On Metrization of the Topologies Induced by Fuzzy Metrics

Since Zadeh [1] first proposed fuzzy set theory in 1965, many researchers have defined concepts of fuzzy metric spaces and studied their properties in different ways [2–4]. Inspired by the notion of probabilistic metric spaces, Kramosil and Michalek [5] in 1975 introduced the notion of fuzzymetric, a fuzzy set in the Cartesian product X × X × R satisfying certain conditions. Later, George and Veeramani [6] used the concept of continuous t-norms to modify this definition of fuzzy metric space and showed that every fuzzy metric space generates a Hausdorff first-countable topology. So far the GV-fuzzy metric theory has been developed by many researchers. Many important topics about the classical metric spaces were developed to fuzzy metric spaces. In this process, it was found that the theory of fuzzy metric was very different from the classical theory of metric. For example, Gregori and Romaguera [7] proved that there exists a GVfuzzy metric space which is not completable. Gregori and Romaguera [8] characterized the class of completable strong fuzzy metric spaces. Recently, some good related results about fixed point in fuzzy metric spaces were introduced. For example, one can see the works [9–11]. Meanwhile, the fuzzy metrics have been applied to domain theory, color image processing, and analysis of algorithms (see [12–17]). In 2000, Gregori and Romaguera [18] obtained a somewhat surprising stronger result. )ey proved that every GV-fuzzy metric generates a metrizable topology. )is important result connects the GV-fuzzy metric and the classical metric. However, the form of the metric function has not been explored in existing literature. It is just the main goal of the present paper. In this paper, we first introduce the concept of a stratified function in a fuzzy metric space which is slightly different from the GV-fuzzy metric space and show that the topology induced by the family of stratified functions is compatible with the metrizable topology. )en, under some special conditions, we give the concrete metric function whose topology coincides with the metrizable topology. )e structure of the paper is as follows. In the next section, we give the preliminary notions on fuzzy metrics, with which we deal. In Section 3, we show our main results. Finally, we give our concluding remarks in Section 4.


Introduction
Since Zadeh [1] first proposed fuzzy set theory in 1965, many researchers have defined concepts of fuzzy metric spaces and studied their properties in different ways [2][3][4]. Inspired by the notion of probabilistic metric spaces, Kramosil and Michalek [5] in 1975 introduced the notion of fuzzy metric, a fuzzy set in the Cartesian product X × X × R satisfying certain conditions. Later, George and Veeramani [6] used the concept of continuous t-norms to modify this definition of fuzzy metric space and showed that every fuzzy metric space generates a Hausdorff first-countable topology. So far the GV-fuzzy metric theory has been developed by many researchers. Many important topics about the classical metric spaces were developed to fuzzy metric spaces. In this process, it was found that the theory of fuzzy metric was very different from the classical theory of metric. For example, Gregori and Romaguera [7] proved that there exists a GVfuzzy metric space which is not completable. Gregori and Romaguera [8] characterized the class of completable strong fuzzy metric spaces. Recently, some good related results about fixed point in fuzzy metric spaces were introduced. For example, one can see the works [9][10][11]. Meanwhile, the fuzzy metrics have been applied to domain theory, color image processing, and analysis of algorithms (see [12][13][14][15][16][17]).
In 2000, Gregori and Romaguera [18] obtained a somewhat surprising stronger result. ey proved that every GV-fuzzy metric generates a metrizable topology. is important result connects the GV-fuzzy metric and the classical metric. However, the form of the metric function has not been explored in existing literature. It is just the main goal of the present paper.
In this paper, we first introduce the concept of a stratified function in a fuzzy metric space which is slightly different from the GV-fuzzy metric space and show that the topology induced by the family of stratified functions is compatible with the metrizable topology.
en, under some special conditions, we give the concrete metric function whose topology coincides with the metrizable topology. e structure of the paper is as follows. In the next section, we give the preliminary notions on fuzzy metrics, with which we deal. In Section 3, we show our main results. Finally, we give our concluding remarks in Section 4.

Preliminaries
In this section, we first introduce some basic concepts and properties of fuzzy metric spaces.
Definition 1 (see [19] (1) * is associative and commutative (2) * is continuous e following continuous t-norms are used in this paper: (1) Also, we say the t-norm Δ ′ is stronger than the t-norm In such case, we denote it as Δ ′ ≥ Δ ″ . It is easy to see that Δ 3 ≥ Δ 2 ≥ Δ 1 .
In the sense of Gregori and Veeramani [6], a GV-fuzzy metric is defined by the follows.
Definition 2. Let X be a nonempty set and * be a continuous t-norm. A fuzzy metric M on the set X is a mapping M: X 2 × (0, ∞) ⟶ (0, 1] satisfying the following conditions: for all x, y, z ∈ X, s, t > 0: If M is a GV-fuzzy metric on X, then the 3-tuple (X, M, * ) is said to be a GV-fuzzy metric space. In that case, if confusion is not possible, we call X a GV-fuzzy metric space for short. e following is a well-known result.
Gregori and Veeramani proved in [7] that every GV-fuzzy metric M on X generates a topology τ M which has as a base where for all x ∈ X, r ∈ (0, 1), and t > 0. ey proved that for each Also, by using Kelley metrization lemma [21], they also proved that τ M is a metrizable topology.

Main Results
First, we introduce the concept of a stratified function in a GV-fuzzy metric space.
Definition 3. (X, M, * ) is a GV-fuzzy metric space. Let r ∈ (0, 1) and x, y ∈ X; set en, d r is called a r-stratified function with respect to (X, M, * ), d r : 0 < r < 1 , the family of stratified functions.
To avoid the occurrence of the empty set, by a fuzzy metric in the rest of this paper, we mean a GV-fuzzy metric satisfying (4) e function M(x, y, ·) is strictly increasing for the fixed points x, y∈ X, if and only if for any r ∈ (0, 1), . From the arbitrariness of y, we know that It is easy to see that erefore, d r 0 (x, y) > inf t > 0: M(x, y, t) ≥ 1 − r 0 , which conflicts with (8).
Conversely, suppose M(x, y, ·) is strictly increasing. Let Obviously, inf t > 0: Now, from Lemma 2 (3), it is easy to see that the topology τ M can be induced by the family of stratified functions. at is, we obtain the following theorem. Theorem 1. Let D � d r : 0 < r < 1 be the family of stratified functions with respect to a fuzzy metric space (X, M, * ), N r (x, t) be defined by (8), and Then, (1) B x is a base of neighborhoods at x ∈ X.
(2) e topology τ D generated by B x : x ∈ X coincides with the topology τ M .
Generally, a stratified function is not a pseudometric. In fact, we have the following result.
Proof. For any r ∈ (0, 1), it is obvious that d r (x, y) ≥ 0, d r (x, y) � d r (y, x), and d r (x, y) � 0 when x � y. us, to complete the proof, we only have to prove that d r (x, y) ≤ d r (x, z) + d r (z, y) if and only if M satisfies condition (15).
Now, we explore the metric which induces the topology τ M .
for any t 1 , t 2 ∈ R + , and f is left continuous and increasing. (20) If one of the following conditions is satisfied: then d is a metric on X.
Next, we prove d is a metric on X, that is, d satisfies the following properties: for any x, y, z ∈ X, Journal of Mathematics e conclusion (M 1 ) is obvious. For the conclusion (M 2 ), it is easy to see that d(x, y) � 0 if x � y. Now, we suppose d(x, y) � 0; however, x ≠ y. en, there exists t 0 > 0 such that M(x, y, t 0 ) ≠ 1, that is, M(x, y, t 0 ) < 1. Since K is continuous at 0, there exists 0 < r 0 < G(t 0 ) such that K(r 0 ) < 1 − M(x, y, t 0 ). is is in direct contradiction to (23). us, d(x, y) � 0 implies that x � y.
To prove (M 3 ), we take r 1 > d(x, z) and r 2 > d(z, y) arbitrarily. From (23), we know (24) Now, suppose that G(t) > r 1 + r 2 . Let Obviously, t 0 ≤ t. It is easy to prove that G(t 0 ) ≤ r 1 . In fact, if G(t 0 ) > r 1 , from the left continuity of G, there exists By the left continuity of G again, we know there exists ε > 0 such that By the definition of t 0 , there exists 0 < s 1 ≤ t such that G(s 1 ) > r 1 and s 1 ≤ t 0 + ε. Noting that G is increasing, we obtain that G(t 0 + ε) ≥ G(s 1 ) > r 1 . From (i) and (ii), we have Combining conditions (I) and (II), we get M(x, y, t) ≥ 1 − K(r 1 + r 2 ).
By the definition of d, we know d(x, y) ≤ r 1 + r 2 . Since r 1 > d(x, z) and r 2 > d(z, y), we obtain directly. (29) Then, ρ is a metric on X and the topology τ ρ induced by ρ coincides with the topology τ M .

Conclusions
In the present paper, we investigate some metric structures in a fuzzy metric space. Especially, we give the concrete form of metric function with respect to the metrizable topology for a fuzzy metric under two special cases. Based on the results in the paper, interesting future research studies about the related topics may be prospective. Moreover, the technique used in this paper is suggestive to discuss the related problem in the general case.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.

Authors' Contributions
Jianrong Wu was responsible for conceptualization, methodology, funding acquisition, and review and editing. Hao Yang contributed to formal analysis, investigation, and original draft preparation. All authors have read and agreed to the published version of the manuscript.