Generated fuzzy metric spaces and some topological properties

1 Introduction

In 1965, the theory of fuzzy sets was discovered by Zadeh [16]. Later, many writings on the fuzzy logic and fuzzy sets have been published as in [4, 14]. Making some modifications George and Veeramani redefined the fuzzy metric space, which was introduced by Kramosil and Michalek in [10], and gived some new results on the fuzzy metric spaces [5, 6]. On the other hand in 1972 Grossman and Katz established a new calculus, called non-Newtonian calculus, alternatively to the classical calculus [8]. Creating this calculus, they used increasing generators. As a special case of generators the exponential function exp gene-rated multiplicative (or geometric) calculus. Bashirov et al. studied on the multiplicative calculus and restated some classical properties of derivatives and integrals in the realm of multiplicative calculus [2]. Çakmak and Başar constructed the field ℝ ( N ) of non-Newtonian real numbers and the concept of non-Newtonian metric [5]. Binbaşıoğlu et.al [3] and Kirişci [9] defined some topological properties related to non-Newtonian metric spaces.

Thinking any generator may not be increasing, we reinterpret the definition of non-Newtonian metric and call generated metric. After that, we generalize the concept of fuzzy metric space by presenting the definition of generated fuzzy metric and generated fuzzy bonded metric. In this case, it is obviously that every fuzzy metric is a generated fuzzy metric when we take the generator as the identity function I. We also give the concept of α-defective diameter for any subset of generated fuzzy metric space based on [11, 12]. Finally, we examine some theorems about convergence of sequences and Baire’s theorem which are located in functional analysis and topological spaces [1, 13] for generated fuzzy metric spaces, and prove with help of the concept of α-defective diameter as different from [6] and [7].

2 Preliminaries

A generator α is a one-to-one map whose domain is ℝ and whose range is ℝ α (or ℝ ( N ) ) a subset of ℝ . Each generator generates exactly one arithmetic, and conversely each arithmetic is generated by exactly one generator. There are four basic binary operations on ℝ α generated by α. The operations α-addition and α-multiplication on ℝ α are defined by x + α y = α ( α - 1 ( x ) + α - 1 ( y ) ) and x · α y = α ( α - 1 ( x ) α - 1 ( y ) ) for all x , y ∈ ℝ α , respectively. For every x ∈ ℝ α , let - α x = α ( - 1 ) · α x = α ( - α - 1 ( x ) ) . Then the operation α-subtraction is defined by x - α y = x + α ( - α y ) = α ( α - 1 ( x ) - α - 1 ( y ) ) for all x , y ∈ ℝ α . Let k _α  = α (k) for all k ∈ ℝ . For every x ∈ ℝ α ∖ { 0 α } , we define x - 1 α = 1 α / α x = α ( 1 / α - 1 ( x ) ) . Then the operation α-division is defined by x / α y = x · α y - 1 α = α ( α - 1 ( x ) / α - 1 ( y ) ) for all x ∈ ℝ α and y ∈ ℝ α ∖ { 0 α } . Let the generator α be monotone. We define a total order (called α-order) ⩽ α with x ⩽ α y ⇔ { α - 1 ( x ) ⩽ α - 1 ( y ) if α is increasing α - 1 ( y ) ⩽ α - 1 ( x ) if α is decreasing for all x , y ∈ ℝ α . Then the range ℝ α is an ordered field. Hence, the generator α generates an arithmetic on ℝ α called α-arithmetic.

Remark 2.1. If the generator α is monotone, then (1) it is continuous on ℝ and (2) for any two elements x and y in ℝ α , x ⩽ α y iff x ⩽ y. Despite this, we will remain faithful to the definition of α-order.

Example 2.2. (a) The identity map I generates classical arithmetic on ℝ I = ℝ .(b) The exponential map exp generates geometric arithmetic on ℝ exp = ℝ + , and ℝ + is an ordered field with the operations x + exp y = xy x · exp y = x ln y = y ln x and with the total order x ⩽ exp y ⇔ x ⩽ y for all x , y ∈ ℝ + .(c) The inverse tangent map arctan generates a arithmetic on ℝ arctan = ( - π 2 , π 2 ) such that ( - π 2 , π 2 ) is an ordered field with the operations x + arctan y = arctan ( tan x + tan y ) , x · arctan y = arctan ( tan x tan y ) and with the total order x ⩽ arctan y ⇔ x ⩽ y for all x , y ∈ ( - π 2 , π 2 ) .(d) Some of the infinite number of arithmetics on ℝ are called q -arithmetics by Grossman and Katz [8]. For any nonzero real number p, the generator q from ℝ to itself defined as q ( x ) = { x 1 / p , x > 0 0 , x = 0 - ( - x ) 1 / p , x < 0 for all x ∈ ℝ . If p = 1 then the q-arithmetic is the classical arithmetic. For p = 2, the map q generates quadratic arithmetic and then ℝ is an ordered field with the operations x + q y = x 2 + y 2 x · q y = xy and with the total order x ⩽ q y ⇔ x ⩽ y for all x , y ∈ ℝ . (e) Let α be a map on ℝ defined as α ( x ) = x 1 + | x | for all x ∈ ℝ . The image ℝ α = ( - 1 , 1 ) is an ordered field with the operations   x + α y = ( x 1 - | x | + y 1 - | y | ) / ( 1 + | x 1 - | x | + y 1 - | y | | ) , x · α y = xy / ( 1 + 2 | xy | - | x | - | y | ) and with the total order x ⩽ α y ⇔ x ⩽ y for all x, y ∈ (-1, 1).

In situation that the α is not monotone, an α-order can also be defined on ℝ α .

Example 2.3. (a) In Example 2.2(d), for p = -1 the generator q defined as q ( x ) = { 1 / x , x ≠ 0 0 , x = 0 generates harmonic arithmetic and then ℝ is an ordered field with the operations   x + q y = { xy x + y if x + y ≠ 0 0 if x + y = 0 , x · q y = xy and with the total order   x ⩽ q y ⇔ { y ⩽ x if x , y < 0 or x , y > 0 x ⩽ y if x ⩽ 0 and y ⩾ 0 for all x , y ∈ ℝ . (b) A map α defined as α ( x ) = { - x if x ∈ [ - 1 , 1 ] x others generate an arithmetic on ℝ with the operations x + α y = { - x - y if x , y ∈ [ - 1 , 1 ] x - y if { x ∉ [ - 1 , 1 ] and y ∈ [ - 1 , 1 ] y - x if { x ∈ [ - 1 , 1 ] and y ∉ [ - 1 , 1 ] x + y if x , y ∉ [ - 1 , 1 ] , x · α y = { - xy if { x ∈ [ - 1 , 1 ] , y ∉ [ - 1 , 1 ] or x ∉ [ - 1 , 1 ] , y ∈ [ - 1 , 1 ] xy if x , y ∈ [ - 1 , 1 ] or x , y ∉ [ - 1 , 1 ] with the total order x ⩽ α y ⇔ { x ⩽ y if x , y ∉ [ - 1 , 1 ] x ⩾ y if x , y ∈ [ - 1 , 1 ] .

Depending on the monotonicity of α, the intervals of ℝ α may be different from those of ℝ . For any x , y ∈ ℝ α ; x ⩽ α y , we define α-open interval with ( x , y ) α = { ( α - 1 ( x ) , α - 1 ( y ) ) if α is increasing ( α - 1 ( y ) , α - 1 ( x ) ) if α is decreasing , and similarly, we define α-closed interval with [ x , y ] α = { [ α - 1 ( x ) , α - 1 ( y ) ] if α is increasing [ α - 1 ( y ) , α - 1 ( x ) ] if α is decreasing . We say that α- lim x n = ∞ α if for every ɛ ⩾ α 0 α there is an integer n₀ = n₀ (ɛ, α)∈ such that x n > α ɛ whenever n > n₀. If the generator α is monotone, then α- lim α ( n ) = lim α ( n ) = ∞ α . The set of α-positive real numbers is ℝ α + = { x ∈ ℝ α : x > α 0 α } = ( 0 α , ∞ α ) α . The Table 1 describes these concepts for some generators.

The α-square of a number x in ℝ α is denoted by x 2 α = x · α x . Since for any x ∈ ℝ α ∖ { 0 α } x - 2 α = 1 α / α x 2 α = 1 α / α α ( [ α - 1 ( x ) ] 2 ) = α ( [ α - 1 ( x ) ] - 2 ) , for any integer m, by induction the m-th α-power of x is x ^{m _α}  = α ([α^-1 (x)]  ^m ). The square α-root is denoted by x α = α ( α - 1 ( x ) ) for all x ∈ ℝ α + ∪ { 0 α } . Generally, for any positive integer n ⩾ 2, the n-th α-root (if there exists) of a number x in ℝ α is x n α = x ( 1 / n ) α = x 1 / n α = α ( α - 1 ( x ) n ) . Besides, the α-absolute value of x in ℝ α is | x | α = x 2 α α = α ( | α - 1 ( x ) | ) = { x , x ⩾ α 0 α - α x , x ⩽ α 0 α .

Consider the generator α may not be increasing, let us update the definition of non-Newtonian metric in [5] with new notations.

Definition 2.4. Let X be a non-empty set. The map d α : X × X → ℝ α is called generated metric (or α-metric) if it satisfies the following properties: (gm1) d α ( x , y ) ⩾ α 0 α , (gm2) d ^α  (x, y) =0 _α if and only if x = y, (gm3) d ^α  (x, y) = d ^α  (y, x) and (gm4) d α ( x , y ) ⩽ α d α ( x , z ) + α d α ( y , z ) for all x, y, z ∈ X . Also the pair (X, d ^α ) is said to be a generated metric space.

Each metric is a generated metric while the generator α is the unit function I.

Remark 2.5. Non-negativity of any α-metric is directly related to the order ⩽ α given. For example, let -I be the inverse of unit function I and X = ℝ - I = ℝ . Then the map d^-I defined as d^-I (x, y) = - |x - y| for all x, y is an -I-metric. Since -I is an decreasing generator, ℝ - I + = ( - ∞ , 0 ) and d - I ( x , y ) ⩾ - I - I ( 0 ) = 0 while d - I ( x , y ) ⩽ 0 for all x , y ∈ ℝ .

We start with the definitions required for the issue.

Definition 3.1. Let α and β be two generator. The function f from ℝ α to ℝ β is called (α, β)-continuous if for every ɛ > β 0 β and for every t ∈ ℝ α there exists δ = δ ( ɛ , t ) > α 0 α such that s ∈ ℝ α and} | s - α t | α < α δ implies | f ( s ) - β f ( t ) | β < β ɛ . In particular, when β = I, then the function f is called α-continuous.

Definition 3.2. [9] A binary operation ∗ : [0, 1] ² → [0, 1] is a t-norm if it is commutative, associative, increasing and 1 is its neutral element, i.e., a ∗ 1 = 1 ∗ a = a for all a ∈ [ 0 , 1 ] .

The above definition is a special case following definition when it be taken α = I.

Definition 3.3. A binary operation ∗ α : [ 0 α , 1 α ] α 2 → [ 0 α , 1 α ] α is a generated t-norm if it is commutative, associative, increasing and 1 _α is its neutral element, i.e., a ∗ α 1 α = 1 α ∗ α a = a for all a ∈ [ 0 α , 1 α ] α .

The double of α from ℝ 2 to ℝ α 2 defined by Λ (a, b) = (α (a) , α (b)) for all a , b ∈ ℝ . Let us give the following result without proof.

Corollary 3.4. For any generator α on ℝ (1) If ∗ α is a generated t-norm, then ∗ = α - 1 ∘ ∗ α ∘ Λ ∣ [ 0 , 1 ] is a t-norm, (2) If   ∗   is a t-norm, then ∗ α = α ∘ ∗ ∘ Λ - 1 ∣ [ 0 α , 1 α ] α is a generated t-norm.

Let ∗ α be a generated t-norm. For the t-norm ∗ = α - 1 ∘ ∗ α ∘ Λ ∣ [ 0 , 1 ] we have α ( a ∗ b ) = α ( a ) ∗ α α ( b ) for all a, b ∈ [0, 1]. Similarly, let   ∗   be a t-norm. For the generated t-norm ∗ α = α ∘ ∗ ∘ Λ - 1 ∣ [ 0 α , 1 α ] α we have α - 1 ( a ∗ α b ) = α - 1 ( a ) ∗ α - 1 ( b ) for all  a, b ∈ [0 _α , 1 _α ]  _α .

Example 3.5. Let α be the exponential function exp. (a) For the product t-norm a ∗ b = ab, the map ∗ exp defined by x ∗ exp y = exp ∘ ∗ ∘ Λ - 1 ( x , y ) = exp ( ln x ∗ ln y ) = x ln y is a generated t-norm on [1, e] ². (b) The binary operation a ∗ exp b = max { e , a . b / e } is a generated t-norm on [1, e] ². The map ∗ defined by x ∗ y = ln ∘ ∗ exp ∘ Λ ( x , y ) = ln ( exp x ∗ exp exp y ) = max { 1 , x + y - 1 } is a t-norm (Łukasiewicz t-norm).

Now let us give the fuzzy metric space in the sense of George and Veeramani[5].

Definition 3.6. Let X be a non-empty set and ∗ be a continuous t-norm. The triple (X, M, ∗) is called fuzzy metric space if the fuzzy set M on X × X × ℝ + satisfies the following properties: (fm1) M (x, y, t) >0, (fm2) M (x, y, t) =1 if and only if x = y, (fm3) M (x, y, t) = M (y, x, t) , (fm4) M (x, y, t) ∗ M (y, z, s) ⩽ M (x, z, t + s) and (fm5) M ( x , y , . ) : ℝ + → [ 0 , 1 ] is continuous for all x, y, z ∈ X and all t, s > 0. If (X, M, ∗) is a fuzzy metric space, we will say that (M, ∗) or simply M is a fuzzy metric on X.

Let ℝ α be an ordered field generated by a generator α on ℝ . We define the generated fuzzy metric space and obtain some topological properties of this space.

Definition 3.7. Let X be a non-empty set and ∗ be a continuous t-norm. The triple (X, M ^α , ∗) is called generated fuzzy metric space (or fuzzy α-metric space) if the fuzzy set M ^α on X × X × ℝ α + satisfies the following properties:

(gfm1) M ^α  (x, y, t) >0, (gfm2) M ^α  (x, y, t) =1 if and only if x = y, (gfm3) M ^α  (x, y, t) = M ^α  (y, x, t) , (gfm4) M α ( x , y , t ) ∗ M α ( y , z , s ) ⩽ M α ( x , z , t + α s ) , (gfm5) M α ( x , y , . ) : ℝ α + → [ 0 , 1 ] is α-continuous for all x, y, z ∈ X and all t , s ∈ ℝ α + . If (X, M ^α , ∗) is a generated fuzzy metric space, we will say that (M ^α , ∗) or simply M ^α is a generated fuzzy metric on X.

As similar to the above definition, we can define a type of fuzzy metric for any α-membership function α M : X × X × ℝ α + → [ 0 α , 1 α ] α .

Definition 3.8. Let X be a non-empty set and ∗ α be a continuous generated t-norm. The triple ( X , α M , ∗ α ) is called generated fuzzy bonded metric space if the fuzzy set ^α M on X × X × ℝ α + satisfies the following properties:

(gfbm1) α M ( x , y , t ) > α 0 α ,

(gfbm2) ^α M (x, y, t) =1 _α if and only if x = y, (gfbm3) ^α M (x, y, t)   = ^αM (y, x, t) ,

(gfbm4) α M ( x , y , t ) ∗ α α M ( y , z , s ) > α α M ( x , z , t + α s ) ,

(gfbm5) α M ( x , y , . ) : ℝ α + → [ 0 α , 1 α ] α is (α, α)-continuous for all x, y, z ∈ X and all t , s ∈ ℝ α + .

If ( X , α M , ∗ α ) is a generated fuzzy bonded metric space, we will say that ( α M , ∗ α ) or simply ^α M is a generated fuzzy bonded metric on X .

Lemma 3.9. Let (M ^α , ∗) be a generated fuzzy metric and let ( α M , ∗ α ) be a generated fuzzy bonded metric on X. Then the M ^α  (x, y,.) and the ^α M (x, y,.) are increasing for all x, y ∈ X. If the generator α is increasing and α ∘ ∗ = ∗ α ∘ Λ ∣ [ 0 , 1 ] , then (1) (α^-1 ∘ ^αM, ∗) is a generated fuzzy metric, (2) ( α ∘ M α , ∗ α ) is a generated fuzzy bonded metric.

Proof. First suppose that M ^α  (x, y,.) and ^α M (x, y,.) are not increasing; namely, for 0 < t < s, let M ^α  (x, y, t) > M ^α  (x, y, s) and let α M ( x , y , t ) > α α M ( x , y , s ) . Then M α ( x , y , t ) ∗ M α ( y , y , s - t ) ⩽ M α ( x , y , s ) < M α ( x , y , t ) , α M ( x , y , t ) ∗ α α M ( y , y , s - α t ) ⩽ α M ( x , y , s ) < α α M ( x , y , t ) hold. By Def. 3.7(gfm2) and Def. 3.8(gfbm2), M α ( y , y , s - t ) = 1 and α M ( y , y , s - α t ) = 1 α , respectively. Hence, the assumptions are wrong since M α ( x , y , t ) < M α ( x , y , t ) and α M ( x , y , t ) < α α M ( x , y , t ) . (1) Since the α-membership function M : X × X × ℝ α + → [ 0 α , 1 α ] α satisfies the conditions Def.?? (gfbm1)-(gfbm5) and the generator α is increasing (and hence the inverse generator α^-1 is increasing), then for all x, y, z ∈ X and t , s ∈ ℝ α + the followigs hold: (gfm1) 0 < α^-1 ( ^α M (x, y, t)) = (α^-1 ∘ ^αM) (x, y, t) if and only if 0 α < α α M ( x , y , t ) . (gfm2) Let x = y. Since ^α M (x, y, t) =1 _α , then (α^-1 ∘ ^αM) (x, y, t) = α^-1 (1 _α ) =1 . To converse, let (α^-1 ∘ ^αM) (x, y, t) =1. Since ^α M (x, y, t) = α [(α^-1 ∘ ^αM) (x, y, t)] = α (1) =1 _α ,   then x = y. (gfm3) Since ^αM(x, y, t) = ^αM (y, x, t), then   (α^-1 ∘ ^αM) (x, y, t) = (α^-1 ∘ ^αM) (y, x, t) . (gfm4) Since α - 1 ( a ∗ α b ) = α - 1 ( a ) ∗ α - 1 ( b ) for all a, b ∈ [0 _α , 1 _α ]  _α , we have ( α - 1 ∘ α M ) ( x , y , t ) ∗ ( α - 1 ∘ α M ) ( y , z , s ) = α - 1 ( α M ( x , y , t ) ∗ α α M ( y , z , s ) ) ⩽ α - 1 ( α M ( x , z , t + α s ) ) = ( α - 1 ∘ α M ) ( x , z , t + α s ) . (gfm5) Since the generator α one to one and increasing, then α and so α^-1 are continuous. Hence, lim t → t 0 ( α - 1 ∘ α M ) ( x , y , t ) = α - 1 ( lim t → t 0 α M ( x , y , t ) ) = ( α - 1 ∘ α M ) ( x , y , t 0 ) . (2) It is proved in a similar way to (1).

Example 3.10. For the generator exp, let (X, d^exp) be a generated metric space and denote a generated t-norm ∗ exp with a ∗ exp b = a ln b for all a, b ∈ [1, e]. The generated fuzzy bonded metric induced by d^exp on X × X × (1, ∞) is defined as exp M d exp ( x , y , t ) = t 1 / ln ( t . d exp ( x , y ) ) .

Then the triple ( X , exp M d exp , ∗ exp ) is a generated fuzzy bonded metric space. On the other hand, by Example 3.5(1) and Lemma 3.9(1), the map M d exp exp defined as M d exp exp ( x , y , t ) = ln t ln ( t . d exp ( x , y ) ) on X × X × (1, ∞) is the generated fuzzy metric induced by d^exp and so ( X , M d exp exp , ∗ exp ) is a generated fuzzy bonded metric space.

Theorem 3.11. Let X be a non-empty set. For any generator α on ℝ let u be a map satisfying u (x, y, k) = (x, y, α (k)) for all x, y ∈ X and k > 0. (1) If (M ^α , ∗) is a generated fuzzy metric on X, then (M ^α  ∘ u, ∗) is a fuzzy metric on X, (2) If (M, ∗) is a fuzzy metric on X, then (M ∘ u^-1, ∗) is a generated fuzzy metric on X.

Proof. (1) Let M ^α be a generated fuzzy metric. One can easily see that for the map M ^α  ∘ u the conditions (fm1), (fm2), (fm3) and (fm5). We only prove the triangular inequality (fm4). For all x, y, z ∈ X and t, s > 0 ( M α ∘ u ) ( x , y , t ) ∗ ( M α ∘ u ) ( y , z , s ) = M α ( u ( x , y , t ) ) ∗ M α ( u ( y , z , s ) ) = M α ( x , y , α ( t ) ) ∗ M α ( y , z , α ( s ) ) ⩽ M α ( x , z , α ( t ) + α α ( s ) ) = M α ( x , z , α ( t + s ) ) = ( M α ∘ u ) ( x , y , t + s ) (2) Let M be a fuzzy metric. M ∘ u^-1 The conditions (gfm1), (gfm2), (gfm3) and (gfm5) are easy hold for M ∘ u^-1. Let us prove (gfm4). For all x, y, z ∈ X and t , s ∈ ℝ α + M ∘ u - 1 ( x , y , t ) ∗ M ∘ u - 1 ( y , z , s ) = M ( x , y , α - 1 ( t ) ) ∗ M ( y , z , α - 1 ( s ) ) ⩽ M ( x , z , α - 1 ( t ) + α - 1 ( s ) ) = M ( u - 1 ( x , z , α [ α - 1 ( t ) + α - 1 ( s ) ] ) ) = ( M ∘ u - 1 ) ( x , z , t + α s ) ) .

The following result is the outcome discussed above.

Corollary 3.12. Let X be a non-empty set. For a generated fuzzy metric (M ^α , ∗) and a generated fuzzy bonded metric ( α M , ∗ α ) on X, let α ∘ ∗ = ∗ α ∘ Λ ∣ [ 0 , 1 ] . If the generator α is increasing, then (1) (α^-1 ∘ ^αM ∘ u, ∗) is a fuzzy metric, (2) ( α ∘ M ∘ u - 1 , ∗ α ) is a generated fuzzy bonded metric.

Now, let us illustrate some topological concepts in a generated fuzzy metric space.

Definition 3.13. Let (X, M ^α , ∗) be a generated fuzzy metric space. We define open ball B ^α  (x, r, t) for t ∈ ℝ α + with centre x ∈ X and radius r, 0 < r < 1 as B α ( x , r , t ) = { y ∈ X : M α ( x , y , t ) > 1 - r }

Now, similarly we describe the closed ball as

B α = { y ∈ X : M α ( x , y , t ) ⩾ 1 - r } .

In the following definition we present a novel concept, which is called α-defective diameter with the aid of [15].

Definition 3.14. Let (X, M ^α , ∗) be a generated fuzzy metric space and A is non-empty subset of X. We define α-defective diameter D A α for t ∈ ℝ α + as

D A α = sup t { inf M α ( x , y , t ) : x , y ∈ A } . As a particular case, if we take α as the identity function I, D A α is called defective diameter of A. The set A is called F (α)-bounded if D A α = 1 .

Remark 3.15. (1) If A is a singleton set, then D A α = 1 . Unlike in crisp set, we see that when D A α = 1 , the set A does not have to be a singleton set. Let α be the identity function I. Suppose (X, M _d , ∗) be a fuzzy metric space where M _d is defined by M _d  (x, y, t) = t/(t + d (x, y)) for all x, y ∈ X and t > 0 with a metric d. For a two-element subset A = {x₀, y₀} of X D A I = sup t t t + d ( x 0 , y 0 ) = 1 . (2) If the set A is F (α)-bounded, then for every x, y ∈ A and t ∈ ℝ α + there exists an r ∈ (0, 1) such that M ^α  (x, y, t) ⩾1 - r, that is, for every t ∈ ℝ α + and for every n ∈ ℕ + there exists an r ∈ (0, 1) such that A ⊂ B ^α  [x, r, t].

Definition 3.16. (a) A sequence (x _n ) in a generated fuzzy metric space (X, M ^α , ∗) is converges to x ∈ X, if for every ɛ ∈ (0, 1) and t ∈ ℝ α + there exists n 0 = n 0 ( ɛ ) ∈ ℕ such that M ^α  (x _n , x, t) >1 - ɛ for all n ⩾ n₀ and it is denoted by

lim n → ∞ M α ( x n , x , t ) = 1 or x n → α , M x (b) A sequence (x _n ) in a generated fuzzy metric space (X, M ^α , ∗) is called Cauchy if for each ɛ ∈ (0, 1) and t ∈ ℝ α + there exists n 0 ∈ ℕ such that M ^α  (x _n , x _m , t) >1 - ɛ for all n, m ⩾ n₀. (c) A generated fuzzy metric space (X, M ^α , ∗) is called complete if every Cauchy sequence in X converges. (d) A subset Y of a generated fuzzy metric space (X, M ^α , ∗) is called closed if (x _n ) ⊂ Y and x n → α , M x imply x ∈ Y .

Theorem 3.17. Let (X, M ^α , ∗) be a generated fuzzy metric space and (x _n ) ⊂ X be a convergent sequence. Then, (1) (x _n ) is F (α)-bounded and its limit x is unique, (2) (x _n ) is a Cauchy sequence in X, (3) Every subsequence of (x _n ) converges to x.

Proof. (1) First, we show that the convergent sequence (x _n ) is F (α) -bounded. Then, for every ɛ ∈ (0, 1) and t ∈ ℝ α + there exists n 0 ∈ ℕ such that M ^α  (x _n , x, t/2) >1 - ɛ for all n ⩾ n₀. Now for x₀ ∈ X and 0 < s < 1, let M α ( x 0 , x , t / 2 ) > 1 - s and for 0 < k < 1, let min { M α ( x n , x , t / 2 ) : n = 1 , 2 , . . . n 0 } = 1 - k . Then we can find a r ∈ (0, 1) such that min { ( 1 - s ) ∗ ( 1 - k ) , ( 1 - s ) ∗ ( 1 - ɛ ) } = 1 - r . Hence, for all n ∈ ℕ + we have M α ( x 0 , x n , t ) ⩾ M α ( x 0 , x , t / 2 ) ∗ M α ( x n , x , t / 2 ) ⩾ 1 - r . Thus for all n ∈ ℕ + (x _n ) ⊂ B ^α  [x₀, r, t], that is, (x _n ) is bounded. Now we show that the limit of (x _n ) is unique. We suppose that the convergent sequence (x _n ) has two different limits x and y. Let t ∈ ℝ α + and ɛ ∈ (0, 1). Then we can find a r ∈ (0, 1) such that (1 - r) ∗ (1 - r) ⩾1 - ɛ. Let ɛ = 1 - M ^α  (x, y, t). According to our assumption there exists n 1 ∈ ℕ such that M ^α  (x _n , x, t/2) >1 - r for all n ⩾ n₁ and there exists n 2 ∈ ℕ such that M ^α  (x _n , y, t/2) >1 - r for all n ⩾ n₂. When we take n₀ = max {n₁, n₂}, then for n ⩾ n₀ M α ( x , y , t ) ⩾ M α ( x n , x , t / 2 ) ∗ M α ( x n , y , t / 2 ) > ( 1 - r ) ∗ ( 1 - r ) ⩾ 1 - ɛ = M α ( x , y , t ) , that is, M ^α  (x, y, t) > M ^α  (x, y, t), which is not possible. (2) Let t ∈ ℝ α + and ɛ ∈ (0, 1). We can find a r ∈ (0, 1) such that (1 - r) ∗ (1 - r) > (1 - ɛ). Since (x _n ) is a convergent sequence, there exists n 0 ∈ ℕ such that M ^α  (x _n , x, t/2) >1 - r for all n ⩾ n₀. For all n, m ⩾ n₀ we get M α ( x n , x m , t ) ⩾ M α ( x n , x , t / 2 ) ∗ M α ( x , x m , t / 2 ) ⩾ ( 1 - r ) ∗ ( 1 - r ) > ( 1 - ɛ ) . (3) Let (x _{n k} ) ⊂ (x _n ). Since (x _n ) is a convergent sequence, there exists n 0 ∈ ℕ for each ɛ ∈ (0, 1) and t ∈ ℝ α + such that M ^α  (x _n , x, t/2) >1 - ɛ for all n ⩾ n₀. If k ⩾ n₀, since n₀ ⩽ k ⩽ n _k , we get M ^α  (x _{n k} , x, t) >1 - ɛ.

Given two generated fuzzy metric spaces ( X , M 1 α , ∗ ) and ( X , M 2 α , ∗ ) , for (x₁, x₂) , (y₁, y₂) ∈ X₁ × X₂, t > 0, if we define M α ( ( x 1 , x 2 ) , ( y 1 , y 2 ) , t ) = M 1 α ( x 1 , y 1 , t ) ∗ M 2 α ( x 2 , y 2 , t ) then it easy to prove that M ^α is generated fuzzy metric on X₁ × X₂.

Theorem 3.18. Let ( X 1 , M 1 α , ∗ ) and ( X 2 , M 2 α , ∗ ) be generated fuzzy metric spaces. If (x _n ) ⊂ X₁, (y _n ) ⊂ X₂ are convergent sequences such that x n → α , M 1 x and y n → α , M 2 y , then ( x n , y n ) → α , M ( x , y ) .

Proof. Let t ∈ ℝ α + and ɛ ∈ (0, 1). We can find a r ∈ (0, 1) such that (1 - r) ∗ (1 - r) > (1 - ɛ). Since (x _n ) ⊂ X₁ and (y _n ) ⊂ X₂ are convergent sequences, there exists n 1 ∈ ℕ such that M 1 α ( x n , x , t ) > 1 - r for all n ⩾ n₁ and there exists n 2 ∈ ℕ such that M 2 α ( y n , y , t ) > 1 - r for all n ⩾ n₂. Using the definition above mentioned for n₀ = max {n₁, n₂} we obtain M α ( ( x n , y n ) , ( x , y ) , t ) = M 1 α ( x n , x , t ) ∗ M 2 α ( y n , y , t ) > ( 1 - r ) ∗ ( 1 - r ) > ( 1 - ɛ ) . This completes the proof of the theorem.

Theorem 3.19. For the generated metric space (X, M ^α , ∗) the followings hold. (1) Every convergent sequence is a Cauchy sequence. (2) Every Cauchy sequence is F (α)-bounded. (3) If a Cauchy sequence (x _n ) has a subsequence (x _{n k} ) such that x n k → α , M x then x n → α , M x .

Proof. (1) Let t ∈ ℝ α + and ɛ ∈ (0, 1). We can find a number r ∈ (0, 1) such that (1 - r) ∗ (1 - r) >1 - ɛ. Since (x _n ) a convergent sequence in (X, M ^α , ∗) there exists n 0 ∈ ℕ such that M ^α  (x _n , x, t/2) >1 - r for all n ⩾ n₀. Hence, for n, m ⩾ n₀ we have M α ( x n , x m , t ) ⩾ M α ( x n , x , t / 2 ) ∗ M α ( x m , x , t / 2 ) > ( 1 - r ) ∗ ( 1 - r ) > 1 - ɛ . (2) Let (x _n ) be a Cauchy sequence. Then, for every ɛ ∈ (0, 1) and t ∈ ℝ α + there exists n 0 ∈ ℕ such that M ^α  (x _n , x _m , t) >1 - ɛ for all n, m ⩾ n₀. So, for n ⩾ n₀ we have M ^α  (x _n , x _{n 0} , t) >1 - ɛ. Let r = 1 - min { M α ( x n , x n 0 , t ) : n = 1 , 2 , . . . n 0 - 1 } . If s = max {ɛ, r}, then {x _n  : n = 1, 2,. . . ⊂ B ^α  [x _{n 0} , s, t], that is, (x _n ) is bounded. (3) Let t ∈ ℝ α + and ɛ ∈ (0, 1). We can find r ∈ (0, 1) such that (1 - r) ∗ (1 - r) ⩾1 - ɛ. Since (x _n ) is a Cauchy sequence, there exists n 0 ∈ ℕ such that M ^α  (x _n , x _m , t/2) >1 - r for all m, n ⩾ n₀. Since x n k → α , M x , there exists positive integer i _p such that i _p  > n₀, M ^α  (x _{i p} , x, t/2) >1 - r. For n ⩾ n₀ we have M α ( x n , x , t ) ⩾ M α ( x n , x i p , t / 2 ) ∗ M α ( x i p , x , t / 2 ) > ( 1 - r ) ∗ ( 1 - r ) ⩾ 1 - ɛ . Then x n → α , M x and hence (X, M ^α , ∗) is complete.

Definition 3.20. Let (X, M ^α , ∗) be a generated fuzzy metric space and (A _n ) be a sequence of non-empty subsets of X .We say that (A _n ) has appearing α-defective diameter if lim n → ∞ D A n α = 1 . a sequence (A _n ) has appearing α-defective diameter, then for every r ∈ (0, 1) and t ∈ ℝ α + there exists n ∈ ℕ such that M ^α  (x, y, t) >1 - r for all x, y ∈ A _n .

Theorem 3.21. Let (X, M ^α , ∗) be a generated fuzzy metric space and (A _n ) be any decreasing sequence of non-empty closed subsets of X with appearing α-defective diameter. X is complete if and only if there is exactly one point x ∈ X such that ⋂ n = 1 ∞ A n = { x } .

Proof. Let the generated fuzzy metric space X be complete. We can form a sequence (x _n ) by taking a point x _n  ∈ A _n for each n ∈ ℕ . If we choose m ⩾ n, we get A _m  ⊂ A _n , so that all points {x _m  : m ⩾ n} of the sequence belong to the set A _n . So for each t ∈ ℝ α + we obtain M α ( x m , x n , t ) ⩾ D A n α for all m ⩾ n and consequently lim n , m → ∞ M α ( x m , x n , t ) = 1 . In this case (x _n ) is a Cauchy sequence. Since X is complete, there exists a point x ∈ X, such that lim n → ∞ M α ( x n , x , t ) = 1 . We take a set A _{n 0} . The limit of the sequence {x _n  : n ⩾ n₀} ⊂ A _{n 0} is of course the same point x. Futhermore, since A _{n 0} is closed, x ∈ A _{n 0} . This means that x belongs to every member of the sequence (A _n ).Hence we get x ∈ ⋂ n = 1 ∞ A n . Now we consider another point x ′ ∈ ⋂ n = 1 ∞ A n . Then, for every t ∈ ℝ α + the relation M α ( x , x ′ , t ) ⩾ D A n α holds and this implies that M ^α  (x, x′, t) =1. This means that ⋂ n = 1 ∞ A n = { x } because of x = x′.

Conversely, let (x _n ) be a Cauchy sequence in X and for every n ∈ ℕ let A _n  = {x _m  : m ⩾ n} be a non-empty closed subset of X. Since the sequence (A _n ) is decreasing and since (x _n ) is a Cauchy sequence, we have lim n → ∞ D A n α = 1 . To our assumption the intersection of all these sets contains only a single point of the space X. Suppose that ⋂ n = 1 ∞ A n = { x } . For every ɛ ∈ (0, 1) there exists n 0 ∈ ℕ such that D A n 0 α > 1 - ɛ . Since x ∈ A _{n 0} , for every ɛ ∈ (0, 1) and t ∈ ℝ α + we have M ^α  (x _n , x, t) >1 - ɛ for all n ⩾ n₀. Therefore, the Cauchy sequence (x _n ) converges to the point x. Hence, the generated fuzzy metric space X is complete.

Now we give the Baire theorem for generated fuzzy metric spaces.

Theorem 3.22. (Baire’s theorem) Let X be a generated complete fuzzy metric space and let { A n ⊂ X : n ∈ ℕ } be a countable family of open subsets which are dense in X. Then the set ⋂ n = 1 ∞ A n is also dense in X.

Proof. In order to prove the theorem we have to show that for every x ∈ X, r ∈ (0, 1) and t ∈ ℝ α + , the intersection of open balls B ^α  (x, r, t) and the set ⋂ n = 1 ∞ A n is non-empty. First we take the set A₁. Since A₁ is dense in X, B ^α  (x, r, t) ∩ A₁ is open and non-empty. Let x₁ ∈ B ^α  (x, r, t) ∩ A₁, then there exist r₁ ∈ (0, 1) and t 1 ∈ ℝ α + such that B ^α  [x₁, r₁, t₁] ⊂ B ^α  (x, r, t) ∩ A₁. Let B₁ = B ^α  (x₁, r₁, t₁). Since A₂ is dense in X, B₁ ∩ A₂ is open and non-empty. Let x₂ ∈ B₁ ∩ A₂, then there exist r₂ ∈ (0, 1/2) and t 2 ∈ ℝ α + such that B ^α  [x₂, r₂, t₂] ⊂ B₁ ∩ A₂. Let B _n  = B ^α  (x _n , r _n , t _n ) for all n ⩾ 2. If we proceed inductively, we obtain a sequence (x _n ) in X and a sequence (r _n ) of positive real numbers such that B α [ x n + 1 , r n + 1 , t n + 1 ] ⊂ B n ∩ A n + 1 ⊂ B α [ x n , r n , t n ] and r _n  ∈ (0, 1/n) for all n. Thus, the Theorem 3.21 guarantees that ⋂ n = 1 ∞ B α [ x n , r n , t n ] is a singleton set. From ⋂ n = 1 ∞ B α [ x n , r n , t n ] ⊂ B α ( x , r , t ) ∩ ⋂ n = 1 ∞ A n we obtain that B α ( x , r , t ) ∩ ⋂ n = 1 ∞ A n ≠ ∅ .