On statistical convergence in fuzzy metric spaces

1 Introduction

Many authors have defined several concepts of fuzzy metric space in different ways [7, 34]. In particular, to make the topology induced by a fuzzy metric to be Hausdorff, George and Veeramani [13] introduced the concept of fuzzy metric space with the help of continuous t-norms. Whereafter, Gregori and Romaguera [29] proved that the topological space induced by a fuzzy metric is metrizable. This version of fuzzy metric determines that are tightly connected with the class of metrizable topological spaces. Furthermore, some well-known results in metric spaces are adapted to the realm of the version of fuzzy metric space. So it is interesting to explore the version of fuzzy metric [1, 46]. Recently, several convergences in fuzzy metric spaces, as p-convergence, s-convergence, st-convergence and std-convergence, were studied by Gregori et al. [16, 22– 24]. In addition, they also gave much progress to the study of completeness of fuzzy metric spaces [17, 31].

The concept of statistical convergence in real spaces was initially defined by Fast [8] and Steinhaus [43] in the same year 1951, respectively. In [9], Fridy gave further research to this topic. Later, some authors became interested in the topic and gave various applications in different fields of mathematics [4, 40].

In [33], Kostyrko et al. first gave the concept of the statistical convergence in metric spaces. Subsequently, Bilalov and Nazarova [2] studied equivalence of the statistical convergence in complete metric spaces. It is well known that the completeness of fuzzy metrics is very different from that of metrics. Indeed, there exist fuzzy metric spaces which are non-completable [30]. This shows that the theory of fuzzy metrics seem to be richer than that of metrics. It is a nature problem to generalize the concept of statistical convergence to the realm of fuzzy metrics. In this paper we do it. We will study statistically convergent sequences, statistically Cauchy sequences and statistically completeness in fuzzy metric spaces. We show that convergent (Cauchy) sequences are all statistically convergent (statistically Cauchy) in a fuzzy metric space, but the converse is not true. At the same time we prove that statistically completeness implies completeness in a fuzzy metric space. Furthermore, we give several examples.

The rest of the paper is organized as follows. In section 2 preliminaries are provided. In section 3 the concept of statistical convergence in fuzzy metric spaces is introduced and studied. In section 4 statistical completeness of fuzzy metric spaces is explored. At the end, conclusions are given in section 5.

2 Preliminaries

From now on, ℕ and ℝ shall denote the set of all positive integer numbers and the set of all real numbers, respectively. Our basic reference for general topology is [6].

Definition 2.1. [13] A binary operation * : [0, 1] × [0, 1] → [0, 1] is a continuous t-norm if it satisfies the following conditions:

* is associative and commutative;

Notice that a * b = min {a, b} and a * b = a · b are two common examples of continuous t-norms.

Obviously, Definition 2.1 shows that, if 1 ≥ r > s ≥ 0, then there exists a δ ∈ (s, 1) such that r * δ > s.

Definition 2.2. [13] An ordered triple (X, M, *) is said to be a fuzzy metric space if X is a nonempty set, * is a continuous t-norm and M is a fuzzy set on X × X × (0, ∞) satisfying the following conditions for all x, y, z ∈ X and s, t ∈ (0, ∞):

M (x, y, t) >0;

If (X, M, *) is a fuzzy metric space, then we will call (M, *), or simply M, a fuzzy metric on X.

Definition 2.3. [13] Let (X, M, *) be a fuzzy metric space and let r ∈ (0, 1) , t > 0 and x ∈ X. The set B M ( x , r , t ) = { y ∈ X | M ( x , y , t ) > 1 - r } is called the open ball with center x and radius r with respect to t.

George and Veeramani [13] proved that {B _M  (x, r, t) |x ∈ X, t > 0, r ∈ (0, 1)} forms a base of a topology τ _M in X.

Let us recall [13] that a sequence in a fuzzy metric space (X, M, *) is said to be convergent if it is convergent with respect to τ _M . And the next characterization of convergent sequences was also provided in [13].

Proposition 2.4. Let (X, M, *) be a fuzzy metric space. A sequence {x _n } in X converges to x₀ ∈ X if and only if lim n → ∞ M ( x n , x 0 , t ) = 1 for all t > 0.

It is easy to see that {x _n } converges to x₀ which is equivalent to the following:

For each r ∈ (0, 1) and each t > 0, there exists n 0 ∈ ℕ such that M (x _n , x₀, t) >1 - r for all n > n₀.

Definition 2.5. [13] Let (X, M, *) be a fuzzy metric space. A sequence {x _n } in X is called Cauchy if for each r ∈ (0, 1) and each t > 0, there exists n 0 ∈ ℕ such that M (x _n , x _m , t) >1 - r for all n, m ≥ n₀. (X, M, *) (or simply, X) is called complete if every Cauchy sequence in X is convergent with respect to τ _M .

Definition 2.6. [30] Let (X₁, M₁, * ₁) and (X₂, M₂, * ₂) be two fuzzy metric spaces.

A mapping f : X₁ → X₂ is called an isometry if for every x, y ∈ X₁ and t > 0, M₁ (x, y, t) = M₂ (f (x) , f (y) , t).

In the following, for a subset B of a set X, |B| will denote the cardinality of B.

Definition 2.7. [8, 43] Let A ⊆ ℕ . For any n ∈ ℕ , put A (n) = {k ≤ n|k ∈ A}. Define δ (A) by δ ( A ) = lim n → ∞ | A ( n ) | n and call it an asymptotic (or natural) density of A. It is straightforward to show that δ (A) ∈ [0, 1] and δ ( ℕ - A ) = 1 - δ ( A ) if δ (A) exists. A is said to be statistically dense provided that δ (A) =1.

Definition 2.8. [8, 43] A sequence {a _n } in ℝ is said to be statistically convergent to a 0 ∈ ℝ if δ ( { n ∈ ℕ | | a n - a 0 | < ɛ } ) = 1 for each ɛ > 0.

3 Statistical convergence in fuzzy metric spaces

In this section we investigate statistically convergent sequences in fuzzy metric spaces.

Definition 3.1. Let (X, M, *) be a fuzzy metric space. A sequence {x _n } in X is said to be statistically convergent to x₀ ∈ X if δ ( { n ∈ ℕ | M ( x n , x 0 , t ) > 1 - r } ) = 1 for all r ∈ (0, 1) and t > 0.

In such a case we say that {x _n } is sts-convergent to x₀ (or, {x _n } sts-converges to x₀). It is obvious that

δ ( { n ∈ ℕ | M ( x n , x 0 , t ) > 1 - r } ) = 1 ⇔

lim n → ∞ | { k ≤ n | M ( x k , x 0 , t ) > 1 - r } | n = 1 .

Theorem 3.2. Let {x _n } be a sequence in a fuzzy metric space (X, M, *). If {x _n } is convergent to x₀, then {x _n } is sts-convergent to x₀.

Proof. Suppose that {x _n } is convergent to x₀. Let r ∈ (0, 1) and t > 0. Then there exists n 0 ∈ ℕ such that M (x _n , x₀, t) >1 - r for all n > n₀. Hence | {k ≤ n|M (x _n , x₀, t) >1 - r} | ≥ n - n₀. It follows that

lim n → ∞ | { k ≤ n | M ( x k , x 0 , t ) > 1 - r } | n

≥ lim n → ∞ n - n 0 n = 1 .

Therefore δ ( { n ∈ ℕ | M ( x n , x 0 , t ) > 1 - r } ) = 1 . We complete the proof.□

Now we shall show that the converse of the preceding theorem is false, in general.

Example 3.3. Let X = [1, 3]. Denote a * b = a · b for all a, b ∈ [0, 1]. Define M by M ( x , y , t ) = t t + | x - y | for all x, y ∈ X and t > 0. Then (X, M, *) is a fuzzy metric space (see [13]). Now, take the sequence {x _n } in X, where

x n = { 2 , n = m ( m + 3 ) 2 , m ∈ ℕ , 1 , otherwise .

It is immediate to see that {x _n } is not convergent to 1.

We claim that {x _n } is sts-convergent to 1. In fact, let r ∈ (0, 1) and t > 0. Put A = { n ∈ ℕ | M ( x n , 1 , t ) > 1 - r } . In case r ∈ ( 1 t + 1 , 1 ) . We obtain that M (x _n , 1, t) =1 > 1 - r if n ≠ m ( m + 3 ) 2 for all m ∈ ℕ and M ( x n , 1 , t ) = t t + 1 = 1 - 1 t + 1 > 1 - r if n = m ( m + 3 ) 2 for some m ∈ ℕ . It follows that M (x _n , 1, t) >1 - r for all n ∈ ℕ . Hence lim n → ∞ | A ( n ) | n = lim n → ∞ n n = 1 . In case r ∈ ( 0 , 1 t + 1 ] . We get that M (x _n , 1, t) =1 > 1 - r if n ≠ m ( m + 3 ) 2 for all m ∈ ℕ and M ( x n , 1 , t ) = t t + 1 = 1 - 1 t + 1 ≤ 1 - r if n = m ( m + 3 ) 2 for some m ∈ ℕ . Now, take n ∈ ℕ . If n = m 0 ( m 0 + 3 ) 2 for an m 0 ∈ ℕ , then

lim n → ∞ | A ( n ) | n = lim m 0 → ∞ m 0 ( m 0 + 3 ) 2 - m 0 m 0 ( m 0 + 3 ) 2

= lim m 0 → ∞ m 0 + 1 m 0 + 3

= lim m 0 → ∞ 1 + 1 m 0 1 + 3 m 0 = 1 .

If n ≠ m ( m + 3 ) 2 for all m ∈ ℕ , then we can find an m 1 ∈ ℕ such that n = m 1 ( m 1 + 3 ) 2 - i with i ∈ ℕ and 1 ≤ i ≤ m₁. Note that 1 m 1 2 ≤ i m 1 2 ≤ 1 m 1 . Hence lim m 1 → ∞ i m 1 2 = 0 . It follows that

lim n → ∞ | A ( n ) | n = lim m 1 → ∞ m 1 ( m 1 + 3 ) 2 - i - ( m 1 - 1 ) m 1 ( m 1 + 3 ) 2 - i

= lim m 1 → ∞ m 1 2 + m 1 - 2 i + 2 m 1 2 + 3 m 1 - 2 i

= lim m 1 → ∞ 1 + 1 m 1 - 2 i m 1 2 + 2 m 1 2 1 + 3 m 1 - 2 i m 1 2 = 1 .

Consequently, δ ( { n ∈ ℕ | M ( x n , 1 , t ) > 1 - r } ) = 1 for all r ∈ (0, 1) and t > 0.

Proposition 3.4. Let {x _n } be a sequence in a fuzzy metric space (X, M, *). If {x _n } is sts-convergent to a and b, then a = b.

Proof. Assume that a ≠ b. Let t > 0. Then M (a, b, t) ∈ (0, 1). Take r₀ ∈ (M (a, b, t) , 1). Then there exists r₁ ∈ (r₀, 1) such that r₁ * r₁ ≥ r₀. We claim that

B M ( a , 1 - r 1 , 1 2 t ) ∩ B M ( b , 1 - r 1 , 1 2 t ) = ∅ .

Indeed, let

c ∈ B M ( a , 1 - r 1 , 1 2 t ) ∩ B M ( b , 1 - r 1 , 1 2 t ) = ∅ .

Then

M ( a , b , t ) ≥ M ( a , c , 1 2 t ) ) * M ( c , b , 1 2 t ) )

≥r₁ * r₁

≥r₀ > M (a, b, t) .

A contradiction occurs. Hence

B M ( b , 1 - r 1 , 1 2 t ) ⊂ { x ∈ X | M ( x , a , 1 2 t ) ≤ r 1 } .

Therefore

{ k ≤ n | M ( x k , b , 1 2 t ) > r 1 }

⊆ { k ≤ n | M ( x k , a , 1 2 t ) ≤ r 1 } .

Observe that

lim n → ∞ | { k ≤ n | M ( x k , a , 1 2 t ) ≤ r 1 } | n

= 1 - lim n → ∞ | { k ≤ n | M ( x k , a , 1 2 t ) > r 1 } | n

= 1 -1 = 0 .

We deduce that

1 = lim n → ∞ | { k ≤ n | M ( x k , b , 1 2 t ) > r 1 } | n

≤ lim n → ∞ | { k ≤ n | M ( x k , a , 1 2 t ) ≤ r 1 } | n = 0 ,

which is a contradiction. We are done.□

With Theorem 3.2 and Proposition 3.4, we obtain the next corollary immediately.

Corollary 3.5. Let {x _n } be a sequence in a fuzzy metric space (X, M, *). If {x _n } sts-converges to x₀ and it is convergent, then {x _n } converges to x₀.

Corollary 3.6. Let {x _n } be a sequence in a completable fuzzy metric space (X, M, *). If {x _n } is a Cauchy sequence in X and it sts-converges to x₀ ∈ X, then {x _n } converges to x₀.

Proof. Let (X₁, M₁, * ₁) be the completion of (X, M, *). Then there exists x₁ ∈ X₁ such that {x _n } converges to x₁. Notice that M₁ (x _n , x₀, t) = M (x _n , x₀, t) for all t > 0 and n ∈ ℕ . Now, let r ∈ (0, 1) and t > 0. Since δ ( { n ∈ ℕ | M ( x n , x 0 , t ) > 1 - r } ) = 1 , we conclude that δ ( { n ∈ ℕ | M 1 ( x n , x 0 , t ) > 1 - r } ) = 1 , which means that {x _n } sts-converges to x₀ ∈ X₁ with respect to M₁. It follows from Corollary 3.5 that x₁ = x₀.□

Remark 3.7. Given a subsequence {x _{n i} } of a sequence {x _n } in a fuzzy metric space (X, M, *). It is well known that, if {x _n } is convergent to x₀ ∈ X, then {x _{n i} } is also convergent to x₀. However, if {x _n } is sts-convergent to x₀ ∈ X, then {x _{n i} } need not be sts-convergent to x₀. The next example illustrates this fact.

Example 3.8. Consider Example 3.3. As we have just seen, {x _n } sts-converges to 1. Now, take the subsequence {x _{n i} } of {x _n }, where n i = i ( i + 3 ) 2 ( i ∈ ℕ ) . Note that {x _{n i} } is a constant sequence. We get that {x _{n i} } converges to 2. Due to Theorem 3.2, we conclude that {x _{n i} } sts-converges to 2. It follows from Proposition 3.4 that {x _{n i} } is not sts-convergent to 1.

The next theorem will be useful in obtaining our succeeding results.

Theorem 3.9. Let {x _n } be a sequence in a fuzzy metric space (X, M, *). Then {x _n } sts-converges to x₀ if and only if there exists an increasing index sequence {n _i } of the positive integer numbers such that {x _{n i} } converges to x₀ and δ (A) =1, where A = { n i | i ∈ ℕ } .

Proof. Suppose that {x _n } sts-converges to x₀. Let A j = { ℕ , j = 1 , { n ∈ ℕ | M ( x n , x 0 , 1 j ) > 1 - 1 j } , j ≥ 2 , where j ∈ ℕ . Then, for each j ∈ ℕ , we have that δ (A _j ) =1 and A_j+1 ⊆ A _j .

Now, take n 1 ′ ∈ A 1 with n 1 ′ > 1 . Since δ (A₂) =1, we can choose n 2 ′ ∈ A 2 with n 2 ′ > n 1 ′ such that | { k ≤ n | M ( x k , x 0 , 1 2 ) > 1 2 } | n > 1 2 whenever n ≥ n 2 ′ . Further, since δ (A₃) =1, we can choose n 3 ′ ∈ A 3 with n 3 ′ > n 2 ′ such that | { k ≤ n | M ( x k , x 0 , 1 3 ) > 2 3 } | n > 2 3 whenever n ≥ n 3 ′ , and so on. Thus, we can construct, by induction, an increasing index sequence { n j ′ } of the positive integer numbers such that n j ′ ∈ A j for every j ∈ ℕ and the following statement holds: | { k ≤ n | M ( x k , x 0 , 1 j + 1 ) > 1 - 1 j + 1 } | n > 1 - 1 j + 1 whenever n ≥ n j + 1 ′ . Put A = { n ∈ ℕ | 1 ≤ n < n 1 ′ } ∪ ( ⋃ { n ∈ A j | n j ′ ≤ n < n j + 1 ′ , j ∈ ℕ } ) . Then we can rewrite A = { n i | i ∈ ℕ } such that {n _i } is an increasing index sequence of the positive integer numbers. Next, we will show that δ (A) =1. Take n > n 2 ′ . Then there exists j 0 ∈ ℕ such that n j 0 ′ ≤ n < n j 0 + 1 ′ . Hence

| { k ≤ n | k ∈ A } | n ≥ | { k ≤ n | k ∈ A j 0 } | n

= | { k ≤ n | M ( x k , x 0 , 1 j 0 ) > 1 - 1 j 0 } | n

> 1 - 1 j 0 .

Since j₀→ ∞ when n→ ∞, we get that

lim n → ∞ | { k ≤ n | k ∈ A } | n = 1 , i.e., δ (A) =1. Finally, we need to show that {x _{n i} } converges to x₀. Let r ∈ (0, 1) and t > 0. Then we can take N 0 > n 2 ′ large enough that, for some l 0 ∈ ℕ , n l 0 ′ ≤ N 0 < n l 0 + 1 ′ with 1 l 0 < t and 1 - 1 l 0 > 1 - r . Let n _m  ≥ N₀ with n _m  ∈ A. Then there exists l ∈ ℕ such that n l ′ ≤ n m < n l + 1 ′ with n _m  ∈ A _l . Note that l ≥ l₀. Then we deduce that

M ( x n m , x 0 , t ) ≥ M ( x n m , x 0 , 1 l 0 )

≥ M ( x n m , x 0 , 1 l )

> 1 - 1 l

≥ 1 - 1 l 0 > 1 - r .

So {x _{n i} } converges to x₀.

Conversely, suppose that there exists an increasing index sequence {n _i } of the positive integer numbers such that {x _{n i} } converges to x₀ and δ (A) =1, where A = { n i | i ∈ ℕ } . Let r ∈ (0, 1) and t > 0. Then there exists N 1 ∈ ℕ such that M (x _{n i} , x₀, t) >1 - r for all n _i  ≥ N₁. Therefore {n _i  ∈ A|M (x _{n i} , x₀, t) ≤1 - r} is a finite set, which implies that δ ({n _i  ∈ A|M (x _{n i} , x₀, t) ≤1 - r}) =0. Since δ (A) =1, we deduce that δ ({n _i  ∈ A|M (x _{n i} , x₀, t) >1 - r}) =1. Note that

{ n ∈ ℕ | M ( x n , x 0 , t ) > 1 - r }

⊇ {n _i  ∈ A|M (x _{n i} , x₀, t) >1 - r} .

We conclude that δ ( { n ∈ ℕ | M ( x n , x 0 , t ) > 1 - r } ) = 1 . The proof is finished.□

4 Statistically complete fuzzy metric spaces

We start this section with the following definition.

Definition 4.1. Let (X, M, *) be a fuzzy metric space. A sequence {x _n } in X is called a statistically Cauchy sequence, if for every r ∈ (0, 1) and t > 0, there exists N 0 ∈ ℕ such that δ ( { n ∈ ℕ | M ( x n , x N 0 , t ) > 1 - r } ) = 1 .

In such a case we say that {x _n } is sts-Cauchy.

Theorem 4.2. Let {x _n } be a sequence in a fuzzy metric space (X, M, *). If {x _n } is sts-convergent, then {x _n } is sts-Cauchy.

Proof. Assume that {x _n } is sts-convergent to x₀. Let r ∈ (0, 1) and t > 0. Then there exists r₁ ∈ (0, r) such that (1 - r₁) * (1 - r₁) >1 - r. According to Theorem 3.9, we can find a subsequence {x _{n i} } of {x _n } such that the sequence {x _{n i} } converges to x₀, where {n _i } is an increasing index sequence of the positive integer numbers. Thus there exists n i 0 ∈ { n i | i ∈ ℕ } such that M ( x n i , x 0 , t 2 ) > 1 - r 1 for all n _i  ≥ n _{i 0} . Let l ∈ { n ∈ ℕ | M ( x n , x 0 , t 2 ) > 1 - r 1 } . Then

M ( x l , x n i 0 , t ) ≥ M ( x l , x 0 , t 2 ) * M ( x n i 0 , x 0 , t 2 )

≥(1 - r₁) * (1 - r₁)

>1 - r .

Hence l ∈ { n ∈ ℕ | M ( x n , x n i 0 , t ) > 1 - r } . It follows that

{ n ∈ ℕ | M ( x n , x 0 , t 2 ) > 1 - r 1 }

⊆ { n ∈ ℕ | M ( x n , x n i 0 , t ) > 1 - r } .

Since δ ( { n ∈ ℕ | M ( x n , x 0 , t 2 ) > 1 - r 1 } ) = 1 , we get that

δ ( { n ∈ ℕ | M ( x n , x n i 0 , t ) > 1 - r } )

≥ δ ( { n ∈ ℕ | M ( x n , x 0 , t 2 ) > 1 - r 1 } ) = 1 .

So {x _n } is sts-Cauchy.□

The converse of the above theorem is not true. We illustrate this fact with the next example.

Example 4.3. Let X = { x n | n ∈ ℕ } , where x n = 1 - 1 n + 1 ( n ∈ ℕ ) . Denote a * b = min {a, b} for all a, b ∈ [0, 1] and let M be a fuzzy set on X × X × (0, ∞) defined as follows: M ( x , y , t ) = { 1 , x = y , min { x , y } , x ≠ y , for all x, y ∈ X and t > 0. Then (X, M, *) is a fuzzy metric space (see [16]). Now, consider the sequence {x _n } in (X, M, *). Then {x _n } is sts-Cauchy, but it is not sts-convergent.

Indeed, let r ∈ (0, 1) and t > 0. Then there exists m 0 ∈ ℕ such that 1 m 0 + 1 < r . Hence M ( x n , x m 0 , t ) = x m 0 = 1 - 1 m 0 + 1 > 1 - r for all n > m₀. Therefore | { k ≤ n | M ( x k , x m 0 , t ) > 1 - r } | > n - m 0 for all n > m₀. Thus

δ ( { n ∈ ℕ | M ( x n , x m 0 , t ) > 1 - r } )

≥ lim n → ∞ n - m 0 n = 1 ,

which means that {x _n } is sts-Cauchy.

Let x ∈ X. Then there exists n 0 ∈ ℕ such that x = x n 0 = 1 - 1 n 0 + 1 . Now, fix t 0 = r 0 = 1 2 ( n 0 + 1 ) . Then

M (x _n , x, t₀) = M (x _n , x _{n 0} , t₀)

= x _{n 0}

= 1 - 1 n 0 + 1

≤1 - r₀

for all n > n₀. Hence

δ ( { n ∈ ℕ | M ( x n , x , t 0 ) ≤ 1 - r 0 } ) = 1 ,

which implies that

δ ( { n ∈ ℕ | M ( x n , x , t 0 ) > 1 - r 0 } ) = 0 .

So {x _n } is not sts-convergent.

Remark 4.4. It is not hard to verify that, if a sequence in a fuzzy metric space is Cauchy, then it is sts-Cauchy. However, the converse is false.

The proof of the next theorem is similar to Theorem 3.9.

Theorem 4.5. Let {x _n } be a sequence in a fuzzy metric space (X, M, *). Then {x _n } is sts-Cauchy if and only if there exists an increasing index sequence {n _i } of the positive integer numbers such that {x _{n i} } is Cauchy and δ (A) =1, where A = { n i | i ∈ ℕ } .

Definition 4.6. The fuzzy metric space (X, M, *) is called statistically complete if every statistically Cauchy sequence in X is statistically convergent.

In such a case we say that (X, M, *) (or simply, X) is sts-complete.

Theorem 4.7. Let (X, M, *) be a fuzzy metric space. If X is sts-complete, then it is complete.

Proof. Suppose that {x _n } is a Cauchy sequence in X. Let r ∈ (0, 1) and t > 0. Then there exists r₁ ∈ (0, r) such that (1 - r₁) * (1 - r₁) >1 - r. Therefore there exists N 0 ∈ ℕ such that M ( x n , x m , t 2 ) > 1 - r 1 for all n, m ≥ N₀. Due to Remark 4.4, we obtain that {x _n } is sts-Cauchy. Since X is sts-complete, we conclude that {x _n } is sts-convergent to a point x₀ ∈ X. Thus, By Theorem 3.9, there exists a subsequence {x _{n i} } of {x _n } such that the sequence {x _{n i} } converges to x₀, where {n _i } is an increasing index sequence of the positive integer numbers. So there exists n i 0 ∈ { n i | i ∈ ℕ } with n _{i 0}  ≥ N₀ such that M ( x n i , x 0 , t 2 ) > 1 - r 1 for all n _i  ≥ n _{i 0} . Hence

M ( x n , x 0 , t ) ≥ M ( x n , n i 0 , t 2 ) * M ( x n i 0 , x 0 , t 2 )

≥(1 - r₁) * (1 - r₁)

>1 - r

for all n ≥ n _{i 0}  ≥ N₀. So {x _n } is convergent to x₀. Consequently, X is complete.□

Problem 4.8. Let (X, M, *) be a fuzzy metric space. If X is complete, then is it sts-complete?

According to Remark 4.4, an sts-Cauchy sequence is not always Cauchy in a fuzzy metric space. However, whether Cauchy sequences can be derived from sts-Cauchy sequences in complete fuzzy metric spaces is unknown. To continue this work we propose the problem as above.

5 Conclusion

We have presented a new contribution to the study of fuzzy metrics by introducing the concepts of sts-convergence, sts-Cauchy and sts-complete in fuzzy metric spaces. Moreover, we have given an equivalent characterization of sts-convergent (sts-Cauchy) sequences in fuzzy metric spaces. At last, we have proven that sts-completeness implies completeness in fuzzy metric spaces. On the other hand, completeness is a specific feature of fuzzy metric spaces since there are fuzzy metric spaces which are not completable. In this paper, we have studied relationships between convergent (Cauchy) sequences and statistically convergent (Cauchy) sequences in fuzzy metric spaces. In this sense, our work can be consider as a continuation of completeness of fuzzy metric spaces.

In a further work we will continue discussing the implication relation between completeness and sts-completeness, since it is unknown that sts-completeness agrees with completeness in fuzzy metric spaces. In addition, we will introduce several new concepts, like statistical completion and statistically completable, and we will explore some characterizations of those fuzzy metric spaces that are statistically completable.