Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 88 https://internationalpubls.com Topological Study on Revised Fuzzy Metric Spaces and Their Generalization 1Thangathamizh R, 2Abdul Razak, 3Rajalakshmi S, 4Gnanabala K, 5Thangammal R, 6Shanmugavel P Jeppiaar Institute of Technology, Sriperumpudhur, Kanchipuram, email: thamizh1418@gmail.com., K.Ramakrishnan College of Engineering, Samayapuram, Trichy, email: arrazak76@gmail.com., Mahalashmi Women’s College of Arts and Science, Paruthipattu, Avadi, Chennai, email: gbalamaths@gmail.com., Chennai Institute of Technology, Chennai, email: rajalakshmi301991@gmail.com., Selvam college of Technology (Autonomous), Namakkal, email: rthangam1981@gmail.com. Selvamm arts and science college (Autonomous), Namakkal, email: p.sham1988@gmail.com. Article History: Received: 12-11-2024 Revised:24-12-2024 Accepted:09-01-2025 Abstract: Introduction In this paper, we explore the concept of metric functions within a revised fuzzy metric space. The study focuses on understanding the relationships between these metric functions and the topological structures they generate. Specifically, we introduce the concept of a stratified function within this framework and investigate its implications. Objectives The main objectives of this paper are: 1. To define and analyze stratified functions in a revised fuzzy metric space. 2. To demonstrate that the topology generated by the family of stratified functions coincides with the topology generated by the revised fuzzy metric. 3. To derive the concrete form of the metric function under specific conditions. Methods We approach these objectives by first introducing the notion of a stratified function in a revised fuzzy metric space. Using this concept, we prove that the topology generated by the family of stratified functions is identical to the topology generated by the revised fuzzy metric. Additionally, we explore the conditions under which a specific form of the metric function can be determined. Results Our findings show that the topology generated by the stratified functions indeed coincides with the topology generated by the revised fuzzy metric. Moreover, under certain special conditions, we can obtain a concrete representation of the metric function. Conclusion This paper provides a deeper understanding of the structure of revised fuzzy metric spaces. The introduction of stratified functions serves as a key tool for analyzing the topology of these spaces, and our results offer a concrete form for the metric function under specific conditions, contributing to the broader study of fuzzy metric spaces. Keywords: t-conorm, Revised fuzzy metric 2020 Mathematical classification: 37C25, 46S40, 46N20, 47H10 1. Introduction Many scholars have created ideas of fuzzy metric spaces and examined their characteristics in various ways since Zadeh [30] initially suggested fuzzy set theory in 1965. In 1975, Kramosil and Michalek mailto:thamizh1418@gmail.com https://cran.r-project.org/web/classifications/MSC-2010.html#code:37C25 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 89 https://internationalpubls.com [8] developed the concept of fuzzy metric, which is a fuzzy set in the Cartesian product that meets specific requirements. This concept was inspired by the idea of probabilistic metric spaces. Subsequently, George and Veeramani [2] modified this notion of fuzzy metric space by introducing the idea of continuous t-norms and shown that all fuzzy metric spaces produce a Hausdorff first- countable topology. The theory of GV-fuzzy metric has been established thus far by several academics. A great deal of knowledge on classical metric spaces was extended to fuzzy metric spaces. It was discovered throughout this procedure that the fuzzy metric theory differed greatly from the traditional theory of metric. As an illustration, Gregori and Romaguera [3] demonstrated the existence of an incompletable GV fuzzy metric space. A classification of the class of completable strong fuzzy metric spaces was provided by [2, 4-7, 12-13, 23, 28-29]. Jainrong Wu and Hao Yang [11] found a stronger result in 2000, which was a little unexpected. They demonstrated that a metrizable topology is produced by each GV-fuzzy metric. This crucial finding establishes a link between the classical metric and the GV-fuzzy metric. Nonetheless, the metric function's shape hasn't been examined in any of the previous research works. That is only the current paper's primary objective. Most interesting motivations is the introduction of Revised fuzzy metric spaces by Alexandar sostak [1]. Later on, Olga Grigorenko [17], Juan jose Minana, Alexander Sostak, Oscar Valero introduced “On t-conorm based Fuzzy (Pseudo) metrics”, they develop the basics of the theory of CB-fuzzy (pseudo) metrics and compare them with “classic” fuzzy (pseudo) metrics [2020]. After that Muraliraj and Thangathamizh [21] proved the existence of fixed points in Revised fuzzy metric space. Good, related results about fixed point in fuzzy metric spaces were introduced recently [14, 16, 26-27] In this study, we first present the idea of a stratified function in a revised fuzzy metric space, which differs somewhat from the RGV-fuzzy metric space. We next demonstrate that the metrizable topology may coexist with the topology produced by the family of stratified functions. The concrete metric function whose topology agrees with the metrizable topology is then provided, subject to certain restrictions. The paper is organized as follows. We address the early ideas on revised fuzzy metrics in the next section. Section 3 presents our primary findings. Lastly, we conclude in Section 4 with some last thoughts. 2. Preliminaries In this section, we first introduce some basic concepts and properties of revised fuzzy metric spaces. Definition 1[30]. A binary operation ⨁: [0, 1]2 ⟶ [0, 1] is a continuous t-conorm if it satisfies the following conditions: (1) ⨁ is associative and commutative (2) ⨁ is continuous (3) 0 ⨁ 𝕡 = 𝕡 for each 𝕡 ∈ [0,1] (4) 𝕡 ⨁ 𝕢 ≤ 𝕣 ⨁ 𝕤 whenever 𝕡 ≤ 𝕣 and 𝕢 ≤ 𝕤 with 𝕡, 𝕢, 𝕣, 𝕤 ∈ [0, 1]. The following continuous t-conorms are used in this paper: Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 90 https://internationalpubls.com 𝕡 ⋁1 𝕢 = 𝑚𝑎𝑥{𝕡, 𝕢}, 𝕡 ⋁2 𝕢 = 𝕡 + 𝕢 − 𝑎𝕢, 𝕡 ⋁3 𝕢 = 𝑚𝑖𝑛{𝕡 + 𝕢, 1}. (1) In the sense of Alexander sostack, a GV-fuzzy metric is defined by the follows. Definition 2[1]. Let 𝔘 be a nonempty set and ⨁ be a continuous t-conorm. A revised fuzzy metric 𝕎 on the set 𝔘 is a mapping 𝕎: 𝔘2 × (0, ∞) ⟶ (0, 1] satisfying the following conditions: for all 𝕒, 𝕓, 𝕔 ∈ 𝔘, 𝑎𝑛𝑑 𝓉, 𝓈 > 0: (𝓡𝓖𝓥 𝟏) 𝕎(𝕒, 𝕓, 𝓉) < 1, (𝓡𝓖𝓥 𝟐) 𝕎(𝕒, 𝕓, 𝓉) = 0 if and only if 𝕒 = 𝕓, (𝓡𝓖𝓥 𝟑) 𝕎(𝕒, 𝕓, 𝓉) = 𝕎(𝕓, 𝕒, 𝓉), (2) (𝓡𝓖𝓥 𝟒) 𝕎(𝕒, 𝕓, 𝓉)⨁ 𝕎(𝕓, 𝕔, 𝓈) ≥ 𝕎(𝕒, 𝕓, 𝓉), (𝓡𝓖𝓥 𝟓) 𝕎(𝕒, 𝕓, −): (0, ∞) ⟶ (0,1] is continuous. If 𝕎 is a 𝓡𝓖𝓥 – revised fuzzy metric on 𝔘, then the 3-tuple (𝔘, 𝕎, ⨁) is said to be a 𝓡𝓖𝓥 – revised fuzzy metric space. In that case, if confusion is not possible, we call 𝔘 a 𝓡𝓖𝓥 – revised fuzzy metric space for short. The following is a well-known result. Lemma 1. Let 𝕎(𝕒, 𝕓, −) is non-increasing for all 𝕒, 𝕓 ∈ 𝔘. Alexander sostack in that every 𝓡𝓖𝓥 – revised fuzzy metric 𝕎 on 𝔘 generates a topology 𝜏𝕎 which has as a base {𝔅𝕎(𝕒, 𝕣, 𝓉): 𝕒 ∈ 𝔘, 𝕣(0,1), 𝓉 > 0} (3) were 𝔅𝕎(𝕒, 𝕣, 𝓉) = {𝕓 ∈ 𝔘: 𝕎(𝕒, 𝕓, 𝓉) < 𝕣}, for all 𝕒 ∈ 𝔘, 𝕣 ∈ (0,1), and 𝓉 > 0. (4) They proved that for each 𝕒 ∈ 𝔘, the family {𝔅𝕎 (𝕒, ( 1 𝕟 ) , ( 1 𝕟 )) : 𝕟 ∈ ℕ} is a local base at 𝕒. A sequence {𝕒𝕟} in {𝔘, 𝜏𝕎)} converges to 𝕒 ∈ 𝔘 if and only if log 𝑛 ⟶ ∞ 𝕎(𝕒𝕟, 𝕒, 𝓉) = 0 for all 𝓉 > 0. Also, by using Kelley metrization lemma [21], they also proved that 𝜏𝕎 is a metrizable topology. 3. Main Results First, we introduce the concept of a stratified function in a 𝓡𝓖𝓥 – revised fuzzy metric space. Definition 3. Let (𝔘, 𝕎, ⨁) is a 𝓡𝓖𝓥 – revised fuzzy metric space. Let 𝕣 ∈ (0,1) and 𝕒, 𝕓 ∈ 𝔘; set 𝕕𝕣(𝕒, 𝕓) = 𝑠𝑢𝑝{𝓉 > 0: 𝕎(𝕒, 𝕓, 𝓉) < 𝕣}. (5) then, 𝕕𝕣 is called a 𝕣 -stratified function with respect to (𝔘, 𝕎, ⨁), {𝕕𝕣: 0 < 𝕣 < 1}, the family of stratified functions. To avoid the occurrence of the empty set, by a revised fuzzy metric in the rest of this paper, we mean an RGV-fuzzy metric satisfying (𝓡𝓖𝓥 𝟔) log 𝑛 ⟶ ∞ 𝕎(𝕒, 𝕓, 𝓉) = 0, ∀ 𝕒, 𝕓 ∈ 𝔘. (6) Lemma 2. Let (𝔘, 𝕎, ⨁) be a revised fuzzy metric space, 𝕣 ∈ (0,1), 𝓉 > 0, 𝕒, 𝕓 ∈ 𝔘. then, Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 91 https://internationalpubls.com (1) For any 𝕕𝕣(𝕒, 𝕓) < 𝜆, 𝕎(𝕒, 𝕓, 𝜆) ≤ 𝕣 (2) The function 𝕕𝕣 is non-decreasing with respect to 𝕣 ∈ (0,1) (3) 𝔅𝕎(𝕒, 𝕣, 𝓉) = ℕ𝕎(𝕒, 𝓉), were ℕ𝕣(𝕒, 𝓉) = 𝕓 ∈ 𝔘: 𝕕𝕣(𝕒, 𝕓) < 𝑡. (7) (4) The function 𝕎(𝕒, 𝕓, −) is strictly non-increasing for the fixed points 𝕒, 𝕓 ∈ 𝔘, if and only if for any 𝕣 ∈ (0,1), 𝕕𝕣(𝕒, 𝕓) = 𝑠𝑢𝑝{𝓉 > 0: 𝕎(𝕒, 𝕓, 𝓉) < 𝕣}. (8) Proof (1) Let 𝜆 > 𝕕𝕣(𝕒, 𝕓). From Definition 3, there exists 0 < 𝓉 < 𝜆 such that 𝕎(𝕒, 𝕓, 𝓉) ≤ 𝕣. So, 𝕎(𝕒, 𝕓, 𝜆) ≤ 𝕣. (2) It follows from Lemma 1 directly. (3) Let 𝕓 ∈ 𝔅𝕎(𝕒, 𝕣, 𝓉), that is, 𝕎(𝕒, 𝕓, 𝓉) ≤ 𝕣. Since 𝕎(𝕒, 𝕓, −) is continuous and non-increasing, there exists 0 < 𝓉1 < 𝓉 such that 𝕎(𝕒, 𝕓, 𝓉1) ≤ 𝕣. From (5), we know 𝕕𝕣(𝕒, 𝕓) ≤ 𝓉1 < 𝑡. So, 𝕓 ∈ 𝕕𝕣(𝕒, 𝓉). From the arbitrariness of 𝕓, we know that 𝔅𝕎(𝕒, 𝕣, 𝓉) ⊆ ℕ𝕣(𝕒, 𝓉). (9) Let 𝕔 ∈ ℕ𝕣(𝕒, 𝓉); then, 𝕕𝕣(𝕒, 𝕔) < 𝓉. From (5), there exists 0 < 𝓉2 < 𝓉 such that 𝕎(𝕒, 𝕓, 𝓉2) ≤ 𝕣. therefore, 𝕎(𝕒, 𝕔, 𝓉) ≤ 𝕎(𝕒, 𝕓, 𝓉2) ≤ 𝕣. So, 𝕔 ∈ 𝔅𝕎(𝕒, 𝕣, 𝓉). From the arbitrariness of 𝕔, we know that ℕ𝕣(𝕒, 𝓉) ⊆ 𝔅𝕎(𝕒, 𝕣, 𝓉). (4) Suppose that (8) holds; however, 𝕎(𝕒, 𝕓, −) is not strictly decreasing. Then, there exist 𝓉1, 𝓉2 ∈ {𝓉 > 0: 0 < 𝕎(𝕒, 𝕓, 𝓉) < 1} such that 𝓉1 < 𝓉2 and 𝕎(𝕒, 𝕓, −) ≡ 𝕣0 on [𝓉1, 𝓉2]. thus, 𝑠𝑢𝑝{𝓉 > 0: 𝕎(𝕒, 𝕓, 𝓉) ≥ 𝕣0} ≥ 𝓉2 > 𝓉1 ≥ 𝑖𝑛𝑓{𝓉 > 0: 𝕎(𝕒, 𝕓, 𝓉) ≤ 𝕣0}. (10) It is easy to see that 𝑠𝑢𝑝{𝓉 > 0: 𝕎(𝕒, 𝕓, 𝓉) ≥ 𝕣0} = 𝑖𝑛𝑓{𝓉 > 0: 𝕎(𝕒, 𝕓, 𝓉) < 𝕣0}. (11) therefore, 𝕕𝕣0 (𝕒, 𝕓) > 𝑖𝑛𝑓{𝓉 > 0: 𝕎(𝕒, 𝕓, 𝓉) < 𝕣0}, which conflicts with (8). Conversely, suppose 𝕎(𝕒, 𝕓, −) is strictly decreasing. Let, 𝓉0 = 𝕕𝕣(𝕒, 𝕓) = 𝑖𝑛𝑓{𝓉 > 0: 𝕎(𝕒, 𝕓, 𝓉) < 𝑟}. (12) Obviously, 𝑖𝑛𝑓{𝓉 > 0: 𝕎(𝕒, 𝕓, 𝓉) ≤ 𝕣} ≤ 𝓉0. If 𝑖𝑛𝑓{𝓉 > 0: 𝕎(𝕒, 𝕓, 𝓉) < 𝑟} < 𝓉0, then there is 0 < 𝓉2 < 𝓉1 such that 𝕎(𝕒, 𝕓, 𝓉2) ≤ 𝕣, so 𝕎(𝕒, 𝕓, 𝓉2) < 𝑟. Since 𝕎(𝕒, 𝕓, −) is Right continuous at 𝓉0, there is 𝛿 > 0 such that 𝕎(𝕒, 𝕓, 𝓉0 − 𝛿) < 𝑟, which conflicts with the definition of 𝓉0. Thus, 𝑖𝑛𝑓{𝓉 > 0: 𝕎(𝕒, 𝕓, 𝓉) ≤ 𝕣} ≥ 𝓉0. So, 𝕕𝕣(𝕒, 𝕓) = 𝓉0 = 𝑖𝑛𝑓{𝓉 > 0: 𝕎(𝕒, 𝕓, 𝓉) ≤ 𝕣}. (13) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 92 https://internationalpubls.com Now, from Lemma 2 (3), it is easy to see that the topology 𝜏𝕎 can be induced by the family of stratified functions. That is, we obtain the following theorem. Theorem 1. Let 𝔐 = {𝕕𝕣: 0 < 𝕣 < 1} be the family of stratified functions with respect to a revised fuzzy metric space (𝔘, 𝕎, ⨁), ℕ𝕣(𝕒, 𝓉) be defined by (8), and 𝔅𝕒 = {ℕ𝕣(𝕒, 𝓉): 𝕣 ∈ (0, 1), 𝓉 > 0}. (14) Then, (1) 𝔅𝕒 is a base of neighborhoods at 𝕒 ∈ 𝔸. (2) The topology 𝜏𝕎 generated by{ 𝔅𝕒: 𝕒 ∈ 𝔸} coincides with the topology 𝜏𝕎. Generally, a stratified function is not a pseudo metric. In fact, we have the following result. Theorem 2. Let (𝔘, 𝕎, ⨁) be a revised fuzzy metric space. A stratified function (𝕕𝕣(𝕣 ∈ (0, 1))) is a pseudometric on 𝔸 if and only if 𝕎 satisfies the following condition: for any 𝕒, 𝕓, 𝕔 ∈ 𝔸, 𝓉1, 𝓉2 > 0, if 𝕎(𝕒, 𝕔, 𝓉1) < 𝑟, 𝕎(𝕔, 𝕓, 𝓉2) < 𝑟, then 𝕎(𝕒, 𝕓, 𝓉1 + 𝓉2) < 𝑟. (15) Proof. For any 𝕣 ∈ (0, 1), it is obvious that 𝕕𝕣(𝕒, 𝕓) ≥ 0, 𝕕𝕣(𝕒, 𝕓) = 𝕕𝕣(𝕓, 𝕒), and 𝕕𝕣(𝕒, 𝕓) = 0 when 𝕒 = 𝕓. Thus, to complete the proof, we only must prove that 𝕕𝕣(𝕒, 𝕓) ≤ 𝕕𝕣(𝕒, 𝕔) + 𝕕𝕣(𝕔, 𝕓) if and only if 𝕎 satisfies condition (15). Sufficiency. For any 휀 > 0, from Lemma 2 (1), we obtain 𝕎 (𝕒, 𝕔, 𝕕𝕣(𝕒, 𝕔) + 𝜀 2 ) < 𝑟, 𝕎 (𝕔, 𝕓, 𝕕𝕣(𝕔, 𝕓) + 𝜀 2 ) < 𝑟. (16) From (15), we have 𝕎 (𝕒, 𝕓, 𝕕𝕣(𝕒, 𝕔) + 𝕕𝕣(𝕔, 𝕓) + 𝜀 2 ) < 𝑟. (17) therefore, 𝕕𝕣(𝕒, 𝕓) ≤ 𝕕𝕣(𝕒, 𝕔) + 𝕕𝕣(𝕔, 𝕓) + 휀. (18) From the arbitrariness of 휀 > 0, we know 𝕕𝕣(𝕒, 𝕓) ≤ 𝕕𝕣(𝕒, 𝕔) + 𝕕𝕣(𝕔, 𝕓). Necessity. Suppose that 𝕎(𝕒, 𝕔, 𝓉1) < 𝑟, 𝕎(𝕔, 𝕓, 𝓉2) < 𝑟. By (RGV5), there exists 𝛿 > 0 such that 𝕎(𝕒, 𝕔, 𝓉1 − δ) < 𝑟, 𝕎(𝕔, 𝕓, 𝓉2 − δ) < 𝑟, (19) So, 𝕕𝕣(𝕒, 𝕔) ≤ 𝓉1 − 𝛿 and 𝕕𝕣(𝕔, 𝕓) ≤ 𝓉2 − 𝛿. Since 𝕕𝕣(𝕒, 𝕓) ≤ 𝕕𝕣(𝕒, 𝕔) + 𝕕𝕣(𝕔, 𝕓), we have 𝕕𝕣(𝕒, 𝕓) ≤ 𝓉1 + 𝓉2 − 2𝛿 < 𝓉1 + 𝓉2. By the definition of 𝕕𝕣(𝕒, 𝕓), there exists 𝓉0 < 𝓉1 + 𝓉2 such that 𝕎(𝕒, 𝕓, 𝓉0) < 𝑟. Thus, 𝕎(𝕒, 𝕓, 𝓉1 + 𝓉2) < 𝑟. Remark 1. Let 𝕎 satisfies (15) if ⨁ = ⋁3. Now, we explore the metric which induces the topology 𝜏𝕎. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 93 https://internationalpubls.com Definition 4. Let ℝ+ = [0, ∞). We call function 𝒢: ℝ+ ⟶ ℝ+ satisfies (i) the condition ℭ1, if 𝒢(0) = 0, 𝒢(𝓉) ≡ 0, and 𝒢 is non increasing and continuous at 0. (ii) the condition ℭ2, if 𝒢(0) = 0, 𝒢(0) > 0 as 𝓉 > 0, lim 𝓉 ⟶ +∞ 𝒢(𝓉) = +∞, 𝒢(𝓉1 + 𝓉2) ≤ 𝒢(𝓉1) + 𝒢(𝓉2) for any 𝓉1 + 𝓉2 ∈ ℝ+, and 𝒢 is right continuous and non-increasing. Theorem 3. Let (𝔘, 𝕎, ⨁) is revised fuzzy metric space; the functions 𝒦 and 𝒢 satisfy the conditions ℭ1 and ℭ2, respectively. Define a function 𝕕 on 𝔘2 as 𝕕(𝕒, 𝕓) = 𝑖𝑛𝑓{𝓈 > 0: if 𝒢(𝓉) > 𝓈, 𝑡ℎ𝑒𝑛 𝕎(𝕒, 𝕓, 𝓉) + 𝒦(𝓉) ≤ 0, ∀𝕒, 𝕓 ∈ 𝔘}. (20) If one of the following conditions is satisfied: (I) 𝕎 satisfies condition (15) (II) ⨁ ≥ 𝛥1, ∀𝓇1, 𝓇2 ∈ [0, ∞) 𝒦(𝓇1) + 𝒦(𝓇2) ≤ 𝒦(𝓇1) + 𝓇2, (21) then 𝕕 is a metric on 𝔘. Proof. First, we prove the following fact: if 𝒢(𝓉) < 𝓇 < 𝕕(𝕒, 𝕓), (22) then 𝕎(𝕒, 𝕓, 𝓉) < 𝓇 ≤ 𝒦(𝓇). (23) In fact, if 𝒢(𝓉) < 𝓇 < 𝕕(𝕒, 𝕓), from the definition of 𝕕, we obtain there exists 0 < 𝓈 < 𝓇 such that if 𝒢(𝓉) > 𝓈, then 𝕎(𝕒, 𝕓, 𝓉) + 𝒦(𝓇) ≤ 0. Therefore, if 𝒢(𝓉) > 𝓇, then 𝒢(𝓉) > 𝓈, and hence, 𝕎(𝕒, 𝕓, 𝓉) + 𝒦(𝑟) ≤ 𝕎(𝕒, 𝕓, 𝓉) + 𝒦(𝓈) ≤ 0. That is, 𝕎(𝕒, 𝕓, 𝓉) ≤ 𝒦(𝑟). Next, we prove 𝕕 is a metric on 𝔘, that is, 𝕕 satisfies the following properties: for any 𝕒, 𝕓, 𝕔 ∈ 𝔘, (M1) 𝕕(𝕒, 𝕓) ≥ 0, 𝕕(𝕒, 𝕓) = 𝕕(𝕓, 𝕒), (M2) 𝕕(𝕒, 𝕓) = 0 if and only if 𝕒 = 𝕓, (M3) 𝕕(𝕒, 𝕓) ≤ 𝕕(𝕒, 𝕔) + 𝕕(𝕔, 𝕓). the conclusion (M1) is obvious. For the conclusion (M2), it is easy to see that 𝕕(𝕒, 𝕓) = 0 if 𝕒 = 𝕓. Now, we suppose 𝕕(𝕒, 𝕓) ≥ 0; however, 𝕒 ≠ 𝕓. then, there exists 𝓉0 > 0 such that 𝕎(𝕒, 𝕓, 𝓉0) ≠ 0, that is, 𝕎(𝕒, 𝕓, 𝓉0) ≥ 0. Since 𝒦 is continuous at 0, there exists 0 < 𝓇 < 𝒢(𝓉) such that 𝒦(𝓇0) > 𝕎(𝕒, 𝕓, 𝓉0). This is indirect contradiction to (23). Thus, 𝕕(𝕒, 𝕓) = 0 implies that 𝕒 = 𝕓. To prove (M3), we take 𝕕(𝕒, 𝕔) < 𝓇1 and 𝕕(𝕔, 𝕓) < 𝓇2 arbitrarily. From (23), we know (i) If 𝒢(𝓉) > 𝓇1, then 𝕎(𝕒, 𝕔, 𝓉) ≤ 𝒦(𝓇1) ≤ 𝒦(𝓇1) + 𝓇2. (24) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 94 https://internationalpubls.com (ii) If 𝒢(𝓉) > 𝓇2, then 𝕎(𝕔, 𝕓, 𝓉) ≤ 𝒦(𝓇2) ≤ 𝒦(𝓇1) + 𝓇2. (25) Now, suppose that 𝓇1 + 𝓇2 < 𝒢(𝓉). Let 𝓉0 = 𝑠𝑢𝑝{𝓈: 0 < 𝓈 ≤ 𝓉, 𝒢(𝓈) > 𝓇1}. (26) Obviously, 𝓉0 ≤ 𝓉. It is easy to prove that 𝒢(𝓉0) ≤ 𝓇1. In fact, if 𝒢(𝓉0) ≤ 𝓇1, from the right continuity of 𝒢, there exists η > 0, such that 𝒢(𝓉0 − 𝜂) ≤ 𝓇1. By the definition of 𝓉0, we conclude that 𝓉0 ≤ 𝓉0 − 𝜂, a contradiction. The fact 𝒢(𝓉0) ≤ 𝓇1 implies that 𝒢(𝓉 − 𝓉0) ≤ (𝒢𝓉 − 𝒢𝓉0) ≤ 𝓇2 By the right continuity of 𝒢 again, we know there exists 휀 > 0 such that 𝒢(𝓉 − (𝓉0 + 휀)) ≤ 𝒢(𝓉 − (𝓉0 − 휀)) < 𝓇2. By the definition of 𝓉0, there exists 0 < 𝓈1 ≤ 𝑡 such that 𝒢(𝓈1) ≤ 𝓇1 and 𝓈1 ≤ 𝓉0 + 휀. Noting that G is increasing, we obtain that 𝒢(𝓉 − 휀)) ≤ 𝒢(𝓈1) < 𝓇1. From (i) and (ii), we have 𝕎(𝕔, 𝕓, (𝓉0 + 휀)) ≤ 𝒦(𝓇1) ≤ 𝒦(𝓇1 + 𝓇2) 𝕎(𝕔, 𝕓, 𝓉 − (𝓉0 + 휀)) ≤ 𝒦(𝓇2) ≤ 𝒦(𝓇1) + 𝓇2. (27) Combining conditions (I) and (II), we get 𝕎(𝕒, 𝕓, 𝓉) ≤ 𝒦(𝓇1 + 𝓇2). By the definition of 𝕕, we know 𝕕(𝕒, 𝕓) ≤ 𝓇1 + 𝓇2. Since 𝕕(𝕒, 𝕔) < 𝓇1 and 𝕕(𝕔, 𝕓) < 𝓇2, we obtain 𝕕(𝕒, 𝕓) ≤ 𝕕(𝕒, 𝕔) + 𝕕(𝕔, 𝕓). (28) directly. Theorem 4. (𝔘, 𝕎, ⨁) is a revised fuzzy metric space. 𝕎 satisfies condition (15) or ⨁ ≥ 𝛥1. Let 𝜌(𝕒, 𝕓) = 𝑠𝑢𝑝{𝕩 > 0: 𝕎(𝕒, 𝕓, 𝕩) + 𝕩 ≤ 0}, ∀𝕒, 𝕓 ∈ 𝔘. (29) Then, 𝜌 is a metric on 𝔘 and the topology 𝜏𝜌 induced by 𝜌 coincides with the topology 𝜏𝕎. Proof. Let 𝒢(𝓉) = 𝓉, 𝒦(𝓉) = 𝓉, ∀t ≥ 0. Then, 𝒦 and 𝒢 satisfy the conditions ℭ1 and ℭ2, respectively. Besides, 𝒦 satisfies condition (21). From Theorem 3, we know that 𝕕(𝕒, 𝕓) = 𝑠𝑢𝑝{𝓈 > 0: 𝑖𝑓 𝓉 > 𝓈, 𝕎(𝕒, 𝕓, 𝓉) + 𝓈 ≤ 0}, ∀𝕒, 𝕓 ∈ 𝔘. (30) is a metric on 𝔘. Thus, to show that 𝜌 is a metric, we only need to show 𝕕(𝕒, 𝕓) ≤ 𝜌(𝕒, 𝕔), ∀𝕒, 𝕓 ∈ 𝔘. In fact, if 𝕩 > 0, 𝕎(𝕒, 𝕓, 𝕩) + 𝕩 ≤ 0 and 𝓉 > 𝕩, then 𝕎(𝕒, 𝕓, 𝓉) + 𝕩 ≤ 𝕎(𝕒, 𝕓, 𝕩) + 𝕩 ≤ 0. Therefore, {𝕩 > 0: 𝕎(𝕒, 𝕓, 𝕩) + 𝕩 ≤ 0 } ⊆ {𝓈 > 0: 𝑖𝑓 𝓉 > 𝓈, 𝑡ℎ𝑒𝑛 𝕎(𝕒, 𝕓, 𝓉) + 𝓈 ≤ 0}. (31) From (29) and (30), we get 𝜌(𝕒, 𝕓) ≥ 𝕕(𝕒, 𝕓). On the other hand, from (30), for any 휀 > 0, there exists 𝓈 > 0 such that 𝓈 > 𝕕(𝕒, 𝕓) + 휀 and 𝕎(𝕒, 𝕓, 𝓉) + 𝓈 ≤ 0 when 𝓉 > 𝓈. thus, Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 95 https://internationalpubls.com 𝕎(𝕒, 𝕓, 𝕕(𝕒, 𝕓) + 휀) + 𝓈 ≤ 0. (32) and hence 𝕎(𝕒, 𝕓, 𝕕(𝕒, 𝕓) + 휀) + 𝕕(𝕒, 𝕓) + 휀) < 0. (33) From (29), we get 𝜌(𝕒, 𝕓) ≤ 𝕕(𝕒, 𝕓) + 휀. From the arbitrariness of ε > 0, we have 𝜌(𝕒, 𝕓) ≤ 𝕕(𝕒, 𝕓). thus, 𝜌(𝕒, 𝕓) = 𝕕(𝕒, 𝕓). Let 𝓇 ∈ (0,1), 𝕒 ∈ 𝔘. We put 𝒰(𝕒, 𝓇) = {𝕓 ∈ 𝔘: 𝜌(𝕒, 𝕓) < 𝓇}. It is easy to show that 𝒰(𝕒, 𝓇) ⊆ ℬ𝕎(𝕒, 𝓇, 𝓇) ⊆ 𝒰(𝕒, 2𝓇). (34) In fact, for any 𝕓 ∈ 𝒰(𝕒, 𝓇), 𝑠𝑢𝑝{𝕩 > 0: 𝕎(𝕒, 𝕓, 𝕩) + 𝕩 ≤ 0} < 𝓇. Thus, there is 𝕩 > 0 such that 𝕩 < 𝓇 and 𝕎(𝕒, 𝕓, 𝕩) + 𝕩 ≤ 0. So, 𝕎(𝕒, 𝕓, 𝓇) + 𝓇 ≤ 0, that is, 𝕓 ∈ ℬ𝕎(𝕒, 𝓇, 𝓇). Hence, 𝒰(𝕒, 𝓇) ⊆ ℬ𝕎(𝕒, 𝓇, 𝓇). (35) On the other hand, for any 𝕔 ∈ ℬ𝕎(𝕒, 𝓇, 𝓇), 𝕎(𝕒, 𝕔, 𝓇) + 𝓇 < 0. From (29), we get 𝜌(𝕒, 𝕔) ≤ 𝓇 < 𝓇. So, 𝕔 ∈ 𝒰(𝕒, 2𝓇). Hence, ℬ𝕎(𝕒, 𝓇, 𝓇) ⊆ 𝒰(𝕒, 𝓇). (36) And (34) holds, which implies that 𝜏𝜌 = 𝜏𝕎 directly. Lemma 3. Let (𝔘, 𝕎, ⨁) is revised fuzzy metric space; the functions 𝒦 and 𝒢 satisfy the conditions ℭ1 and ℭ2, respectively. then, the function d defined by (20) can be represented as follows, ∀ 𝕒, 𝕓 ∈ 𝔘: 𝕕(𝕒, 𝕓) = 𝑖𝑛𝑓{𝓈 > 0: 𝑖𝑓 𝒢(𝓉) > 𝓈, 𝑡ℎ𝑒𝑛 𝕎(𝕒, 𝕓, 𝓉) + 𝒦(𝓈) < 0}. (37) Proof. In fact, we only need to show that 𝑖𝑛𝑓𝔸 = 𝑠𝑢𝑝𝔹, Where, 𝔸 = {𝕩 > 0: 𝑖𝑓 𝒢(𝓉) > 𝕩, 𝑡ℎ𝑒𝑛 𝕎(𝕒, 𝕓, 𝕩) + 𝒦(𝕩) ≤ 0}, 𝔹 = {𝕪 > 0: 𝑖𝑓 𝒢(𝓉) > 𝕪, 𝑡ℎ𝑒𝑛 𝕎(𝕒, 𝕓, 𝕩) + 𝒦(𝕪) ≤ 0}. (38) Take 𝕩 ∈ 𝔸, 𝕪 ∈ 𝔹 arbitrarily. Suppose 𝕪 > 𝕩. If 𝒢(𝓉) > 𝕪, then 𝒢(𝓉) > 𝕩. From the definitions of 𝔸 and 𝔹, we obtain 𝕎(𝕒, 𝕓, 𝓉) + 𝒦(𝕩) ≤ 0 > 𝕎(𝕒, 𝕓, 𝓉) + 𝒦(𝕪). (39) this is in direct contradiction to the condition that 𝒦 is non increasing. So, 𝕩 ≥ 𝕪, and hence, 𝑖𝑛𝑓𝔸 ≥ 𝑠𝑢𝑝𝔹. Now, let 𝑖𝑛𝑓𝔸 = 𝛼, 𝑠𝑢𝑝𝔹 = 𝛽. For any 𝛿 > 0, we know 𝛽 + 𝛿 ∉ 𝔹, that is, 𝛽 + 𝛿 ∈ 𝔸, which implies that inf 𝔸 ≤ 𝛽 + 𝛿. By the arbitrariness of 𝛿, we know that 𝑖𝑛𝑓𝔸 ≤ 𝛽 = 𝑠𝑢𝑝𝔹. This completes the proof. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 96 https://internationalpubls.com Theorem 5. Let (𝔘, 𝕎, ⨁) is a revised fuzzy metric space. If 𝕎 satisfies condition (15) or ⨁ ≥ 𝛥1, then ∀𝕒, 𝕓 ∈ 𝔘: 𝜌(𝕒, 𝕓) = 𝑖𝑛𝑓{𝓉 > 0: 𝕎(𝕒, 𝕓, 𝑡) + 𝓉 < 0}, (40) where ρ is defined by (29). Proof. Let 𝒢(𝓉) = 𝓉, 𝒦(𝓉) = 𝓉, ∀t ≥ 0. Then, (40) follows from Lemma 3 directly. 4. Conclusions We study several metric structures in a revised fuzzy metric space in current research. We provide the explicit form of the metric function about the metrizable topology for a revised fuzzy metric in two exceptional scenarios. 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