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 Open Access
An approach version of fuzzy metric spaces including an ad hoc fixed point theorem
 AntonioFrancisco RoldánLópezdeHierro^{1}Email author,
 Manuel de la Sen^{2},
 Juan MartínezMoreno^{1} and
 Concepción RoldánLópezdeHierro^{3}
https://doi.org/10.1186/s1366301502767
© RoldánLópezdeHierro et al.; licensee Springer. 2015
 Received: 15 November 2014
 Accepted: 30 January 2015
 Published: 27 February 2015
Abstract
In this paper, inspired by two very different, successful metric theories such us the real viewpoint of Lowen’s approach spaces and the probabilistic field of Kramosil and Michálek’s fuzzy metric spaces, we present a family of spaces, called fuzzy approach spaces, that are appropriate to handle, at the same time, both measure conceptions. To do that, we study the underlying metric interrelationships between the above mentioned theories, obtaining six postulates that allow us to consider such kind of spaces in a unique category. As a result, the natural way in which metric spaces can be embedded in both classes leads to a commutative categorical scheme. Each postulate is interpreted in the context of the study of the evolution of fuzzy systems. First properties of fuzzy approach spaces are introduced, including a topology. Finally, we describe a fixed point theorem in the setting of fuzzy approach spaces that can be particularized to the previous existing measure spaces.
1 Introduction
The notion of metric has always been intimately related to a spatial conception. However, it was not until the end of the first decade of the twentieth century that Fréchet presented the currently accepted axioms on which this theory is based. After the appearance of the concept of metric space, and mainly due to its potential applications, many researchers have been interested in extending it to wider fields of knowledge. In this sense, in order to model realworld experiments, many theories have been introduced over the last century. In this manuscript we pay especial attention to two of them.
Some of the most productive extensions of the concept of metric were introduced in the field of probability. Convinced of the difficulty of accurately measuring the distance between two points in real life (for instance, between two subatomic particles), Menger proposed in 1942 the idea of a statistical metric space (see [1]). His main contribution was to replace the distance between two points, which had traditionally been a real number, by a random variable. Taking into account that, under uncertain environment, classical approaches are unable to give a complete answer to many real problems in several areas, Zadeh presented the notion of fuzzy set [2]. A number between 0 and 1 was employed to express the imprecision/vagueness that arises in many real situations. Subsequently, many authors, inspired by the notion of probabilistic metric space given by Schweizer and Sklar [3], have introduced their particular points of view about the definition of fuzzy metric space (see [4–8], among others), to which we will refer as FMspace. See also [9–11].
In 1975, Kramosil and Michálek [6] modified the axioms used until then and established a new class of FMspaces provided with a Hausdorff topology. Later, other authors have considered certain modifications of this concept which are sufficient to verify new peculiar qualities (see [7]). FMspaces have many applications. For example, in [12], the authors used FMspaces in order to predict access histories working on variations of the fuzzy construction. In [13], Colubi and GonzálezRodríguez analyzed a fuzzifying process of finitely valued random variables by means of triangular fuzzy sets. Several studies of fuzzy regression analysis can be found in [14, 15]. The concept of fuzzy number supposes a generalization of the notion of real number and there exists a complete arithmetic among them (see [5, 16]). Furthermore, there exists a complete theoretic and practical analysis of fuzzy linear systems. In this field, inspired by Banach’s theorem, fixed point theory has been intensively studied [17–31], even in frameworks like intuitionistic fuzzy metric spaces [32–36].
With respect to nonprobabilistic extensions of the notion of metric space, the perspective can be highlighted introduced by Lowen through his approach spaces (in the sequel, Aspaces). Although they can seem, at first sight, simple generalizations of the necessary conditions that distances between point and subsets must satisfy, these spaces have turned out to have such good properties that they solved the problem of how to treat, in a unified way, such different notions as metrics, topologies, and uniformities, especially from the categorical viewpoint (see [37, 38]). In this sense, the main purpose for their introduction was to fill some gaps concerning categorical aspects of metrizable spaces. They are useful in many areas such as probability theory, functional analysis and function spaces. In his referential book [38], Lowen offered at least seven equivalent formulations of Aspaces.
With the aim of finding interrelationships between both successful theories, this paper introduces the notion of fuzzy approach space, which is a mixture of the best advantages of both conceptions. We also show some basic properties of this kind of spaces, and how the categories of quasimetric spaces, Aspaces, and FMspaces (in the sense of Kramosil and Michálek) can be naturally embedded in the supercategory of fuzzy approach spaces. Moreover, under some conditions, we provide these spaces with a topology. Finally, the study of contractions between fuzzy approach spaces leads to a first result in the field of fixed point theory involving this new class of spaces, which illustrates that some metric questions naturally arise in this ambiance.
2 Preliminaries
Let \(\mathbb{N}=\{0,1,2,\ldots\}\), ℝ and \(\overline{\mathbb {R}}= [ \infty,\infty ] \) denote the set of all nonnegative integers, the set of all real numbers, and the extended real line, respectively (for simplicity, +∞ will be denoted as ∞). Given a real number \(r\in [ 0,\infty [ \), we use the notation \(t\geq r\) if \(t\in [ r,\infty ] \), \(t>r\) if \(t\in \,] r,\infty ] \), and \(t<\infty \) if \(t\in [ 0,\infty [ \).
In the sequel, X will be a nonempty set and \(\mathcal{P}(X)\) will denote the sets of all subsets of X. A fuzzy set F on X is a map \(F:X\rightarrow [ 0,1 ] \). A fuzzy set F is crisp if its image is included in \(\{ 0,1 \} \).
 M1.:

\(\mathtt{d} ( p,p ) =0\) for all \(p\in X\).
 M2.:

\(\mathtt{d} ( p,q ) \neq0\) if \(p\neq q\) for all \(p,q\in X\).
 M3.:

\(\mathtt{d} ( p,q ) =\mathtt{d} ( q,p ) \) for all \(p,q\in X\).
 M4.:

Triangular inequality: \(\mathtt{d} ( p,q ) \leq\mathtt{d} ( p,x ) +\mathtt{d} ( x,q ) \) for all \(p,q,x\in X\).
If M3 is omitted, the function d is a quasimetric (briefly, qMspace). The topology of a qMspace \(( X,\mathtt{d} ) \) is completely determined by the distance function between two points. In a metric space, the distance between a point \(p\in X\) and a subset A of X is the infimum of the distances between p and all the points of the subset A. The analysis of the essential properties of a function \(\delta:X\times \mathcal{P} ( X ) \rightarrow\overline{\mathbb{R}}\) such as the previous one (considering the empty infimum as ∞) leads to the following spaces.
Definition 2.1
(Lowen [37])
 A1.:

\(\delta ( p,\{p\} ) =0\) for all \(p\in X\).
 A2.:

\(\delta ( p,\varnothing ) =\infty\) for all \(p\in X\).
 A3.:

\(\delta ( p,A\cup B ) =\min(\delta ( p,A ) ,\delta ( p,B ) )\) for all \(p\in X\) and \(A,B\in \mathcal{P} ( X ) \).
 A4.:

\(\delta ( p,A ) \leq\delta(p,A^{ ( s ) })+s\) for all \(p\in X\), \(A\in\mathcal{P} ( X ) \) and \(s\in [ 0,\infty ] \), where \(A^{(s)}=\{ q\in X / \delta ( q,A ) \leq s \} \).
 A4′.:

\((A^{(\alpha)})^{ ( \beta ) }\subseteq A^{(\alpha+\beta)}\) for all \(A\in\mathcal{P}(X)\) and all \(\alpha,\beta\in [ 0,\infty ] \).
Lemma 2.1
(qMspace ↪ Aspace)
A binary operation \(\ast: [ 0,1 ] \times [ 0,1 ] \rightarrow [ 0,1 ] \) is a continuous tnorm if \(( [ 0,1 ] ,\ast ) \) is a topological monoid with unity 1 such that \(a\ast b\leq c\ast d\) whenever \(a\leq c\) and \(b\leq d\) for \(a,b,c,d\in [ 0,1 ] \).
Definition 2.2
 FM1.:

\(M ( x,y,0 ) =0\).
 FM2.:

\(x=y\) if, and only if, \(M ( x,y,t ) =1\) for all \(0< t<\infty\).
 FM3.:

\(M ( x,z,t+s ) \geq M ( x,y,t ) \ast M ( y,z,s ) \) for all \(0\leq s,t<\infty\).
 FM4.:

\(M ( x,y,\cdot ) : [ 0,\infty [\, \rightarrow [ 0,1 ] \) is leftcontinuous.
Remark 2.1
In order to define a Hausdorff topology and to prove some existing results (including Baire’s theorem), George and Veeramani [7] slightly modified the concept of fuzzy metric space introduced by Kramosil and Michálek. It is well known that all fuzzy (quasi)metric spaces in the sense of George and Veeramani are also fuzzy (quasi)metric spaces in the sense of Kramosil and Michálek. Hence, the results we will prove also include fuzzy metric spaces in the sense of George and Veeramani (see also [40]).
Proposition 2.1
(Grabiec [41])
For all \(x,y\in X\), \(M ( x,y,\cdot ) \) is a nondecreasing function on \([ 0,\infty [ \).
Proof
If \(z=y\) and \(s>0\) in FM3, we have \(M ( x,y,t+s ) \geq M ( x,y,t ) \ast M ( y,y,s ) =M ( x,y,t ) \ast1=M ( x,y,t ) \). □
Lemma 2.2
(qMspace ↪ F_{ q }Mspace)
Proof
3 Fuzzy approach spaces
In this section, the notion of approach space is generalized to the fuzzy setting and fuzzy approach spaces are presented.
Definition 3.1
 FA1.:

\(F ( x,A,0 ) =0\).
 FA2.:

\(F ( x,\varnothing,t ) =0\) for all \(t<\infty\).
 FA3.:

\(F(x,\{x\},t)=1\) for all \(t>0\).
 FA4.:

\(F(x,A,\cdot): [ 0,\infty [\, \rightarrow [ 0,1 ] \) is a leftcontinuous function.
 FA5.:

\(F(x,A\cup B,t)\geq\max ( F(x,A,t),F(x,B,t) )\).
 FA6.:

If we denote, for all \(r\in [ 0,\infty [ \),then$$\begin{aligned} & A^{(r)}= \bigl\{ y\in X / F(y,A,t)=1, \forall t>r \bigr\} \quad\mbox{and}\\ & A^{(\infty)}= \bigl\{ y\in X / F(y,A,\infty)=1 \bigr\} , \end{aligned}$$$$\begin{aligned} & F ( x,A,t+s ) \geq F\bigl(x,A^{(r)},t\bigr) \quad\mbox{for all }r\in [ 0,s[ \quad\mbox{and}\\ & F ( x,A,\infty ) \geq F\bigl(x,A^{(\infty)},t\bigr). \end{aligned}$$
If FA6 is omitted, then \((X,F)\) is a fuzzy semiapproach space (briefly, F _{ s } Aspace). The map F is symmetric (or the F_{ s }Aspace \((X,F)\) is symmetric) if \(F(x,\{y\},t)=F(y,\{x\},t)\) for all \(x,y\in X\) and all \(t>0\). Finally, \(( X,F ) \) is a crisp FAspace (or a crisp F _{ s }Aspace) if the image of F is in \(\{ 0,1 \} \).
Remark 3.1
Remark 3.2
The original definition of approach space establishes equality in condition A3. However, it will be replaced in FA5 with inequality. In fact, it would be possible to develop a similar theory for approach spaces by setting in A3 only the inequality ≥.
Kramosil and Michálek considered \(M(x,y,t)\) defined only for \(t\in [0,\infty [ \). In our definition of a fuzzy approach space, ∞ must be included trying to generalize the approach spaces. But it is not a problem because we can consider, in Definition 2.2, \(M(x,y,\infty)=1\) for all \(x,y\in X\).
When \(A=\{y\}\) is a single point, we denote \(F(x,\{y\},t)\) by \(F(x,y,t)\), and, similarly, we will use this notation with other maps.
The following examples are given to illustrate a possible interpretation of the previous conditions in order to study the dynamic of a fuzzy system that can appear in fuzzy decision making.
Example 3.1
Let X be a set of experts that express their opinions about one or more questions. For instance, one may consider the level of satisfaction with respect to a product that is going to come on to the market. This level can be expressed as a number between 0 and 1. Experts discuss among themselves and, in the course of time, they reach agreement about their points of view. In this way, an expert’s opinion is not constant in time and it changes according the other experts’ opinions. Then it seems reasonable to measure the ‘degree of agreement (or disagreement) between an expert \(x\in X\) and a experts’ committee \(A\subseteq X\) in the instant t’. This uncertainty can be interpreted as the difference between 1 and the minor distance between the expert’s opinion x (a number between 0 and 1) and the set of opinions of the rest of members of A.

Property FA1 means that, at the beginning, there is not agreement between the experts.

Property FA2 establishes that it is necessary, at least, for a person in the committee to reach an agreement.

Property FA3 says us that an expert’s opinion is always coherent with himself/herself.

Property FA4 shows that an expert’s opinion depends continuously on his/her opinion in the past, but not in the future.

Property FA5 guarantees that there being involved more participants, it will be easier to persuade an expert to reach an agreement with the rest.
The next example is inspired by the previous one.
Example 3.2
4 Two families of FAspaces and a commutative diagram
Firstly, we will show that there are two families of spaces satisfying the above six properties.
Theorem 4.1
(Aspace ↪ FAspace)
Proof
Theorem 4.2
(F_{ q }Mspace ↪ FAspace)

If \(x,y\in X\) verify that \(F_{M} ( x,y,t ) =1\) for every \(0< t<\infty\), then \(x=y\).
Proof
 FA7.:

If \(x,y\in X\) are such that \(F ( x,y,t ) =1\) for every \(0< t<\infty\), then \(x=y\).
Corollary 4.1
Proof
5 Characterization of FAspaces
Now, we ask about the fuzzy maps \(F:X\times\mathcal{P}(X)\times{}[ 0,\infty]\rightarrow{}[0,1]\) that could provide X with an FAspace structure. In these cases, it could be interesting to weaken the conditions FA5 and FA6.
Proposition 5.1

\(F(x,A,\infty)=1\) for all \(x\in X\) and all \(A\in\mathcal{P}(X)\).

\(F ( x,A,t ) =1\) for all \(x\in A\subseteq X\) and all \(t>0\).
 FA5′.:

For every nonempty subsets \(A,B\subseteq X\) such that \(A\neq B\) and \(A\cup B\subset X\), for all \(t>0\) and for all \(x\in X\diagdown ( A\cup B ) \), we have \(F ( x,A\cup B,t ) \geq\max ( F ( x,A,t ) ,F ( x,B,t ) ) \).
The equality in FA5 is always true if, and only if, the equality is achieved in the previous inequality.
 FA6′.:

For all \(t,s\in \,] 0,\infty [ \), all \(r\in [ 0,s [ \), all \(\varnothing\neq A\subset X\) and all \(x\in X\diagdown A\), the following inequality holds: \(F(x,A,t+s)\geq F(x,A^{(r)},t)\).
Proof
We are going to see that there exist some cases in which, under the above mentioned hypotheses, the inequality (or the equality) trivially holds.
6 First properties of the FAspaces
The following definition is possible even if the set X has not been provided with a topology. In fact, in Section 7, we will show how to consider an appropriate topology on each FAspace depending on the fuzzy map F.
Definition 6.1
The following properties hold even if F does not satisfy the condition FA6.
Lemma 6.1
 (1)
If \(a\in A\), then \(F ( a,A,t ) =1\) for all \(t>0\).
 (2)
If \(a\in A\), then \(F ( x,a,t ) \leq F ( x,A,t ) \) for all \(t\geq0\).
 (3)
If \(A\subseteq B\subseteq X\), then \(F ( x,A,t ) \leq F ( x,B,t ) \) for all \(t\geq0\).
 (4)
If \(A\neq\varnothing\), \(\sup_{a\in A}F ( x,a,t ) \leq F ( x,A,t ) \) for all \(t\geq0\).
 (5)
If \(A\subseteq B\subseteq X\), then \(A^{ ( r ) }\subseteq B^{ ( r ) }\) for all \(r\geq0\).
 (6)If \(0\leq r\leq s\leq\infty\), thenIn particular, \(F ( x,A,t ) \leq F ( x,\overline{A},t ) \) for all \(t\geq0\).$$A\subseteq\overline{A}=A^{ ( 0 ) }\subseteq A^{ ( r ) }\subseteq A^{ ( s ) }\subseteq\cdots\subseteq A^{ ( \infty ) }\subseteq X. $$
 (7)
A is closed if, and only if, \(\overline{A}=A\).
 (8)
\(\varnothing^{ ( s ) }=\varnothing \) for all \(s\in [ 0,\infty [ \).
 (9)\(x\in A^{ ( r ) }\) if, and only if, \(x\in A^{ ( r+s ) }\) for every \(s>0\), i.e.,$$A^{ ( r ) }=\bigcap_{s>r}A^{ ( s ) }= \bigcap_{s>0}A^{ ( r+s ) }. $$
 (10)
If \(\Delta ( x,A ) =\{ s\in [0,\infty [\,/ x\in A^{(s)} \}\) is a nonempty set, then there exists \(a\in [ 0,\infty [ \) such that \(\Delta ( x,A ) = [ a,\infty [ \). If \(\Delta ( x,A ) \neq\varnothing\) then \(\inf\Delta ( x,A ) =\min\Delta ( x,A ) \).
 (11)The following properties are equivalent:
 (a)
\(A^{(\infty)}=X\) for all \(A\in\mathcal{P}(X)\).
 (b)
\(F(x,A,\infty)=1\) for all \(x\in X\) and all \(A\in\mathcal{P}(X)\).
 (c)
\(F(x,\varnothing,\infty)=1\) for all \(x\in X\).
 (d)
\(\varnothing^{(\infty)}=X\).
 (a)
Proof
(2) It follows from condition FA5.
(4) By item (2), we find that if \(a\in A\), then \(F ( x,a,t ) \leq F ( x,A,t ) \). Taking the supremum, \(\sup_{a\in A}F ( x,a,t ) \leq F ( x,A,t ) \).
(5) If \(A\subseteq B\) and \(r<\infty\), by item (3), \(F ( x,A,t ) \leq F ( x,B,t ) \) for each \(t\geq0\). If \(x\in A^{(r)}\), that is, \(F ( x,A,t ) =1\) for each \(t>r\), we have \(F ( x,B,t ) =1\) for each \(t>r\), i.e., \(x\in B^{(r)}\). If \(r=\infty\), a similar argument can be used.
(6) If \(a\in A\), by item (1), \(F ( a,A,t ) =1\) for each \(t>0\). Thus \(a\in\overline{A}\). Let \(r\leq s\). If \(x\in A^{(r)}\), i.e., \(F ( x,A,t ) =1\), for each \(t>r\), then, in particular, \(F ( x,A,t ) =1\) for each \(t>s\). Thus \(x\in A^{(s)}\). A similar argument is valid if \(r=\infty\) or \(s=\infty\). See Remark A.4 in the Appendix.
(7) It is trivial since \(A\subseteq\overline{A}\).
(8) From FA2, \(F ( x,\varnothing,t ) =0\) for all \(t<\infty\). So \(\varnothing^{(s)}= \{ x\in X / F(x,\varnothing,t)=1, \forall t>s \} =\varnothing\) if \(s<\infty\).
(9) By definition, \(x\in A^{ ( r ) }\) if, and only if, \(F ( x,A,t ) =1\) for all \(t>r\). This condition is equivalent to the following: for each \(s\in \,] r,\infty [ \), we see that \(F ( x,A,t ) =1\) holds for all \(t\in \,] r,\infty [ \), i.e., \(x\in A^{ ( r+s ) }\) for all \(s>0\).
(10) Let \(r\in\Delta ( x,A ) \), that is, \(x\in A^{ ( r ) }\). From the previous item, \(x\in \bigcap_{s>r}A^{ ( s ) }\), i.e., \(x\in A^{ ( s ) }\) for all \(s>r\). So \([ r,\infty [\, \subseteq \Delta ( x,A ) \). This proves that \(\Delta ( x,A ) \) is a real nonbounded interval. Let \(a=\inf\Delta ( x,A ) \). It is clear that \(0\leq a<\infty\). For all \(s>a\) we have \(s\in\Delta ( x,A ) \), that is, \(x\in A^{(s)}\). The previous item implies that \(x\in A^{(a)}\), so \(a\in\Delta ( x,A ) \) and we deduce that \(\Delta ( x,A ) = [ a,\infty [ \). Its infimum is a, which is the minimum.
(11) Clearly, (a) ⇔ (b) ⇒ (c) ⇔ (d). To prove (c) ⇒ (b), suppose that \(F(x,\varnothing,\infty)=1\) for all \(x\in X\). Then, for all \(A\in \mathcal{P}(X)\), by item (3), we have \(F(x,A,\infty )\geq F(x,\varnothing,\infty)=1\). □
The following properties refine the previous items in FAspaces using property FA6.
Lemma 6.2
 (1)
\(F ( x,A,t ) =F ( x,\overline {A},t ) \) for all \(t\in [ 0,\infty ] \).
 (2)
\(F ( x,A,\cdot ) \) is a nondecreasing function in \([ 0,\infty ] \).
 (3)
A is closed if, and only if, for each \(x\in X\) such that \(F ( x,A,t ) =1\) for all \(t>0\), we have \(x\in A\).
 (4)
\(( X,F ) \) satisfies property FA7 if, and only if, points are closed subsets, that is, \(\overline{ \{ x \} }=\{x\}\).
 (5)
\((A^{ ( r ) })^{ ( s ) }\subseteq A^{ ( r+s ) }\) for all \(r,s\in [ 0,\infty ] \).
 (6)
\(A^{ ( r ) }\) is closed for all \(r\geq0 \).
 (7)
The closure \(\overline{A}\) of A is a closed set, that is, \(\overline{\overline{A}}=\overline{A}\).
 (8)
\(F(x,A,\infty)=F(x,A^{(r)},\infty)\) for all \(r\in [ 0,\infty ] \).
Proof
(2) Let \(t\in [ 0,\infty [ \) and \(s\in \,] 0,\infty [ \) be arbitrary numbers. Items (3) and (6) of Lemma 6.1 imply that, as \(A\subseteq A^{ ( s/2 ) }\), \(F ( x,A,t ) \leq F(x,A^{ ( s/2 ) },t)\). Property FA6 implies that \(F ( x,A,t+s ) \geq F(x,A^{(s/2)},t)\geq F ( x,A,t ) \). Finally, by FA6 with \(r=0\), we have \(F ( x,A,\infty ) \geq F(x,A^{(0)},t)=F(x,\overline{A},t)=F(x,A,t)\). Therefore, \(F ( x,A,\cdot ) \) is a nondecreasing function in \([ 0,\infty ] \).
(4) Suppose that \(( X,F ) \) satisfies the condition FA7. Let \(y\in\overline{ \{ x \} }\), that is, \(F(y,x,t)=1\) for all \(t>0\). So \(y=x\). The converse is similarly true.
(6) \(\overline{A^{ ( r ) }}=(A^{ ( r ) })^{(0)}\subseteq A^{ ( r+0 ) }=A^{ ( r ) }\).
(7) \(\overline{\overline{A}}=(A^{ ( 0 ) })^{(0)}\subseteq A^{ ( 0+0 ) }=A^{ ( 0 ) }=\overline {A}\).
(8) Taking into account that \(A\subseteq A^{(r)}\), \(F(x,A,t)\leq F(x,A^{(r)},t)\) for all \(t\in [ 0,\infty ] \). The opposite inequality follows from the condition FA6. □
Property (8) of the previous lemma is not intuitive. Property (6) of Lemma 6.1 means that the family of closures \(\{A^{(r)}\}_{r\geq0}\) is increasing. However, F associates the same value to all closures when \(t=\infty\). Although it can be less than 1, the following proposition proves that it is natural that this number takes the value 1.
Proposition 6.1
Proof
7 The induced topology in FAspaces
Proposition 7.1
 (1)
\(A^{ ( s ) }\cup B^{ ( s ) }\subseteq ( A\cup B ) ^{ ( s ) }\).
 (2)
If \(( X,F ) \) verifies the equality in FA5 and \(F ( x,A,\cdot ) \) and \(F ( x,B,\cdot ) \) are nondecreasing in \([ 0,\infty ] \), then \(( A\cup B ) ^{ ( s ) }=A^{ ( s ) }\cup B^{ ( s ) }\).
Proof
(1) As \(A\subseteq A\cup B\) and \(B\subseteq A\cup B\), item (5) of Lemma 6.1 implies \(A^{ ( s ) }\subseteq ( A\cup B ) ^{ ( s ) }\) and \(B^{ ( s ) }\subseteq ( A\cup B ) ^{ ( s ) }\), and then \(A^{ ( s ) }\cup B^{ ( s ) }\subseteq ( A\cup B ) ^{ ( s ) }\).
Theorem 7.1
 (1)
∅, \(X\in\Upsilon_{F}\).
 (2)
The arbitrary intersection of closed subsets of X is also a closed subset of X.
 (3)
If \(( X,F ) \) reaches the equality in FA5, then a finite union of closed subsets of X is also a closed subset of X.
Proof
(2) Let \(\{ A_{i} \} _{i\in I}\) be a family of closed subsets of X and let \(B=\cap_{i\in I}A_{i}\) be the intersection. If \(B=\varnothing\), B is closed. Suppose that \(B\neq \varnothing\) and let \(x\in\overline{B}\) be a point. This is equivalent to \(F ( x,B,t ) =1\) for all \(t>0\). As \(B\subseteq A_{j}\), for each \(j\in I\), item (3) of Lemma 6.1 implies \(F ( x,A_{j},t ) \geq F ( x,B,t ) =1\) for all \(t>0\). As \(F ( x,A_{j},t ) =1\) for all \(t>0\), we have \(x\in\overline{A_{j}}=A_{j}\) for all \(j\in I\). So \(x\in\cap_{i\in I}A_{i}=B\) and B is closed.
(3) Let \(A_{1},A_{2}\subseteq X\) be two closed and nonempty subsets of X. From item (2) of Proposition 7.1, we have \(\overline{A_{1}\cup A_{2}}= ( A_{1}\cup A_{2} ) ^{(0)}=A_{1}^{ ( 0 ) }\cup A_{2}^{ ( 0 ) }=\overline{A_{1}}\cup\overline{A_{2}}=A_{1}\cup A_{2}\), so \(A_{1}\cup A_{2}\) is closed. The rest can be proved by induction. □
Taking the complement in X, the previous result has the following corollary.
Corollary 7.1
If an FAspace \((X,F)\) verifies the equality in FA5 axiom, then the family \(\Upsilon_{F}\) is a topology on X.
In fact, for all \(x\in X\), the collection \(\beta_{x}= \{ \{x\}^{(r)} ,r>0 \} \) is a basis of neighborhoods of \(\Upsilon_{F}\) at the point x.
See Proposition A.6 in the Appendix.
If \(( X,F ) \) is an FAspace which verifies the equality in the FA5 axiom, the (fuzzy) topology in X induced by the fuzzy metric F is the topology \(\Upsilon_{F}\) defined in Corollary 7.1.
8 Fixed point theory in FAspaces
In this section, in order to show that the category of FAspaces is appropriate to study fixed point theorems, we present a simple but interesting result in this setting. Its main advantage is that it can be particularized at the same time to the categories of metric spaces, quasimetric spaces, approach spaces, and fuzzy metric spaces (both in the sense of Kramosil and Michálek and in the sense of George and Veeramani). Before that, we present the following notions, which are the natural definitions associated to the topology \(\Upsilon_{F}\). Notice that F is not necessarily symmetric, so we must distinguish right and left notions.
Definition 8.1

rightCauchy sequence if for all \(r>0\), there exists \(n_{0} \in\mathbb{N}\) such that if \(m>n\geq n_{0}\), then \(x_{n}\in\{x_{m}\}^{(r)}\), in the sense that$$F ( x_{n},x_{m},t ) =1 \quad\mbox{for all }t>r; $$

leftCauchy sequence if for all \(r>0\), there exists \(n_{0}\in\mathbb{N}\) such that if \(m>n\geq n_{0}\), then \(x_{m}\in\{x_{n}\}^{(r)}\), in the sense that$$F ( x_{m},x_{n},t ) =1 \quad\mbox{for all }t>r; $$

Cauchy sequence if it is both rightCauchy and leftCauchy;

convergent sequence to x if, for all \(r>0\), there exists \(n_{0}\in\mathbb{N}\) such that if \(n\geq n_{0}\), then \(x_{n}\in\{x\}^{(r)}\) and \(x\in\{x_{n}\}^{(r)}\).
We say that \(( X,F ) \) is complete if every Cauchy sequence in \(( X,F ) \) is convergent to a point of X.
Remark 8.1
Notice that a subset only containing a pair of points of X must be bounded. But two possibilities arise when a Mspace is seen as a FMspace. From a probabilistic point of view, (2) shows that \(D_{A}(t)\) does not take the value 1 when \(t\in [ 0,\infty [ \). But, from a deterministic point of view, it seems to be reasonable that, when A is bounded, there must exist a point \(t_{0}\) such that \(P ( d(a,b)\leq t_{0} ) =1\) for all \(a,b\in A\). In fact, our way to embed qMspaces as FMspaces is closer to the second viewpoint (see Lemma 2.2). Hence, in the following result, the existence of \(x_{0}\) seems to be natural, because the set \(\{x_{0},Tx_{0}\}\) must be bounded.
Next, we present the main result of this section.
Theorem 8.1
Furthermore, if for all \(x,y\in\operatorname*{Fix}(T)\) we have \(\lim_{t\rightarrow\infty}F(x,y,t)=1\), then T has a unique fixed point.
Proof
The first two arguments of F are intrinsically different. Hence, it makes no sense to impose the requirement that F is symmetric. However, to compensate the lack of symmetry in F, we introduce the following condition.
Definition 8.2
Given \(M>0\), we say that a F_{ s }Aspace \(( X,F ) \) is Msymmetric if \(F(x,y,t)\leq M F(y,x,t)\) for all \(x,y\in X\) and all \(t>0\).
Proposition 8.1
 (1)
\(\{x_{n}\}\) is a rightCauchy sequence.
 (2)
\(\{x_{n}\}\) is a leftCauchy sequence.
 (3)
\(\{x_{n}\}\) is a Cauchy sequence.
Theorem 8.2
 (⊛):

\(( X,F ) \) is Msymmetric and there exist \(x_{0}\in X\) and \(t_{0}\in(0,\infty)\) such that \(F(x_{0},Tx_{0},t_{0})=1\).
Proof
Repeating, step by step, the arguments of the proof of Theorem 8.1, and using \(F(x_{0},Tx_{0},t_{0})=1\), we deduce that \(\{x_{n}\}\) is a rightCauchy sequence in \((X,F)\). As \((X,F)\) is Msymmetric, Proposition 8.1 guarantees that \(\{ x_{n}\}\) is a Cauchy sequence in \(( X,F ) \). Then the proof of Theorem 8.1 can be repeated point by point. □
Example 8.1
Declarations
Acknowledgements
The authors are very grateful to the Spanish Government for Grant DPI201230651 and to the Basque Government for Grant IT37810. This paper has been partially supported by projects FQM178, FQM268 and FQM265 of the Andalusian CICE, Spain.
Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.
Authors’ Affiliations
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