[Retracted] Some Properties of Fuzzy Quasimetric Spaces

Abstract

Some properties of fuzzy quasimetric spaces are studied. We prove that the topology induced by any 𝑀-complete fuzzy-quasi-space is a 𝑑-complete quasimetric space. We also prove Baire's theorem, uniform limit theorem, and second countability result for fuzzy quasi-metric spaces.

1. Introduction and Preliminaries

Zadeh [1] introduced the concept of fuzzy sets as a new way to represent vagueness in our everyday life. Since then, many authors regarding the theory of fuzzy sets and its applications have developed a lot of literatures. Fuzzy Metric Spaces and existence of fixed points in fuzzy metric spaces have been emerged as two of the major of research activities. As natural, several mathematicians have introduced fuzzy metric spaces in different ways (Kramosil and Michálek [2], Erceg [3], Deng [4], Kaleva and Seikkla [5]). The definition proposed by Kramosil and Michalek in 1975 [2] is the most accepted one which is closely related to a class of probabilistic metric spaces [6]. Following this definition, a lot of research have been done on the existence of fixed points for the mappings under different conditions. Many authors have investigated and modified the definition of this concept and defined a Hausdorff topology on this fuzzy metric space. They showed that every metric induces a fuzzy metric, and, conversely, every fuzzy metric space generates a metrizable topology [2, 7–9].

Recently, many authors observed that various topological properties, fixed point theorems, and contraction mappings in metric spaces may be exactly translated into fuzzy metric spaces [10–21].

On the other hand, Künzi in [22] showed that the concepts from the theory of quasimetric spaces can be used as an efficient tool to solve several problems from theoretical computer science, approximation theory, topological algebra, and so forth.

In [23], Gregori and Romaguera introduced and studied the class of fuzzy quasimetric spaces as a natural generalization of the corresponding notion of fuzzy metric space to the quasimetric. Our paper continues to study such spaces and investigates some properties of this class. In particular, we address issues related to compactness, completeness, and Baire's theorem.

A binary operation ∗∶[0,1]×[0,1]→[0,1] is a continuous 𝑡-norm [6] if ∗ satisfies the following conditions:

(1)∗ is associative and commutative.(2)∗ is continuous.(3)𝑎∗1=𝑎 for every 𝑎∈[0,1].(4)𝑎∗𝑏≤𝑐∗𝑑 whenever 𝑎≤𝑐 and 𝑏≤𝑑, with 𝑎,𝑏,𝑐,𝑑∈[0,1].

Example 1.1. The four basic t-norms are the following. (1)The Lukasierviez 𝑡-norm: ∗_𝐿∶𝐼×𝐼→𝐼,𝑎∗_𝐿𝑏=max{𝑎+𝑏−1,0}.(2)The product 𝑡-norm: 𝑎∗_𝑃𝑏=𝑎𝑏.(3)The minimum 𝑡-norm: 𝑎∗_𝑀𝑏=min{𝑎,𝑏}.(4)The weakest 𝑡-norm, the drastic product:∗_𝐷(𝑎,𝑏)=min{𝑎,𝑏}ifmax(𝑎,𝑏)=1,0otherwise. (1.1)

Using pointwise ordering, we have the inequalities ∗_𝐷<∗_𝐿<∗_𝑃<∗_𝑀.

Let 𝑋 be a nonempty set, ∗ a continuous 𝑡-norm, and 𝑀 a fuzzy set in 𝑋×𝑋×[0,+∞). For all 𝑥,𝑦,𝑧∈𝑋 and 𝑡,𝑠>0, consider the following conditions.

(FM-1)𝑀(𝑥,𝑦,0)=0(FM-2)𝑀(𝑥,𝑥,𝑡)=1 for all 𝑡>0(FM-3)𝑥=𝑦 if 𝑀(𝑥,𝑦,𝑡)=𝑀(𝑦,𝑥,𝑡)=1 for all 𝑡>0(FM-3^′)𝑥=𝑦 if 𝑀(𝑥,𝑦,𝑡)=1 for all 𝑡>0(FM-4)𝑀(𝑥,𝑧,𝑡+𝑠)≥𝑀(𝑥,𝑦,𝑡)∗𝑀(𝑦,𝑧,𝑠) for all 𝑡,𝑠≥0(FM-5)𝑀(𝑥,𝑦,⋅)∶[0,+∞)→[0,1] is left continuous. (FM-6)𝑀(𝑥,𝑦,𝑡)=𝑀(𝑦,𝑥,𝑡) for all 𝑡>0.

A fuzzy quasimetric on 𝑋 [23] is a pair (𝑀,∗) satisfying the conditions (FM-1), (FM-2), (FM-3), (FM-5), and (FM-6). If 𝑀 satisfies conditions (FM-1), (FM-2), (FM-4), (FM-5), and (FM-6), then we call (𝑀,∗) a 𝑇₁ fuzzy quasimetric on 𝑋. If 𝑀 satisfies conditions (FM-1), (FM-2), (FM-3), (FM-5), (FM-6), and (FM-7), then we call (𝑀,∗) a fuzzy metric space. A fuzzy quasimetric space is a triple (𝑋,𝑀,∗) such that X is a nonempty set and (𝑀,∗) is a fuzzy quasimetric on 𝑋.

Remark 1.2. (1) It is clear that fuzzy metric ⇒𝑇₁ fuzzy quasimetric ⇒ fuzzy quasimetric.
(2) If (𝑀,∗) is a fuzzy quasimetric on 𝑋, then 𝑀⁻¹ is also fuzzy quasimetric on 𝑋, where 𝑀⁻¹ is the fuzzy-set in 𝑋×𝑋×[0,+∞) such that 𝑀⁻¹(𝑥,𝑦,𝑡)=𝑀(𝑦,𝑥,𝑡).
(3) If (𝑀,∗) is a fuzzy quasimetric on 𝑋, then 𝑀(𝑥,𝑦,𝑡)=min{𝑀(𝑥,𝑦,𝑡),𝑀⁻¹(𝑥,𝑦,𝑡)} is a fuzzy metric on 𝑋.
(4) If (𝑋,𝑀,∗) is a fuzzy quasimetric space, then for each 𝑥,𝑦,∈𝑋 the function 𝑀(𝑥,𝑦,⋅) is nondecreasing.

Proof. Let 𝑥,𝑦∈𝑋 and 0≤𝑡<𝑠. By property (FM-4) 𝑀(𝑥,𝑦,𝑡)=𝑀(𝑥,𝑥,𝑠−𝑡)∗𝑀(𝑥,𝑦,𝑡)≤𝑀(𝑥,𝑦,𝑠).

Example 1.3 (see [23]). Let (𝑋,𝑑) be a quasimetric space. Define 𝑎∗𝑏=𝑎𝑏 for any 𝑎,𝑏,∈[0,1], and let 𝑀_𝑑 be the function defined on 𝑋×𝑋×[0,+∞) by 𝑀_𝑑𝑡(𝑥,𝑦,𝑡)=.𝑡+𝑑(𝑥,𝑦) (1.2) Then (𝑋,𝑀_𝑑,∗) is a fuzzy quasimetric space and 𝑀_𝑑 is called the fuzzy quasimetric induced by 𝑑. Also the topology 𝜏_𝑑 induced by the metric 𝑑 and the topology 𝜏_𝑀𝑑 induced by the fuzzy quasimetric (𝑀,∗) are the same.

We call a fuzzy quasimetric (𝑀,∗) on 𝑋 a non-Archimedean if 𝑀(𝑥,𝑧,𝑡)≥min{𝑀(𝑥,𝑦,𝑡),𝑀(𝑦,𝑧,𝑡)} for all 𝑥,𝑦,𝑧∈𝑋,𝑡>0. It is obvious to see that if 𝑀 is a non-Archimedean fuzzy quasimetric on 𝑋, then (𝑀,∗_𝑀) is a fuzzy quasimetric on 𝑋.

The proof of the following theorem is straightforward. Therefore, it is omitted.

Theorem 1.4. Let 𝑀_𝑑 be the standard fuzzy quasimetric of the quasimetric 𝑑 on 𝑋. Then, 𝑀_𝑑 is non-Archimedean if and only if 𝑑 is non-Archimedean.

Definition 1.5. Let (𝑋,𝑀,∗) be a fuzzy quasimetric space and let 𝑟∈(0,1),𝑡>0, and 𝑥∈𝑋.
The set 𝐵^𝑀_𝑟(𝑥,𝑡)={𝑦∈𝑋∶𝑀(𝑥,𝑦,𝑡)>1−𝑟} is called the open ball with center 𝑥 and radius 𝑟 with respect to 𝑡. It is clear that 𝑥∈𝐵^𝑀_𝑟(𝑥,𝑡).

Lemma 1.6. Let (𝑋,𝑀,∗) be a fuzzy quasimetric space. Than every open ball 𝐵^𝑀_𝑟(𝑥,𝑡) is an open set.

Proof. Let 𝑦∈𝐵^𝑀_𝑟(𝑥,𝑡). Then 𝑀(𝑥,𝑦,𝑡)>1−𝑟. Set 𝑟₀=𝑀(𝑥,𝑦,𝑡₀). Since 𝑟₀>1−𝑟,∃𝑠∈(0,1) such that 𝑟₀>1−𝑠>1−𝑟. Now given 𝑟₀ and 𝑠 such that 𝑟₀>1−𝑠,∃𝑟₁∈(0,1) such that 𝑟₀∗𝑟₁>1−𝑠. Consider the open ball 𝐵^𝑀_1−𝑟1(𝑦,𝑡−𝑡₀). We claim 𝐵^𝑀_1−𝑟1(𝑦,𝑡−𝑡₀)⊂𝐵^𝑀_𝑟(𝑥,𝑡). Let 𝑧∈𝐵^𝑀_1−𝑟1(𝑦,𝑡−𝑡₀), then 𝑀(𝑦,𝑧,𝑡−𝑡₀)>𝑟₁. So, 𝑀(𝑥,𝑧,𝑡)≥𝑀(𝑥,𝑦,𝑡₀)∗𝑀(𝑦,𝑧,𝑡−𝑡₀)≥𝑟₀∗𝑟₁≥1−𝑠>1−𝑟. Thus 𝑧∈𝐵^𝑀_𝑟(𝑥,𝑡).

2. Quasimetrization and Completeness Results

By Remark (2)–(4), if 𝑥∈𝑋,0<𝑟₁≤𝑟₂<1, and 0<𝑡₁≤𝑡₂, then 𝐵^𝑀_𝑟1(𝑥,𝑡₁)⊆𝐵^𝑀_𝑟2(𝑥,𝑡₂). Hence the following theorem and lemma are easy to prove.

Theorem 2.1. Let (𝑋,𝑀,∗) be a fuzzy (𝑇₁) quasimetric space. Then 𝜏_𝑀={𝐴⊂𝑋∶ for each 𝑥∈𝐴,∃𝑡>0 and 𝑟∈(0,1) such that 𝐵^𝑀_𝑟(𝑥,𝑡)⊂𝐴} is a 𝑇₀(𝑇₁) topology on 𝑋.

Lemma 2.2. Let (𝑋,𝑀) be a fuzzy quasimetric space. Then for each 𝑥∈𝑋,{𝐵(𝑥,1/𝑛,1/𝑛)∶𝑛∈ℕ} is a neighborhood base at 𝑋 for the topology 𝜏_𝑀. Moreover, the topology 𝜏_𝑀 is first countable.

Proposition 2.3 (A.H. Frink [24]). 𝑇₁ space admits a quasiuniformity with a countable base if it is quasimetrizable.

We can use a similar proof of [7, Theorem 1] to prove the following theorem.

Theorem 2.4. Let (𝑋,𝑀,∗) be a fuzzy quasimetric space. Then (𝑋,𝜏_𝑀) is a quasimetric space.

Proof. Let 𝑈_𝑛={(𝑥,𝑦)∈𝑋×𝑋∶𝑀(𝑥,𝑦,1/𝑛)>1−(1/𝑛),𝑛∈ℕ}. We claim that {𝑈_𝑛∶𝑛∈ℕ} is a base for quasiuniformity 𝒰 on 𝑋 whose induced topology coincides with 𝜏_𝑀. It is clear that {(𝑥,𝑥)∶𝑥∈𝑋}⊆𝑈_𝑛,𝑈_𝑛+1⊆𝑈_𝑛. Also (by continuity of ∗) for each 𝑛∈ℕ, there is an 𝑚 such that 𝑚>2𝑛 and (1−(1/𝑚))∗(1−(1/𝑚))>1−(1/𝑛). Then 𝑈_𝑚∘𝑈_𝑚⊆𝑈_𝑛. Let (𝑥,𝑦),(𝑦,𝑧)∈𝑈_𝑀. From (Remark (2)–(4), 𝑀(𝑥,𝑧,1/𝑛)≥𝑀(𝑥,𝑧,𝑙/𝑚). So, 𝑀(𝑥,𝑧,1/𝑛)≥𝑀(𝑥,𝑦,1/𝑚)∗𝑀(𝑦,𝑧,1/𝑚)≥(1−(1/𝑚))∗(1−(−(1/𝑚)>1−1/𝑛. Therefore (𝑥,𝑧)∈𝑈_𝑛. Thus {𝑈_𝑛∶𝑛∈ℕ} is a base for a quasiuniformity 𝒰 on 𝑋. Since for each 𝑥∈𝑋 and 𝑛∈ℕ, 𝑈_𝑛(𝑥)={𝑦∈𝑋∶𝑀(𝑥,𝑦,1/𝑛)>1−(1/𝑛)}=𝐵(𝑥,𝑦,1/𝑛), we deduce from Lemma 2.2 that the topology induced by 𝒰 coincides with 𝜏_𝑀. By Proposition 2.3, (𝑋,𝜏_𝑀) is a quasimetrizable space.

Corollary 2.5. A topological space is quasimetrizable if and only if it admits a compatible fuzzy quasimetric.

Definition 2.6. Let (𝑋,𝑀,∗) be a fuzzy quasimetric space. Then (1)a sequence {𝑥_𝑛} in 𝑋 is said to be 𝑀-Cauchy if for each 𝛿>0 and each 𝑡>0, there exists 𝑛₀∈ℕ such that 𝑀(𝑥_𝑛,𝑥_𝑚,𝑡)>1−𝛿, for all 𝑛,𝑚≥𝑛₀; (2)a sequence {𝑥_𝑛} in 𝑋𝑀-converges to 𝑥 if and only if 𝑀(𝑥_𝑛,𝑥,𝑡)→1 as 𝑛→∞ for all 𝑡>0; (3)a sequence {𝑥_𝑛} in 𝑋𝑀-converges to 𝑥 if and only if 𝑀(𝑥_𝑛,𝑥,𝑡)→1 as 𝑛→∞ for all 𝑡>0; (4)(𝑋,𝑀,∗) is 𝑀-complete if every 𝑀-Cauchy sequence is 𝑀-convergent with respect to 𝜏𝑀.

Remark 2.7. It is easy to prove that in a fuzzy quasimetric space (𝑋,𝑀,∗) if a sequence {𝑥_𝑛} is 𝑀-Cauchy, then lim_𝑛→∞𝑀(𝑥_𝑛,𝑥_𝑛+𝑚,𝑡)=1 for each 𝑚∈ℕ and 𝑡>0.

Theorem 2.8. Let (𝑋,𝑀,∗) be a fuzzy quasimetric space such that every 𝑀-Cauchy sequence in 𝑋 has an 𝑀-convergent subsequence. Then (𝑋,𝑀,∗) is 𝑀-complete.

Proof. Suppose {𝑥_𝑛} is a 𝑀-Cauchy sequence and {𝑥_𝑛𝑘} a subsequence of {𝑥_𝑛} such that 𝑀-converges to 𝑥. To prove that 𝑥_𝑛𝑀-converges to 𝑥, let 𝑡>0 and 𝛿∈(0,1). Choose 𝑟∈(0,1) such that (1−𝑟)∗(1−𝑟)≥1−𝛿. Since {𝑥_𝑛} is 𝑀-Cauchy sequence, there is 𝑛₀∈ℕ such that 𝑀(𝑥_𝑚,𝑥_𝑛,𝑡/2)>1−𝑟 for all 𝑚,𝑛≥𝑛₀. Since 𝑥_𝑛𝑘𝑀-converges to 𝑥, there is 𝑘₁∈ℕ such that 𝑘₁>𝑛₀,𝑀(𝑥_𝑘1,𝑥,𝑡/2)>1−𝑟. This implies that if 𝑛≥𝑛₀, then 𝑀(𝑥_𝑛,𝑥,𝑡)≥𝑀(𝑥_𝑛,𝑥_𝑘1,𝑡/2)∗𝑀(𝑥_𝑘1,𝑥,𝑡/2>(1−𝑟)∗(1−𝑟)≥1−𝛿. Therefore, 𝑥_𝑛𝑀-converges to 𝑥 and hence (𝑋,𝑀,∗) is 𝑀-complete.

Let 𝑑 be a quasimetric on 𝑋, 𝑀_𝑑 the corresponding fuzzy quasimetric, and 𝑑=max{𝑑(𝑝,𝑞),𝑑(𝑞,𝑝)} for each 𝑝,𝑞∈𝑋. Since a sequence {𝑥_𝑛} is an 𝑀_𝑑-Cauchy sequence in 𝑋 if and only if {𝑥_𝑛} is a 𝑑-Cauchy sequence in 𝑋, it is not difficult to prove the following lemma; hence we omit the proof.

Lemma 2.9. Let (𝑋,𝑑) be a quasimetric space. Then (𝑋,𝑑) is 𝑑-complete if and only if (𝑋,𝑀_𝑑) is 𝑀_𝑑-complete.

Definition 2.10. Let (𝑋,𝑀,∗) be a fuzzy quasimetric space. A collection {𝐹_𝑛}_𝑛∈ℕ is said to have fuzzy diameter zero if for each 𝑟∈(0,1) and each 𝑡>0, there exists 𝑛₀∈ℕ such that 𝑀(𝑥,𝑦,𝑡)>1−𝑟 for all 𝑥,𝑦∈𝐹_𝑛0.

It is clear that a nonempty subset 𝐹 of a fuzzy quasimetric space 𝑋 has fuzzy diameter zero if and only if 𝐹 is a singleton set.

For self-containment and clarity, we give the proofs of the following theorems even though they share similarities to those in [9].

Theorem 2.11. A fuzzy quasimetric space (𝑋,𝑀,∗) is 𝑀-complete if and only if every nested sequence {𝐹_𝑛}_𝑛∈ℕ of nonempty closed sets with fuzzy diameter zero has nonempty intersection.

Proof. First, suppose that the given condition is satisfied. We claim that (𝑋,𝑀,∗) is 𝑀-complete. Let {𝑥_𝑛} be a 𝑀-Cauchy sequence in 𝑋. Set 𝐵_𝑛={𝑥_𝑘∶𝑘⩾𝑛} and 𝐹_𝑛=𝐵_𝑛, then we claim that {𝐹_𝑛} has a fuzzy diameter zero. For given 𝑠∈(0,1) and 𝑡>0, we choose 𝑟∈(0,1) such that (1−𝑟)∗(1−𝑟)∗(1−𝑟)>1−𝑠. Since {𝑥_𝑛} is 𝑀-Cauchy sequence, there exists 𝑛₀∈ℕ such that 𝑀(𝑥_𝑛,𝑥_𝑚,(1/3)𝑡)>1−𝑟 for all 𝑚,𝑛⩾𝑛₀. Therefore, 𝑀(𝑥,𝑦,(1/3)𝑡)>1−𝑟 for all 𝑥,𝑦∈𝐵_𝑛0. Let 𝑥,𝑦∈𝐹_𝑛0. Then there exist sequences {𝑥^′_𝑛} and {𝑦^′_𝑛} in 𝐵_𝑛0 such that 𝑥^′_𝑛→𝑥 and 𝑦^′_𝑛→𝑦. Hence 𝑥^′_𝑛∈𝐵(𝑥,𝑟,𝑡/3) and 𝑦^′_𝑛∈𝐵(𝑦,𝑟,𝑡/3) for sufficiently large 𝑛. Now we have 𝑀(𝑥,𝑦,𝑡)⩾𝑀(𝑥,𝑥^′_𝑛,𝑡/3)∗𝑀(𝑥^′_𝑛,𝑦^′_𝑛,𝑡/3)∗𝑀(𝑦^′_𝑛,𝑦,𝑡/3)>(1−𝑟)∗(1−𝑟)∗(1−𝑟)>1−𝑠. Therefore, 𝑀(𝑥,𝑦,𝑡)>1−𝑠 for all 𝑥,𝑦∈𝐹_𝑛0. Thus {𝐹_𝑛} has fuzzy diameter zero and hence by hypothesis ⋂_𝑛∈ℕ𝐹_𝑛 is nonempty.
Take 𝑥∈⋂_𝑛∈ℕ𝐹_𝑛. We show that 𝑥_𝑛→𝑥. Then, for 𝑟∈(0,1) and 𝑡>0, there exists 𝑛₁∈ℕ such that 𝑀(𝑥_𝑛,𝑥,𝑡)>1−𝑟 for all 𝑛⩾𝑛₁. Therefore, for each 𝑡>0, 𝑀(𝑥_𝑛,𝑥,𝑡)→1 as 𝑛→∞ and hence 𝑥_𝑛→𝑥. Therefore, (𝑋,𝑀,∗) is 𝑀-complete.
Conversely, suppose that (𝑋,𝑀,∗) is 𝑀-complete and {𝐹_𝑛}_𝑛∈ℕ is nested sequence of non empty closed sets with fuzzy diameter zero. For each 𝑛∈ℕ, choose a point 𝑥_𝑛∈𝐹_𝑛. We claim that {𝑥_𝑛} is a 𝑀-Cauchy sequence. Since {𝐹_𝑛} has fuzzy diameter zero, for 𝑡>0 and 𝑟∈(0,1), there exists 𝑛₀∈ℕ such that 𝑀(𝑥,𝑦,𝑡)>1−𝑟 for all 𝑥,𝑦∈𝐹_𝑛0. Since {𝐹_𝑛} is a nested sequence, 𝑀(𝑥_𝑛,𝑥_𝑚,𝑡)>𝑙−𝑟 for all 𝑛,𝑚⩾𝑛₀. Hence {𝑥_𝑛} is a 𝑀-Cauchy sequence. Since (𝑋,𝑀,∗) is 𝑀-complete, 𝑥_𝑛→𝑥 for some 𝑥∈𝑋. Therefore, 𝑥∈𝐹_𝑛=𝐹_𝑛 for every 𝑛, and hence 𝑥∈⋂_𝑛∈ℕ𝐹_𝑛. This completes the proof.

Remark 2.12. The element 𝑥∈⋂_𝑛∈ℕ𝐹_𝑛 is unique. For if there are two elements 𝑥,𝑦∈⋂_𝑛∈ℕ𝐹_𝑛 since {𝐹_𝑛} has a fuzzy diameter zero, for each fixed 𝑡>0, 𝑀(𝑥,𝑦,𝑡)>1−1/𝑛 for each 𝑛∈ℕ. This implies that 𝑀(𝑥,𝑦,𝑡)=1 and hence 𝑥=𝑦.

Note that the topologies induced by the standard fuzzy quasimetric and the corresponding quasimetric are the same. So we have the following.

Corollary 2.13. Let (𝑋,𝑑) be a quasimetric space and 𝑑=max{𝑑(𝑝,𝑞),𝑑(𝑞,𝑝)} for each 𝑝,𝑞∈𝑋. Then (𝑋,𝑑) is 𝑑-complete if and only if every nested sequence {𝐹_𝑛}_𝑛∈ℕ of nonempty closed sets with diameter tending to zero has a nonempty intersection.

Theorem 2.14. Every separable fuzzy quasimetric space is second countable.

Proof. Let (𝑋,𝑀,∗) be the given separable fuzzy quasimetric space. Let 𝐴={𝑥_𝑛∶𝑛∈ℕ} be a countable dense subset of 𝑋. Consider the family ℬ={𝐵(𝑥_𝑗,1/𝑘,1/𝑘)∶𝑗,𝑘∈ℕ}. Then ℬ is countable. We claim that ℬ is a base for the family of all open sets in 𝑋. Let 𝑈 be any open set in 𝑋 and let 𝑥∈𝑈. Then there exist 𝑡>0 and 𝑟∈(0,𝐼) such that 𝐵(𝑥,𝑟,𝑡)⊂𝑈. Since 𝑟∈(0,1), we can choose a 𝑠∈(0,1) such that (1−𝑠)∗(𝑙−𝑠)>1−𝑟. Take 𝑚∈ℕ such that 1/𝑚<𝑚𝑖𝑛{𝑠,𝑡/2}. Since 𝐴 is dense in 𝑋, there exists 𝑥_𝑗∈𝐴 such that 𝑥_𝑗∈𝐵(𝑥,1/𝑚,1/𝑚). Now, if 𝑦∈𝐵(𝑥_𝑗,1/𝑚,1/𝑚), then 𝑀(𝑥,𝑦,𝑡)⩾𝑀(𝑥,𝑥_𝑗,𝑡/2)∗𝑀(𝑦,𝑥_𝑗,𝑡/2)⩾𝑀(𝑥,𝑥_𝑗,1/𝑚)∗𝑀(𝑦,𝑥_𝑗,1/𝑚)⩾(1−1/𝑚)∗(1−1/𝑚)⩾(1−𝑠)∗(1−𝑠)>1−𝑟. Thus, 𝑦∈𝐵(𝑥,𝑟,𝑡)⊂𝑈 and hence ℬ is a base.

Remark 2.15. Since second countability is a hereditary property and second countability implies separability, we obtain the following: every subspace of a separable fuzzy quasimetric space is separable.

A fuzzy quasimetric (𝑋,𝑀,∗) is said to be totally bounded if for all 𝜖>0,𝑥∈𝑋, there exist 𝑥₁,𝑥₂,𝑥₃,…,𝑥_𝑛∈𝑋 and 𝑖∈{1,2,3,…,𝑛} such that 𝑀(𝑥,𝑥_𝑖,𝑡)>1−𝜖 for all 𝑡>0.

We can use proofs similar to that in [24, Proposition 7.2, Proposition 7.5, and Corollary 7.6] to prove the following theorems.

Theorem 2.16. Let (𝑋,𝑀,∗) be a fuzzy quasimetric space. Then (𝑋,𝜏_𝑀) is a second countable if and only if it is totally bounded.

A fuzzy quasimetric 𝑀 on a set 𝑋 is continuous provided that for each 𝑥∈𝑋 the function 𝑀(𝑥,⋅,𝑡)∶𝑋→[0,1] defined by 𝑀(𝑥,⋅,𝑡)(𝑝)=𝑀(𝑥,𝑝,𝑡) is a continuous function.

Theorem 2.17. Let (𝑋,𝑀,∗) be a fuzzy quasimetric space and let 𝑥∈𝑋,𝑡>0. Then 𝑀(𝑥,⋅,𝑡) is a continuous function if and only if for each 𝛿>0, 𝐵^𝑀_𝛿(𝑥,𝑡)⊂{𝑦∈𝑋∶𝑀(𝑥,𝑦,𝑡)≥1−𝛿}.

Theorem 2.18. Let (𝑋,𝑀,∗) be a fuzzy quasimetric space. If 𝑀(⋅,𝑥,𝑡) is an upper semicontinuous function for each 𝑥∈𝑋,𝑡>0, then (𝑋,𝜏_𝑀) is metrizable.

Definition 2.19. Let 𝑋 be any nonempty set and (𝑌,𝑀,∗) a fuzzy quasimetric space. Then a sequence {𝑓_𝑛} of functions from 𝑋 to 𝑌 is said to converge uniformly to a function 𝑓 from 𝑋 to 𝑌 if given 𝑡>0 and 𝑟∈(0,1), there exists 𝑛₀∈ℕ such that 𝑀(𝑓_𝑛(𝑥),𝑓(𝑥),𝑡)>1−𝑟 for all 𝑛⩾𝑛₀ and for all 𝑥∈𝑋.

Theorem 2.20 (Uniform Limit Theorem). Let 𝑓_𝑛∶𝑋→𝑌 be a sequence of continuous functions from a topological space 𝑋 to a fuzzy quasimetric space (𝑌,𝑀,∗). If {𝑓_𝑛} converges uniformly to f: 𝑋→𝑌, then 𝑓 is continuous.

Proof. Let 𝑉 be on open set of 𝑌 and let 𝑥₀∈𝑓⁻¹(𝑉). We want to find a neighborhood 𝑈 of 𝑥₀ such that 𝑓(𝑈)⊂𝑉. Since 𝑉 is open, there exist 𝑡>0 and 𝑟∈(0,1) such that we choose a 𝑠∈(0,1) such that (1−𝑠)∗(1−𝑠)∗(1−𝑠)>1−𝑟. Since {𝑓_𝑛} converges uniformly to 𝑓, given 𝑡>0 and 𝑠∈(0,1), there exists 𝑛₀∈ℕ such that 𝑀(𝑓_𝑛(𝑥),𝑓(𝑥),𝑡/3)>1−𝑠 for all 𝑛⩾𝑛₀ and for all 𝑥∈𝑋. Since 𝑓_𝑛 is continuous for all 𝑛∈ℕ, there exists a neighborhood 𝑈 of 𝑥₀ such that 𝑓_𝑛(𝑈)⊂𝐵(𝑓_𝑛(𝑥₀),𝑠,𝑡/3). Hence 𝑀(𝑓_𝑛(𝑥),𝑓_𝑛(𝑥₀),𝑡/3)>1−𝑠 for all 𝑥∈𝑈. Now 𝑀(𝑓(𝑥),𝑓(𝑥₀),𝑡)⩾𝑀(𝑓(𝑥),𝑓_𝑛(𝑥),𝑡/3)∗𝑀(𝑓_𝑛(𝑥),𝑓_𝑛(𝑥₀),𝑡/3)∗𝑀(𝑓_𝑛(𝑥₀),𝑓(𝑥₀),𝑡/3)⩾(1−𝑠)∗(1−𝑠)∗(1−𝑠)>1−𝑟. Thus, 𝑓(𝑥)∈𝐵(𝑓(𝑥₀),𝑟,𝑡)⊂𝑉 for all 𝑥∈𝑈. Hence 𝑓(𝑈)⊂𝑉 and so 𝑓 is continuous.

Remark 2.21. Let (𝑋,𝑀,∗) be a fuzzy quasimetric space. It is easy to prove that if 𝑡>0 and 𝑟,𝑠∈(0,1) such that (1−𝑠)∗(1−𝑠)≥(1−𝑟), then 𝐵(𝑥,𝑠,𝑡/2)⊂𝐵(𝑥,𝑟,𝑡).

Lemma 2.22. A subset 𝐴 of a fuzzy quasimetric space (𝑋,𝑀,∗) is nowhere dense if and only if every nonempty open set in 𝑋 contains an open ball whose closure is disjoint from 𝐴.

Proof. Let 𝑈 be a nonempty open subset of 𝑋. Then there exists a nonempty open set 𝑉 such that 𝑉⊂𝑈 and 𝑉∩𝐴≠𝜙. Let 𝑥∈𝑉. Then there exist 𝑟∈(0,1) and 𝑡>0 such that 𝐵(𝑥,𝑟,𝑡)⊂𝑉. Choose 𝑠∈(0,1) such that (1−𝑠)∗(1−𝑠)⩾1−𝑟. By Remark 2.21, 𝐵(𝑥,𝑠,𝑡/2)⊂𝐵(𝑥,𝑟,𝑡). Thus 𝐵(𝑥,𝑠,𝑡/2)⊂𝑈 and 𝐵(𝑥,𝑠,𝑡/2)∩𝐴=𝜙.
Conversely, suppose 𝐴 is not nowhere dense. Then int(𝐴)≠𝜙, so there exists a nonempty open set 𝑈 such that 𝑈⊂𝐴. Let 𝐵(𝑥,𝑟,𝑡) be an open ball such that 𝐵(𝑥,𝑟,𝑡)⊂𝑈. Then 𝐵(𝑥,𝑟,𝑡)∩𝐴≠𝜙. This is a contradiction.

Theorem 2.23 (Baire’s Theorem). Suppose {𝑈_𝑛}_𝑛∈ℕ is a sequence of dense open subsets of a 𝑀-complete fuzzy quasimetric space (𝑋,𝑀,∗). Then ⋂_𝑛∈ℕ𝑈_𝑛 is also dense in 𝑋.

Proof. Let 𝑉 be a nonempty open set of 𝑋. Since 𝑈₁ is dense in 𝑋,𝑉∩𝑈₁≠𝜙. Let 𝑥₁∈𝑉∩𝑈₁. Since 𝑉∩𝑈₁ is open, there exist 𝑟₁∈(0,1) and 𝑡₁>0 such that 𝐵(𝑥₁,𝑟₁,𝑡₁)⊂𝑉∩𝑈₁. Choose 𝑟^′₁<𝑟₁ and 𝑡^′₁=min(𝑡₁,1) such that 𝐵(𝑥₁,𝑟^′₁,𝑡^′₁)⊂𝑉∩𝑈₁. Since 𝑈₂ is dense in 𝑋,𝐵(𝑥₁,𝑟^′₁,𝑡^′₁)∩𝑈₂≠𝜙. Let 𝑥₂∈𝐵(𝑥₁,𝑟^′₁,𝑡^′₁)∩𝑈₂. Since 𝐵(𝑥₁,𝑟^′₁,𝑡^′₁)∩𝑈₂ is open, there exist 𝑟₂∈(0,1) and 𝑡₂>0 such that 𝐵(𝑥₂,𝑟₂,𝑡₂)⊂𝐵(𝑥₁,𝑟^′₁,𝑡^′₁)∩𝑈₂. Choose 𝑟^′₂<𝑟₂ and 𝑡^′₂=min{𝑡₂,1/2} such that 𝐵(𝑥₂,𝑟^′₂,𝑡^′₂)⊂𝐵(𝑥₁,𝑟^′₁,𝑡^′₁)∩𝑈₂. Continuing in this manner, we obtain a sequence {𝑥_𝑛} in 𝑋 and a sequence {𝑡^′_𝑛} such that 0<𝑡^′_𝑛<1/𝑛 and 𝐵(𝑥_𝑛,𝑟^′_𝑛,𝑡^′_𝑛)⊂𝐵(𝑥_𝑛−1,𝑟^′_𝑛−1,𝑡^′_𝑛−1)∩𝑈_𝑛.
Now we claim that {𝑥_𝑛} is a 𝑀-Cauchy sequence. For a given 𝑡>0 and 𝜖>0, choose 𝑛₀∈ℕ such that 1/𝑛₀−𝑡 and 1/𝑛₀<𝜖. Then for 𝑛⩾𝑛₀ and 𝑚⩾𝑛, 𝑀(𝑥_𝑛,𝑥_𝑚,𝑡)⩾𝑀(𝑥_𝑛,𝑥_𝑚,1/𝑛)⩾1−1/𝑛>1−𝜖.
Therefore, {𝑥_𝑛} is a 𝑀-Cauchy sequence. Since 𝑋 is 𝑀-complete, there exists 𝑥∈𝑋 such that 𝑥_𝑛→𝑥. Since 𝑥_𝑘∈𝐵(𝑥_𝑛,𝑟^′_𝑛,𝑡^′_𝑛) for 𝑘⩾𝑛, we obtain 𝑥∈𝐵(𝑥_𝑛,𝑟^′_𝑛,𝑡^′_𝑛). Hence 𝑥∈𝐵(𝑥_𝑛,𝑟^′_𝑛,𝑡^′_𝑛)⊂𝐵(𝑥_𝑛−1,𝑟^′_𝑛−1,𝑡^′_𝑛−1)∩𝑈_𝑛 for all 𝑛. Therefore, 𝑉∩(⋂_𝑛∈ℕ𝑈_𝑛)≠𝜙.
Hence ⋂_𝑛∈ℕ𝑈_𝑛 is dense in 𝑋.

3. Compactness Results

Theorem 3.1. Every 𝑇₁ fuzzy quasimetric space is Hausdorff.

Proof. Let (𝑋,𝑀,∗) be a fuzzy quasimetric space. Let 𝑥 and 𝑦 be two distinct points in 𝑋. Then 0<𝑀(𝑥,𝑦,𝑡)<1. Put 𝑟₁=𝑀(𝑥,𝑦,𝑡) and 𝑟=max{𝑟₁,1−𝑟₁}. For each 𝑟₀∈(𝑟,1), there exists 𝑟₂ such that 𝑟₂∗𝑟₂⩾𝑟₀. Put 𝑟₃=max{𝑟₂,1−𝑟₂} and consider the open balls 𝐵(𝑥,1−𝑟₃,𝑡/2) and 𝐵(𝑦,1−𝑟₃,𝑡/2). Then clearly 𝐵(𝑥,1−𝑟₃,𝑡/2)∩𝐵(𝑦,1−𝑟₃,𝑡/2)=𝜙. For if there exists 𝑧∈𝐵(𝑥,1−𝑟₃,𝑡/2)∩𝐵(𝑦,1−𝑟₃,𝑡/2), then 𝑟₁=𝑀(𝑥,𝑦,𝑡)⩾𝑀(𝑥,𝑧,𝑡/2)∗𝑀(𝑧,𝑦,𝑡/2)⩾𝑟₃∗𝑟₃⩾𝑟₂∗𝑟₂⩾𝑟₀>𝑟₁, which is a contradiction. Hence (𝑋,𝑀,∗) is Hausdorff.

Definition 3.2. Let (𝑋,𝑀,∗) be a 𝑇₁ fuzzy quasimetric space. A subset 𝐴 of 𝑋 is said to be 𝐼𝐹-bounded if there exist 𝑡>0 and 𝑟∈(0,1) such that 𝑀(𝑥,𝑦,𝑡)>1−𝑟 for all 𝑥,𝑦∈𝐴.

Remark 3.3. Let (𝑋,𝑀,∗) be a 𝑇₁ fuzzy quasimetric space induced by a quasimetric 𝑑 on 𝑋. Then 𝐴⊂𝑋 is 𝐼𝐹-bounded if and only if it is bounded.

Theorem 3.4. Every compact subset A of a 𝑇₁ fuzzy quasimetric space (𝑋,𝑀,∗) is 𝐼𝐹-bounded.

Proof. Let 𝐴 be a compact subset of a fuzzy quasimetric space 𝑋. Fix 𝑡>0 and 0<𝑟<1. Consider an open cover {𝐵(𝑥,𝑟,𝑡)∶𝑥∈𝐴} of 𝐴. Since 𝐴 is compact, there exist 𝑥₁,𝑥₂,…,𝑥_𝑛∈𝐴 such that 𝐴⊆⋃^𝑛_𝑖=1𝐵(𝑥_𝑖,𝑟,𝑡). Let 𝑥,𝑦∈𝐴.
Then 𝑥∈𝐵(𝑥_𝑖,𝑟,𝑡) and 𝑦∈𝐵(𝑥_𝑗,𝑟,𝑡) for some 𝑖,𝑗. Thus we have 𝑀(𝑥,𝑥_𝑖,𝑡)>1−𝑟,𝑀(𝑦,𝑥_𝑗,𝑡)>1−𝑟. Now, let 𝛼=min{𝑀(𝑥_𝑖,𝑥_𝑗,𝑡)∶1⩽𝑖,𝑗⩽𝑛}. Then 𝛼>0. Now, we have 𝑀(𝑥,𝑦,3𝑡)⩾𝑀(𝑥,𝑥_𝑗,𝑡)∗𝑀(𝑥_𝑖,𝑥_𝑗,𝑡)∗𝑀(𝑥_𝑗,𝑦,𝑡)⩾(1−𝑟)∗(1−𝑟)∗𝛼>1−𝑠₁ for some 0<𝑠₁<1. Taking 𝑠=max{𝑠₁,1−𝑠₁} and 𝑡^′=3𝑡, we have 𝑀(𝑥,𝑦,𝑡^′)>1−𝑠 for all 𝑥,𝑦∈𝐴. Hence 𝐴 is 𝐼𝐹-bounded.

From Remark 3.3 and Theorems 3.1 and 3.4, we have the following corollary.

Corollary 3.5. In a 𝑇₁ fuzzy quasimetric space, every compact set is closed and bounded.

4. Conclusion

This work studies the concept of fuzzy quasimetric spaces which was introduced by Gregori and Romaguera in 2004 [23] as a natural generalization of quasimetric spaces. A topological space is quasimetrizable if and only if it admits a compatible fuzzy–quasimetric. This has been expressed in our first main result, Corollary 2.5. Moreover, in Corollary 2.13, we have proved that the 𝑀-completeness of quasimetric space can be characterized in terms of a nested sequence of nonempty closed sets with diameter tending to zero which have nonempty intersection in a natural way. Following the proofs of Fletcher and Lindgren [24] of a metrization theorem of quasimetrizable spaces, we can prove a similar result as in Theorem 2.18. We also obtained Baire's Theorem, Uniform Limit Theorem, and a second countability result for fuzzy quasimetric spaces.

One important point which has been left for further study is the behaviour of fuzzy quasimetric spaces under mappings.