfill M\left(x,y,t\right)& \le \underset{n\to \infty }{lim}M\left(x_n,y_n,t+\epsilon \right).\hfill \end{array}$$

Since $\epsilon >0$ is arbitrary, we obtain

$$M\left(x,y,t\right)\le \underset{n\to \infty }{lim}M\left(x_n,y_n,t\right).$$

Similarly, for a given $\epsilon >0$ and $t>0$,

$$\begin{array}{cc}\hfill M\left(x_n,y_n,t\right)\ast M\left(x_n,x,\frac{\epsilon }{2k}\right)& \le M\left(y_n,x,t+\frac{\epsilon }{2}\right)\hfill \\ \hfill M\left(x_n,y_n,t\right)\ast M\left(x_n,x,\frac{\epsilon }{2k}\right)\ast M\left(y_n,y,\frac{\epsilon }{2k}\right)& \le M\left(y_n,x,t+\frac{\epsilon }{2}\right)\ast M\left(y_n,y,\frac{\epsilon }{2k}\right)\hfill \\ \hfill M\left(x_n,y_n,t\right)\ast M\left(x_n,x,\frac{\epsilon }{2k}\right)\ast M\left(y_n,y,\frac{\epsilon }{2k}\right)& \le M\left(x,y,t+\epsilon \right).\hfill \end{array}$$

As $n\to \infty$, we have

$$\begin{array}{cc}\hfill \underset{n\to \infty }{lim}\left[M\left(x_n,y_n,t\right)\ast M\left(x_n,x,\frac{\epsilon }{2k}\right)\ast M\left(y_n,y,\frac{\epsilon }{2k}\right)\right]& \le \underset{n\to \infty }{lim}M\left(x,y,t+\epsilon \right)\hfill \\ \hfill \underset{n\to \infty }{lim}M\left(x_n,y_n,t\right)\ast \underset{n\to \infty }{lim}M\left(x_n,x,\frac{\epsilon }{2k}\right)\ast \underset{n\to \infty }{lim}M\left(y_n,y,\frac{\epsilon }{2k}\right)& \le M\left(x,y,t+\epsilon \right)\hfill \\ \hfill \underset{n\to \infty }{lim}M\left(x_n,y_n,t\right)\ast 1\ast 1& \le M\left(x,y,t+\epsilon \right)\hfill \\ \hfill \underset{n\to \infty }{lim}M\left(x_n,y_n,t\right)& \le M\left(x,y,t+\epsilon \right).\hfill \end{array}$$

Since $\epsilon >0$ is arbitrary, we obtain

$$\underset{n\to \infty }{lim}M\left(x_n,y_n,t\right)\le M\left(x,y,t\right).$$

Therefore, ${lim}_{n\to \infty }M\left(x_n,y_n,t\right)=M\left(x,y,t\right)$ and $M\left(-,-,t\right):X\times X\to \left[0,1\right]$ is continuous. □

Lemma 1.

Let $\left(X,d,k\right)$ be an $sb$-metric space and $\left(X,M_d,·,k\right)$ be the standard fuzzy $sb$-metric space.

(1) 

$x_n\to x$ in $\left(X,d,k\right)$ if and only if $x_n\to x$ in $\left(X,M_d,·,k\right)$.

(2) 

$\left(x_n\right)$ is Cauchy in $\left(X,d,k\right)$ if and only if $\left(x_n\right)$ is Cauchy in $\left(X,M_d,·,k\right)$.

Proof. 

(1) Let $x_n\to x$ in $\left(X,d,k\right)$. Then $d\left(x_n,x\right)\to 0$ and we have

$$M_d\left(x_n,x,t\right)={\displaystyle \frac{t}{t+d\left(x_n,x\right)}}\to 1$$

which implies that $x_n\to x$ in $\left(X,M_d,·,k\right)$. Converse is similar.

(2) Let $\left(x_n\right)$ be Cauchy in $\left(X,d,k\right)$ and $r\in \left(0,1\right)$ and $t>0$ are given. For $\epsilon ={\displaystyle \frac{tr}{1-r}}$, there exists $n_0\in \mathbb{N}$ such that $d\left(x_n,x_m\right)<\epsilon$ for every $n,m\ge n_0$. Then we have

$$\begin{array}{cc}\hfill t+d\left(x_n,x_m\right)& <t+\epsilon =t+{\displaystyle \frac{tr}{1-r}}=\frac{t}{1-r}\hfill \\ \hfill \frac{1}{t+d\left(x_n,x_m\right)}& >{\displaystyle \frac{1-r}{t}}\hfill \\ \hfill \frac{t}{t+d\left(x_n,x_m\right)}& >1-r\hfill \\ \hfill M_d\left(x_n,x_m,t\right)& >1-r\hfill \end{array}$$

for every $n,m\ge n_0$. Hence $\left(x_n\right)$ is Cauchy in $\left(X,M_d,·,k\right)$. On the other hand, let $\left(x_n\right)$ be Cauchy in $\left(X,M_d,·,k\right)$ and $\epsilon >0$ is given. Choose $r={\displaystyle \frac{\epsilon }{t+\epsilon }}\in \left(0,1\right)$. Then for any $t>0$, there exists $n_0\in \mathbb{N}$ such that $M_d\left(x_n,x_m,t\right)>1-r$ for every $n,m\ge n_0$. It follows that

$$\begin{array}{cc}\hfill M_d\left(x_n,x_m,t\right)& =\frac{t}{t+d\left(x_n,x_m\right)}>1-r=1-{\displaystyle \frac{\epsilon }{t+\epsilon }}=\frac{t}{t+\epsilon }\hfill \\ \hfill d\left(x_n,x_m\right)& <\epsilon \hfill \end{array}$$

for every $n,m\ge n_0$ which implies $\left(x_n\right)$ is Cauchy in $\left(X,d,k\right)$. □

Corollary 1.

An $sb$-metric space $\left(X,d,k\right)$ is complete if and only if the standard fuzzy $sb$-metric space $\left(X,M_d,·,k\right)$ is complete.

3. Metrizability of Fuzzy sb-Metric Spaces

Gregori et al. in [32] proved that every fuzzy metric space is metrizable. In the proof they use Kelly metrization lemma.

Lemma 2

([33]). A $T_1$ topological space $(X,\tau )$ is metrizable if and only if it admits a compatible uniformity with a countable base.

Applying the same lemma we prove here that fuzzy $sb$-metric spaces are metrizable as well.

Theorem 3.

Fuzzy $sb$-metric spaces are metrizable.

Proof. 

Let $(X,M,\ast ,k)$ be a fuzzy $sb$-metric space and ${\tau }_M$ be the induced topology by M. Firstly, we will show that

$$\mathcal{B}=\left\{U_n:U_n=\left\{\left(x,y\right)\in X\times X:M\left(x,y,\frac{1}{n}\right)>1-\frac{1}{n}\right\},n\in \mathbb{N}\right\}$$

is a base for a uniformity $\mathcal{U}$ on X. Then we will show that ${\tau }_M$ and the topology induced by $\mathcal{U}$ are the same. To show that $\mathcal{B}$ is a base for a uniformity $\mathcal{U}$ on X we have to verify the following conditions:

(i)

$\mathsf{\Delta }\subseteq U_n$.

(ii)

$U_n=U_n^{-1},$

(iii)

$U_{n+1}\subset U_n.$

Conditions (i) and (ii) are obvious for each $n\in \mathbb{N}$. To show condition (iii) we are reasoning as follows. Let $(x,y)\in U_{n+1}$. Since $n<n+1$, we have

$$1-\frac{1}{n}<1-\frac{1}{n+1}<M(x,y,\frac{1}{n+1})<M(x,y,\frac{1}{n})$$

which implies $(x,y)\in U_n$.

Since * is continuous, referring to Remark 3 we can find sufficiently large $m\in \mathbb{N}$ such that $m>\left(1+k\right)n$ and

$$1-(1/n)<(1-(1/m\left)\right)\ast (1-(1/m\left)\right).$$

Let $(x,y)\in U_m$ and $(y,z)\in U_m$. ${\displaystyle \frac{1+k}{m}}<{\displaystyle \frac{1}{n}}$ implies $M(x,z,{\displaystyle \frac{1+k}{m}})\le M(x,z,{\displaystyle \frac{1}{n}})$ and we have

$$\begin{array}{cc}\hfill M(x,z,{\displaystyle \frac{1}{n}})& \ge M(x,z,{\displaystyle \frac{1+k}{m}})\ge M(x,y,{\displaystyle \frac{1}{m}})\ast M(y,z,{\displaystyle \frac{1}{m}})\hfill \\ \hfill & \ge (1-(1/m\left)\right)\ast (1-(1/m\left)\right)\hfill \\ \hfill & >1-(1/n)\hfill \end{array}$$

and this means that $(x,z)\in U_n$ and $U_m\circ U_m\subset U_n$. Hence $\left\{U_n:n\in \mathbb{N}\right\}$ is a base for a uniformity $\mathcal{U}$ on X. Moreover, since we have

$$U_n\left(x\right)=\left\{y\in X:M\left(x,y,{\displaystyle \frac{1}{n}}\right)>1-\frac{1}{n}\right\}=\left\{B\left(x,{\displaystyle \frac{1}{n}},{\displaystyle \frac{1}{n}}\right):n\in \mathbb{N}\right\}$$

for each $n\in \mathbb{N}$ and $x\in X$, the topology induced by $\mathcal{U}$ is the same as ${\tau }_M$. By Lemma 2, $(X,{\tau }_M)$ is a metrizable. □

The topology ${\tau }_d$ induced by an $sb$-metric d is same as the topology induced by the corresponding standard fuzzy sb-metric $M_d$. Hence, we have the following by the above theorem.

Corollary 2.

Every $sb$-metric space is metrizable.

Corollary 3.

A topological space is metrizable if and only if it admits a compatible fuzzy $sb$-metric.

Theorem 4.

If $\left(X,M,\ast ,k\right)$ is a complete fuzzy $sb$-metric space, then $\left(X,{\tau }_M\right)$ is completely metrizable.

Proof. 

By Theorem 3, $\left(X,M,\ast ,k\right)$ is metrizable and there is a metric space $\left(X,d\right)$ whose induced topology and ${\tau }_M$ are the same. Moreover, the uniformity induced by d coincides with the uniformity $\mathcal{U}$; in its turn the topology induced by $\mathcal{U}$ coincides with the topology ${\tau }_M$. Recalling that

$$U_n=\left\{\left(x,y\right)\in X\times X:M\left(x,y,\frac{1}{n}\right)>1-1/n\right\}$$

is a base for $\mathcal{U},$ we consider a Cauchy sequence $\left(x_n\right)$ in $\left(X,d\right)$. Then $\left(x_n\right)$ is a Cauchy sequence in $\left(X,\mathcal{U}\right)$. For any $r\in \left(0,1\right)$ and $t>0$, we can choose $s\in \mathbb{N}$ such that $1/s\le min\left\{t,r\right\}$ and there is $n_0\in \mathbb{N}$ such that $\left(x_n,x_m\right)\in U_s$ for every $n,m\ge n_0$. Therefore, for every $n,m\ge n_0$,

$$M\left(x_n,x_m,t\right)\ge M\left(x_n,x_m,1/s\right)>1-\frac{1}{s}\ge 1-r$$

and this means that $\left(x_n\right)$ is a Cauchy sequence in $\left(X,M,\ast ,k\right)$. Since $\left(X,M,\ast ,k\right)$ is complete, it is convergent with respect to ${\tau }_M$. Hence, $\left(X,d\right)$ is complete and $\left(X,{\tau }_M\right)$ is completely metrizable. □

For a precise characterization of completely metrizable spaces in terms of complete fuzzy $sb$-metrics, we need the following statement:

Proposition 4.

A topological space is completely metrizable if and only if it admits a compatible complete fuzzy metric [32].

Corollary 4.

A topological space is completely metrizable if and only if it admits a compatible complete fuzzy $sb$-metric.

Proof. 

For a completely metrizable space $\left(X,\tau \right)$, by Proposition 4, it admits a compatible complete fuzzy metric. Since every fuzzy metric is a fuzzy $sb$-metric, $\left(X,\tau \right)$ admits a compatible complete fuzzy $sb$-metric. The converse follows from Theorem 4. □

4. Diameter Zero Sets and Completeness in Fuzzy sb-Metric

Completeness is an important property of metric spaces. For instance, it is needed to establish Baire property in metric spaces; this is crucial in the investigation of the existence and uniqueness of fixed points for mappings of metric spaces and in many other important both from theoretical and practical points of view fields of research. In a similar way, completeness of fuzzy metric spaces showed to be crucial in the study of the analogous problems in fuzzy context. In particular, fuzzy versions of Baire theorem were given for fuzzy metrics in [13] and for fuzzy $sb$-metrics in [12]. In addition, a restricted version of Baire theorem for fuzzy b-metric spaces was proved in [16].

By patterning the results in [34], in this section we define and investigate diameter zero sets in fuzzy sb-metric spaces and characterize their completeness in terms of diameter zero sets.

Definition 5.

Let $\left(X,M,\ast ,k\right)$ be a fuzzy $sb$-metric space and ${\left\{A_n\right\}}_{n\in I}$ be a collection of subsets of X. ${\left\{A_n\right\}}_{n\in I}$ is said to have fuzzy $sb$-diameter zero if for each $r\in \left(0,1\right)$ and $t>0$ there exists $n\in I$ such that $M(x,y,t)>1-r$ for all $x,y\in A_n$.

Remark 4.

One can easily notice that $A\subseteq X$ is a nonempty subset in a fuzzy $sb$-metric space $\left(X,M,\ast ,k\right)$ then A has fuzzy $sb$-diameter zero if and only if A is a singleton set.

Theorem 5.

Let $\left(X,M,\ast ,k\right)$ be a fuzzy $sb$-metric space. $(X,M,\ast ,k)$ is complete if and only if every nested sequence of nonempty closed sets ${\left\{A_n\right\}}_{n=1}^{\infty }$ with fuzzy $sb$-diameter zero has nonempty intersection.

Proof. 

Let $(X,M,\ast ,k)$ be a complete fuzzy $sb$-metric space and ${\left\{A_n\right\}}_{n=1}^{\infty }$ be a nested sequence of nonempty closed sets with fuzzy $sb$-diameter zero. We need to prove that ${\bigcap }_{n=1}^{\infty }{A}_n$ is nonempty. For each $n\in \mathbb{N},$ choose $x_n$ in $A_n$. Then for $t>0$ and $r\in \left(0,1\right)$, there is $n_0\in \mathbb{N}$ such that $M(x,y,t)>1-r$ for all $x,y\in A_{n_0}$. Hence, for all $n,m\ge n_0,$ we have $M(x_n,x_m,t)>1-r$. Notice that $x_n\in A_n\subset A_{n_0}$ and $x_m\in A_m\subset A_{n_0}$. Hence $\left\{x_n\right\}$ is Cauchy and $x_n\to x$ for some $x\in X$. For each $n,x_k\in A_n$ for all $k\ge n$. Hence $x_k\to x$ and $x\in \overline{A_n}$ where $\overline{A_n}$ is the closure of $A_n$. Since $A_n$ is closed $x\in A_n$ for every n means that $x\in {\bigcap }_{n=1}^{\infty }{A}_n$.

Conversely, let every nested sequence of nonempty closed sets ${\left\{A_n\right\}}_{n=1}^{\infty }$ with fuzzy $sb$-diameter zero have nonempty intersection. We shall show that $(X,M,\ast ,k)$ is complete. Let a Cauchy sequence $\left\{x_n\right\}$ in $(X,M,\ast ,k)$ be given. We define $B_n=\{x_n,x_{n+1},x_{n+2},\cdots \}$ and $A_n=\overline{B_n}$. Clearly, ${\left\{A_n\right\}}_{n=1}^{\infty }$ is nested, nonempty and all sets $A_n$ are closed. Now we shall show that ${\left\{A_n\right\}}_{n=1}^{\infty }$ has fuzzy $sb$-diameter zero. Referring to Remark 3, for a given $s\in \left(0,1\right)$, we can find $r\in (0,1)$, such that

$$(l-r)\ast (1-r)\ast (1-r)>(1-s)$$

For $r\in \left(0,1\right)$ and $t>0$, there is $n_0\in \mathbb{N}$ such that for every $m,n\ge n_0,$ $M(x_n,x_m,t/3k)>1-r$. Hence, for every $x,y\in B_{n_0},$ $M(x,y,t/3k)>1-r$. Let $x,y\in A_{n_0}$. Since $A_{n_0}$ is closed, there are sequences $\left\{a_n^{\prime }\right\}$ and $\left\{b_n^{\prime }\right\}$ in $B_{n_0}$ such that $a_n^{\prime }$ converges to x and $b_n^{\prime }$ converges to y. Therefore, $a_n^{\prime }\in B(x,r,t/3)$ and $b_n^{\prime }\in B(y,r,t/3k^2)$ for sufficiently large n. Then we have

$$\begin{array}{cc}\hfill M(x,a_n^{\prime },t/3)\ast M(a_n^{\prime },b_n^{\prime },t/3k)\ast M(b_n^{\prime },y,t/3k^2)& \le M(x,y,t)\hfill \\ \hfill (l-r)\ast (1-r)\ast (1-r)& <\hfill \\ \hfill 1-s& <\hfill \end{array}$$

and this means that $\left\{A_n\right\}$ has fuzzy $sb$-diameter zero. Hence by hypothesis ${\bigcap }_{n=1}^{\infty }{A}_n\ne \varnothing$. Choose $x\in {\bigcap }_{n=1}^{\infty }{A}_n$. For $t>0$ and $r\in \left(0,1\right)$ there is $n_1$ such that for all $n\ge n_1,M(x_n,x,t)>1-r$. This implies that for each $t>0$, $M(x_n,x,t)\to 1$ as $n\to \infty$. Therefore, $x_n\to x$ and $(X,M,\ast ,k)$ is complete. □

Remark 5.

It is clear that in the above theorem the element $x\in {\cap }_{n=1}^{\infty }{A}_n$ is unique.

Now, by Theorem 5, we indirectly characterize the completeness of an sb-metric space in terms of diameter zero sets.

Definition 6.

Let $\left(X,d,k\right)$ be an $sb$-metric space and $A\subseteq X$. Diameter of A is defined as $\delta \left(A\right)=sup\left\{d\left(x,y\right):x,y\in A\right\}$.

Lemma 3.

Let $\left(X,d,k\right)$ be an $sb$-metric space and $\left(X,M_d,·,k\right)$ be the standard fuzzy $sb$-metric space. If ${\left\{A_n\right\}}_{n=1}^{\infty }$ is a nested sequence of nonempty closed sets with diameter tending to zero in $\left(X,d,k\right)$, then ${\left\{A_n\right\}}_{n\in \mathbb{N}}$ has fuzzy $sb$-diameter zero in $\left(X,M_d,·,k\right)$.

Proof. 

$\delta \left(A_n\right)\to 0$ implies that, for each $0<r<1$ and $t>0$ there exists $n\in \mathbb{N}$ such that $\delta \left(A_n\right)<{\displaystyle \frac{tr}{1-r}}$. Then we have $d(x,y)<{\displaystyle \frac{tr}{1-r}}$ which implies that $M_d(x,y,t)>1-r$ for all $x,y\in A_n$. Therefore, $\left\{A_n\right\}$ has fuzzy diameter zero. □

Corollary 5.

An $sb$-metric space $(X,d,k)$ is complete if and only if every nested sequence of nonempty closed sets ${\left\{A_n\right\}}_{n=1}^{\infty }$ with diameter tending to zero has nonempty intersection.

Proof. 

Assume that $(X,d,k)$ is complete. Then $(X,M_d,·,k)$ is also complete. By Lemma 3, every nested sequence of nonempty closed sets $\left\{A_n\right\}$ in $(X,d,k)$ with diameter tending to zero has fuzzy $sb$-diameter zero in $(X,M_d,·,k)$. Hence by Theorem 5, ${\bigcap }_{n=1}^{\infty }{A}_n$ is nonempty.

On the other hand, assume that every nested sequence of nonempty closed sets ${\left\{A_n\right\}}_{n=1}^{\infty }$ in $(X,d,k)$ with diameter tending to zero has nonempty intersection. By Lemma 3, $\left\{A_n\right\}$ has fuzzy $sb$-diameter zero and has a nonempty intersection in $(X,M_d,·,k)$. Therefore, by Theorem 5 $(X,M_d,·,k)$ is complete and by Corollary 1 $(X,d,k)$ is also complete. □

5. Category of Fuzzy sb-Metric Spaces

In this section we make a brief glance on fuzzy sb-metric spaces from the categorical point of view. First of all we must define the morphisms for this category, naturally called continuous mappings. Let * be a fixed t-norm.

Definition 7.

Let $(X,M_X,\ast ,k_1)$ and $(Y,N_Y,\ast ,k_2)$ be fuzzy $sb$-metric spaces. A mapping $f:(X,M_X)\to (Y,M_Y)$ is called continuous if it is continuous as a mapping $f:(X,{\tau }_{M_X})\to (Y,{\tau }_{M_Y})$.

The following characterizations of a continuous function between fuzzy $sb$-metric spaces are obvious since open balls form a base for the corresponding first countable topologies.

Theorem 6.

Let $(X,M_X,\ast ,k_1)$ and $(Y,N_Y,\ast ,k_2)$ be fuzzy $sb$-metric spaces. For a mapping $f:(X,M_X)\to (Y,M_Y)$, the following are equivalent.

1. 

$f:(X,M_X)\to (Y,M_Y)$ is continuous;

2. 

for every $a\in X,r_1\in (0,1)$ and $t_1>0$, there exits $r_2\in (0,1)$ and $t_2>0$ such that $M_Y(f\left(a\right),f\left(x\right),t_1)>1-r_1$ whenever $M_X(a,x,t_2)>1-r_2$;

3. 

if $\left(x_n\right)$ is a sequence converging to a point x in $(X,M_X,\ast ,k_1)$, then the sequence $\left(f\left(x_n\right)\right)$ converges to $f\left(x\right)$ in $(Y,M_Y,\ast ,k_2)$.

Now, we define category of fuzzy $sb$-metric spaces.

Definition 8.

The objects of the category*-Fsb-Metrof fuzzy $sb$-metric spaces are four-tuples $(X,M,\ast ,k)$ where $k\ge 1$. The morphisms of the category*-Fsb-Metrare continuous mappings $f:(X,M_X)\to (Y,M_Y)$.

By*-Fsbk-Metr, we denote the full subcategory of the category*-Fsb-Metrwhose objects are fuzzy $sb$-meric spaces $(X,M,\ast ,k)$ where $k\ge 1$ is a fixed constant.

In the following, we investigate the products and coproducts of fuzzy $sb$-metric spaces. For the products, we distinguish finite and countable cases.

Let $\{(X_i,M_i,\ast ,k_i):i=1\cdots ,n\}$ be a family of fuzzy $sb$-metric spaces and $k=max\{k_1,\cdots k_n\}$. We define $X={\prod }_{i=1}^n{X}_i$ and $M:X\times X\times (0,\infty )\to [0,1]$ by

$$M(x,y,t)=M_1(x_1,y_1,t)\ast M_2(x_2,y_2,t)\ast \cdots \ast M_n(x_n,y_n,t)$$

where $t>0$ and $x=(x_1,\cdots ,x_n),y=(y_1,\cdots ,y_n)\in X$.

Theorem 7.

$(X,M,\ast ,k)$ is the product of the family $\{(X_i,M_i,\ast ,k_i):i=1\cdots ,n\}$ in the category*-Fsb-Metr. Moreover, the topology ${\tau }_M$ induced by M coincides with the product of the topologies ${\tau }_{M_i}$ induced by $M_i$.

Proof. 

Showing that M is a fuzzy $sb$-metric on X can be done repeating the proof of Proposition 2.2 in [12]. Let $(Y,N,\ast ,k^{\prime })$ be a fuzzy $sb$-metric space and $f_i:(Y,N)\to (X_i,M_i)$ be a continuous function for every $i=1,\cdots ,n$. It is clear that, $f:(Y,N)\to (X,M)$ defined by $f\left(y\right)=(f_1\left(y\right),\cdots ,f_n\left(y\right))$ is a continuous function such that $f_i=p_i\circ f$ where $p_i:(X,M)\to (X_i,M_i)$ is the projection. Therefore, $(X,M,\ast ,k)$ is the product of the family $\{(X_i,M_i,\ast ,k_i):i=1\cdots ,n\}$ in the category *-Fsb-Metr. Now, we show that projections are continuous. Let $a\in X$ and $B_{M_i}(a_i,r,t)$ be given for some $r\in (0,1)$ and $t>0$. Consider the corresponding ball $B_M(a,r,t)$ in the product space. Since * is monotone and 1 is its neutral element, for any $y\in B_M(a,r,t)$, we have

$$\begin{array}{cc}\hfill M(a,y,t)& >1-r\hfill \\ \hfill M_1(a_1,y_1,t)\ast \cdots \ast M_i(a_i,y_i,t)\ast \cdots \ast M_n(a_n,y_n,t)& >1-r\hfill \\ \hfill M_i(a_i,y_i,t)& >1-r\hfill \end{array}$$

and $y_i\in B_{M_i}(a_i,r,t)$ and this means that $p_i$ is continuous. Therefore, the topology ${\tau }_M$ induced by M is finer than the product of the topologies ${\tau }_{M_i}$ induced by $M_i$. On the other hand, let $U\subset X$ be open in ${\tau }_M$ and $a\in U$. Then there exists $r\in (0,1)$ and $t>0$ such that $B_M(a,r,t)\subset U$. We fix $s_1,s^{\prime }\in (0,1)$ such that $s_1>s^{\prime }>1-r$. Then there exists $s_2\in (0,1)$ such that $s_1\ast s_2\ge s^{\prime }>1-r$. Continuing in this way we can find $s_1,s_2,\cdots ,s_n$ in $(0,1)$ such that $s_1\ast s_2\ast \cdots \ast s_n\ge s^{\prime }>1-r$. Now consider the ball $B_{M_i}(a_i,r_i,t)$, where $r_i=1-s_i$ for each i. Then for $x\in {\bigcap }_{i=1}^n{p}_i^{-1}\left(B_{M_i}(a_i,r_i,t)\right)$ we have

$$\begin{array}{cc}\hfill M(a,x,t)& =M_1(a_1,x_1,t)\ast \cdots \ast M_n(a_n,x_n,t)\hfill \\ \hfill & >(1-r_1)\ast \cdots \ast (1-r_n)\hfill \\ \hfill & >s_1\ast \cdots \ast s_n\hfill \\ \hfill & >1-r.\hfill \end{array}$$

This means that

$$a\in \bigcap _{i=1}^n{p}_i^{-1}\left(B_{M_i}(a_i,r_i,t)\right)\subset B_M(a,r,t)\subset U$$

and therefore U is an open set in the product topology. □

Coming to the problem of countable products for fuzzy $sb$-metric spaces, we restrict to the case when the constant k is fixed. Besides, in order to provide correctness of the definition of the fuzzy $sb$-metric space, we restrict to the cases when $\ast =·$ or $\ast =\wedge$.

Let $\{(X_i,M_i,·,k):i\in \mathbb{N}\}$ be a countable family of fuzzy $sb$-metric spaces and $\{(X_i,N_i,·,k):i\in \mathbb{N}\}$ be the family of corresponding F-bounded fuzzy $sb$-metric spaces where $N_i(x,y,t)=max\left\{M_i(x_i,y_i,t),1-{\displaystyle \frac{1}{2^i}}\right\}$ (see Proposition 2). We define $X={\prod }_{i\in \mathbb{N}}{X}_i$ and $M:X\times X\times (0,\infty )$ by

$$M(x,y,t)=\prod _{i\in \mathbb{N}}{N}_i(x_i,y_i,t)$$

where $t>0,x,y\in X$ and $x_i,y_i$ are $i^{th}$ coordinates of x and y respectively.

Theorem 8.

$(X,M,·,k)$ is the product of the family $\{(X_i,N_i,·,k):i\in \mathbb{N}\}$ in the category ·-Fsbk-Metr.Moreover, the topology ${\tau }_M$ induced by M coincides with the product of the topologies ${\tau }_{N_i}$ induced by $N_i$.

Proof. 

To make the definition of M meaningful, first we need to show that the infinite product ${\prod }_{i\in \mathbb{N}}{N}_i(x_i,y_i,t)$ is convergent. Since $1-\frac{1}{2^i}\le N_i(x_i,y_i,t)\le 1$ the sequence of partial products ${\prod }_{i=1}^n{N}_i(x_i,y_i,t)$ is decreasing and bounded. Therefore, it converges to the limit ${\prod }_{i\in \mathbb{N}}{N}_i(x_i,y_i,t)$ and hence the product metric is defined correctly.

The validity of axioms $\left(f{m}_1\right)\left(f{m}_2\right),\left(f{m}_3\right),\left(f{m}_4\right)$ is obvious. We prove $\left(fsb{m}_5\right)$ for M as follows. Let $x,y,z\in X$, then

$$\begin{array}{cc}\hfill M(x,z,t+k·s)& =\prod _{i\in \mathbb{N}}{N}_i(x_i,z_i,t+k·s)\ge \prod _{i\in \mathbb{N}}\left(N_i(x_i,y_i,t)·N_i(y_i,z_i,s)\right)\hfill \\ \hfill & \ge \left(\prod _{i\in \mathbb{N}}(N_i(x_i,y_i,t)\right)·\left(\prod _{i\in \mathbb{N}}(N_i(y_i,z_i,s)\right)\hfill \\ \hfill & \ge M(x,y,t)·M(y,z,s).\hfill \end{array}$$

Let $(Y,N,·,k)$ be a fuzzy $sb$-metric space and $f_i:(Y,N)\to (X_i,N_i)$ be a continuous function for every $i\in \mathbb{N}$. It is clear that $f:(Y,N)\to (X,M)$ defined by $f\left(y\right)={(f_i\left(y\right)\in X_i)}_{i\in \mathbb{N}}$ is a continuous function such that $f_i=p_i\circ f$ where $p_i:(X,M)\to (X_i,N_i)$ is the projection. Therefore, $(X,M,·,k)$ is the product of the family $\{(X_i,N_i,·,k):i\in \mathbb{N}\}$ in the category ·-Fsbk-Metr. Now, we show that projections are continuous. Let $a\in X$ and $B_{N_i}(a_i,r,t)$ be given for $r\in (0,1)$ and $t>0$. Consider $B_M(a,r,t)$. Since the product · is monotone and 1 is its neutral element, for any $y\in B_M(a,r,t)$, we have

$$\begin{array}{cc}\hfill M(a,y,t)& >1-r\Rightarrow \prod _{i\in \mathbb{N}}{N}_i(a_i,y_i,t)>1-r\Rightarrow N_i(a_i,y_i,t)>1-r\hfill \end{array}$$

and $y_i\in B_{N_i}(a_i,r,t)$ and this means that $p_i$ is continuous. Therefore, the topology ${\tau }_M$ induced by M is finer than the product of the topologies ${\tau }_{N_i}$ induced by $N_i$. On the other hand, let $U\subset X$ be open in ${\tau }_M$ and $a\in U$. Then there exist $r\in (0,1)$ and $t>0$ such that $B_M(a,r,t)\subset U$. We can find $n\in \mathbb{N}$ such that ${\prod }_{i=n+1}^{\infty }\left(1-{\displaystyle \frac{1}{2^i}}\right)=s^{\prime }>1-r$ since ${lim}_{n\to \infty }{\prod }_{i=n+1}^{\infty }\left(1-{\displaystyle \frac{1}{2^i}}\right)=1$. We fix $s^{″}\in (0,1)$ such that $s^{\prime }>s^{″}>1-r$. In addition, there exist $s_1,s_2,\dots ,s_n\in (0,1)$ such that $s_1·s_2\cdots s_n·s^{\prime }\ge s^{″}>1-r$. For all $i\le n$, consider the balls $B_{N_i}(a_i,r_i,t)$ where $r_i=1-s_i$. Then for $x\in {\bigcap }_{i=1}^n{p}_i^{-1}\left(B_{N_i}(a_i,r_i,t)\right)$ we have

$$\begin{array}{cc}\hfill M(a,x,t)& =\prod _{i=1}^n{N}_i(a_i,x_i,t)·\prod _{i=n+1}^{\infty }{N}_i(a_i,x_i,t)\hfill \\ \hfill & >\prod _{i=1}^n(1-r_i)·\prod _{i=n+1}^{\infty }\left(1-{\displaystyle \frac{1}{2^i}}\right)\hfill \\ \hfill & >s_1·s_2\cdots s_n·s^{\prime }\hfill \\ \hfill & >1-r.\hfill \end{array}$$

This means that

$$a\in \bigcap _{i=1}^n{p}_i^{-1}\left(B_{N_i}(a_i,r_i,t)\right)\subset B_M(a,r,t)\subset U$$

and therefore U is an open set in the product topology. □

Corollary 6.

$(X,M,·,k)$ is the product of the family $\{(X_i,M_i,·,k):i\in \mathbb{N}\}$ in the category ·-Fsbk-Metr.

Coming to the minimum t-norm case let $\{(X_i,M_i,\wedge ,k):i\in \mathbb{N}\}$ be a countable family of fuzzy $sb$-metric spaces and $\{(X_i,N_i,\wedge ,k):i\in \mathbb{N}\}$ be the family of corresponding F-bounded fuzzy $sb$-metric spaces where $N_i(x_i,y_i,t)=max\left\{M_i(x_i,y_i,t),1-{\displaystyle \frac{1}{2^i}}\right\}$. We define $X={\prod }_{i\in \mathbb{N}}{X}_i$ and $M:X\times X\times (0,\infty )$ by

$$M(x,y,t)=\underset{i\in \mathbb{N}}{\bigwedge }{N}_i(x_i,y_i,t)$$

where $t>0,x,y\in X$ and $x_i,y_i$ are the $i^{th}$ coordinates of x and y respectively.

Theorem 9.

$(X,M,\wedge ,k)$ is the product of the family $\{(X_i,N_i,\wedge ,k):i\in \mathbb{N}\}$ in the category ∧-Fsbk-Metr. Moreover, the topology ${\tau }_M$ induced by M coincides with the product of the topologies ${\tau }_{N_i}$ induced by $N_i$.

Proof. 

The validity of axioms $\left(f{m}_1\right)\left(f{m}_2\right),\left(f{m}_3\right),\left(f{m}_4\right)$ is obvious. We prove $\left(fsb{m}_5\right)$ for M as follows. Let $x,y,z\in X$, then

$$\begin{array}{cc}\hfill M(x,z,t+k·s)& =\underset{i\in \mathbb{N}}{\bigwedge }{N}_i(x_i,z_i,t+k·s)\ge \underset{i\in \mathbb{N}}{\bigwedge }\left(N_i(x_i,y_i,t)\wedge N_i(y_i,z_i,s)\right)\hfill \\ \hfill & \ge \left(\underset{i\in \mathbb{N}}{\bigwedge }(N_i(x_i,y_i,t)\right)\wedge \left(\underset{i\in \mathbb{N}}{\bigwedge }(N_i(y_i,z_i,s)\right)\hfill \\ \hfill & \ge M(x,y,t)\wedge M(y,z,s).\hfill \end{array}$$

Let $(Y,N,\wedge ,k)$ be a fuzzy $sb$-metric space and $f_i:(Y,N)\to (X_i,N_i)$ be a continuous function for every $i\in \mathbb{N}$. It is clear that the function $f:(Y,N)\to (X,M)$ defined by $f\left(y\right)={\left(f_i\left(y\right)\right)}_{i\in \mathbb{N}}\in X$ is a continuous function such that $f_i=p_i\circ f$ where $p_i:(X,M)\to (X_i,N_i)$ is the projection. Therefore, $(X,M,\wedge ,k)$ is the product of the family $\{(X_i,N_i,\wedge ,k):i\in \mathbb{N}\}$ in the category ∧-Fsbk-Metr. Now, we show that projections are continuous. Let $a\in X$ and $B_{N_i}(a_i,r,t)$ is given for $r\in (0,1)$ and $t>0$. Consider $B_M(a,r,t)$. Since ∧ is monotone and 1 is its neutral element, for any $y\in B_M(a,r,t)$, we have

$$\begin{array}{cc}\hfill M(a,y,t)& >1-r\Rightarrow \underset{i\in \mathbb{N}}{\bigwedge }{N}_i(a_i,y_i,t)>1-r\Rightarrow N_i(a_i,y_i,t)>1-r\hfill \end{array}$$

and $y_i\in B_{N_i}(a_i,r,t)$ and this means that $p_i$ is continuous. Therefore, the topology ${\tau }_M$ induced by M is finer than the product of the topologies ${\tau }_{N_i}$ induced by $N_i$. On the other hand, let $U\subset X$ be open in ${\tau }_M$ and $a\in U$. Then there exists $r\in (0,1)$ and $t>0$ such that $B_M(a,r,t)\subset U$. We can find $n\in \mathbb{N}$ such that $1-{\displaystyle \frac{1}{2^i}}>1-r$ whenever $i>n$. For all $i\le n$, consider the ball $B_{N_i}(a_i,r,t)$. Then for $x\in {\bigcap }_{i=1}^n{p}_i^{-1}\left(B_{N_i}(a_i,r,t)\right)$ we have

$$\begin{array}{cc}\hfill M(a,x,t)& =\underset{i=1}{\overset{n}{\bigwedge }}{N}_i(a_i,x_i,t)\wedge \underset{i=n+1}{\overset{\infty }{\bigwedge }}{N}_i(a_i,x_i,t)\hfill \\ \hfill & >\underset{i=1}{\overset{n}{\bigwedge }}(1-r)\wedge \underset{i=n+1}{\overset{\infty }{\bigwedge }}(1-{\displaystyle \frac{\lambda }{2^i}})\hfill \\ \hfill & >\underset{i\in \mathbb{N}}{\bigwedge }(1-r)\hfill \\ \hfill & >1-r.\hfill \end{array}$$

This means that

$$a\in \bigcap _{i=1}^n{p}_i^{-1}\left(B_{N_i}(a_i,r,t)\right)\subset B_M(a,r,t)\subset U$$

and therefore U is an open set in the product topology. □

Corollary 7.

$(X,M,\wedge ,k)$ is the product of the family $\{(X_i,M_i,\wedge ,k):i\in \mathbb{N}\}$ in the category ∧-Fsbk-Metr.

Let $\{(X_i,M_i,\ast ,k):i\in I\}$ be an arbitrary family of fuzzy $sb$-metric spaces and $\{(X_i,N_i,\ast ,k):i\in I\}$ be the family of corresponding F-bounded fuzzy $sb$-metric spaces where $N_i(x,y,t)=max\{M_i(x,y,t),\lambda \}$ where $\lambda \in (0,1)$. We define $X={\coprod }_{i\in I}{X}_i$ and $M:X\times X\times (0,\infty )$ by

$$M(x,y,t)=\{\begin{array}{cc}{N}_i(x,y,t)& ,x,y\in X_i\hfill \\ \lambda & ,otherwise\hfill \end{array}$$

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

Theorem 10.

$(X,M,\ast ,k)$ is the coproduct of the family $\{(X_i,N_i,\ast ,k):i\in I\}$ in the category*-Fsbk-Metr. Moreover, the topology ${\tau }_M$ on X induced by M coincides with the coproduct (direct sum) of the topologies ${\tau }_{N_i}$ on $X_i$ induced by $N_i$.

Proof. 

The validity of axioms $\left(f{m}_1\right)\left(f{m}_2\right),\left(f{m}_3\right),\left(f{m}_4\right)$ is obvious for M. For $\left(fsb{m}_5\right)$, we distinguish 3 cases.

Case 1: If $x,y,z\in X_i$, then it is obvious.

Case 2: If $x,y\in X_i$ and $z\in X_j,i\ne j$, then

$$\begin{array}{cc}\hfill M(x,z,t+k·s)=\lambda & \ge N_i(x,y,t)\ast \lambda \hfill \\ \hfill & \ge M(x,y,t)\ast M(y,z,s).\hfill \end{array}$$

Case 3: If $x,z\in X_i$ and $y\in X_j,i\ne j$, then

$$\begin{array}{cc}\hfill M(x,z,t+k·s)=N_i(x,z,t+k·s)\ge \lambda & \ge \lambda \ast \lambda \hfill \\ \hfill & \ge M(x,y,t)\ast M(y,z,s).\hfill \end{array}$$

Therefore, M is a fuzzy $sb$-metric on X. For all $i\in I$, it is obvious that the inclusion mapping $q_i:(X_i,N_i)\to (X,M)$ is continuous. Let $(Y,M^{\prime },\ast ,k)$ be a fuzzy $sb$-metric space and $f_i:(X_i,N_i)\to (Y,M^{\prime })$ be continuous function for all $i\in I$. By setting $f\left(x\right)=f_i\left(x\right)$ if $x\in X_i$, we obtain a continuous function $f:(X,M)\to (Y,M^{\prime })$ such that $f\circ q_i=f_i$. Hence, $(X,M,\ast ,k)$ is the coproduct of the family $\{(X_i,N_i,\ast ,k):i\in I\}$ in the category *-Fsbk-Metr. Finally, we show the equivalence of the topologies. It is easy to see that $X_i\in {\tau }_M$ and ${\tau }_M{|}_{X_i}={\tau }_{N_i}$ since ${M|}_{X_i}=N_i$ for all $i\in I$. Therefore, X can be represented as the union of a family of pairwise disjoint open subsets. By Proposition 2.2.4 in [33], the topology ${\tau }_M$ on X induced by M coincides with the coproduct of the topologies ${\tau }_{N_i}$ on $X_i$ induced by $N_i$. □

Corollary 8.

$(X,M,\ast ,k)$ is the coproduct of the family $\{(X_i,M_i,\ast ,k):i\in I\}$ in the category*-Fsbk-Metr.

6. Conclusions

The concept of a fuzzy $sb$-metric as a strengthening of the concept of a fuzzy b-metric on one side and as a fuzzy version of the notion of a strong b-metric was introduced in Reference [11,12]. In this work, we further develop the study of fuzzy $sb$-metrics. There are five main issues considered in the paper. We study continuity of a fuzzy $sb$-metric $M(x,y,-)$ and prove metrizability of the topology induced by a fuzzy $sb$-metric, investigate diameter zero sets in fuzzy $sb$-metric spaces and use them for characterization of completeness of fuzzy $sb$-metric spaces. Some examples showing that the class of fuzzy $sb$-metric spaces lies strictly between the classes of fuzzy b-metric spaces and fuzzy metric spaces are provided. In the last section we turn to the constructions of products and coproducts of fuzzy $sb$-metric spaces.

Concerning the further development of the research in the area of fuzzy and crisp $sb$-metrics we have vision of both theoretical aspects to be explored and possible practical applications. In the theoretical direction of the research of fuzzy metric spaces we distinguish the following two. First, to develop further the study of categorical properties of fuzzy $sb$-metric spaces. In this paper we have touched only the problem of products and coproducts in the category of fuzzy $sb$-metric spaces. It is interesting to study further the inner properties of this category, such as the existence of initial and final structures, special objects in this category, etc., as well as to investigate deeper the connections of this category with other related categories, such as fuzzy metric and fuzzy b-metric spaces from one side and with the category of cirsp $sb$-metric spaces from the other. The second direction which is interesting from the theoretical point of view and can be important also in studying practical problems is the issue of fixed point property for mappings of such spaces. There are many published works developing methods for extending fixed point properties for continuous mappings from the “classical” crisp case to the case of fuzzy metric spaces. On the other hand, similar extension of most of such methods to the case of fuzzy b-metric spaces seems problematic because of the peculiarity of the induced topology of such spaces (“open” balls need not be open in the induced fuzzy topology, see comments in front of Remark 5 in our paper). We see some prospects for extending these methods to the case of mappings of fuzzy $sb$-metric spaces that are free from this disadvantage.

Another area where fuzzy and crisp $sb$-metrics could be helpful is image processing. Noticing that the full power of a metric in order to describe distance between the images in the problems of pattern matching is not needed and in some situations may be also onerous, Fagin et al. [4] turned to the use of special b-metrics (called nonlinear elastic matching in this paper) for measuring distance between sequences. We foresee, that (fuzzy) $sb$-metrics, being a special kind of (fuzzy) b-metrics, can also be useful in improving methods for color image filtering and pattern matching.