# Some Properties of Fuzzy Quasimetric Spaces

Authors: Abdul Mohamad
Journal: International Journal of Mathematics and Mathematical Sciences
 (2010)

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## Abstract

Some properties of fuzzy quasimetric spaces are studied. We prove that the topology
induced by anyM^-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.


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## Body

## 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 basict-norms are the following. (1)
The Lukasierviezt-norm: *L:I×I→I,a*Lb=max{a+b-1,0}.(2)
The productt-norm: a*Pb=ab.(3)
The minimumt-norm: a*Mb=min{a,b}.(4)
The weakestt-norm, the drastic product:(1.1)*D(a,b)={min{a,b}ifmax(a,b)=1,0otherwise.Using pointwise ordering, we have the inequalities*D<*L<*P<*M.LetX 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 onX [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 M̂(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.
Letx,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. Definea*b=abfor anya,b,∈[0,1], and letMdbe the function defined onX×X×[0,+∞)by(1.2)Md(x,y,t)=tt+d(x,y).
Then (X,Md,*) is a fuzzy quasimetric space and Md is called the fuzzy quasimetric induced by d. Also the topology τd induced by the metric d and the topology τMd 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.
LetMd be the standard fuzzy quasimetric of the quasimetric d on X. Then, Md 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 setBrM(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∈BrM(x,t).Lemma 1.6.
Let(X,M,*) be a fuzzy quasimetric space. Than every open ball BrM(x,t) is an open set.Proof.
Lety∈BrM(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 B1-r1M(y,t-t0). We claim B1-r1M(y,t-t0)⊂BrM(x,t). Let z∈B1-r1M(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∈BrM(x,t).


## 2. Quasimetrization and Completeness Results

By Remark (2)–(4), ifx∈X,0<r1≤r2<1, and 0<t1≤t2, then Br1M(x,t1)⊆Br2M(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 τM={A⊂X: for each x∈A,∃t>0 and r∈(0,1) such that BrM(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 τM. Moreover, the topology τM 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,τM) is a quasimetric space.Proof.
LetUn={(x,y)∈X×X:M(x,y,1/n)>1-(1/n),n∈ℕ}. We claim that {Un:n∈ℕ} is a base for quasiuniformity 𝒰 on X whose induced topology coincides with τM. It is clear that {(x,x):x∈X}⊆Un,Un+1⊆Un. 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 Um∘Um⊆Un. Let (x,y),(y,z)∈UM. 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)∈Un. Thus {Un:n∈ℕ} is a base for a quasiuniformity 𝒰 on X. Since for each x∈X and n∈ℕ, Un(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 τM. By Proposition 2.3, (X,τM) 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{xn} in X is said to be M̂-Cauchy if for each δ>0 and each t>0, there exists n0∈ℕ such that M̂(xn,xm,t)>1-δ, for all n,m≥n0;(2)
a sequence{xn} in XM-converges to x if and only if M(xn,x,t)→1 as n→∞ for all t>0;(3)
a sequence{xn} in XM̂-converges to x if and only if M̂(xn,x,t)→1 as n→∞ for all t>0;(4)
(X,M,*) is M̂-complete if every M̂-Cauchy sequence is M̂-convergent with respect to τM̂.Remark 2.7.
It is easy to prove that in a fuzzy quasimetric space(X,M,*) if a sequence {xn} is M̂-Cauchy, then limn→∞M̂(xn,xn+m,t)=1 for each m∈ℕ and t>0.Theorem 2.8.
Let(X,M,*) be a fuzzy quasimetric space such that every M̂-Cauchy sequence in X has an M̂-convergent subsequence. Then (X,M,*) is M̂-complete.Proof.
Suppose{xn} is a M̂-Cauchy sequence and {xnk} a subsequence of {xn} such that  M̂-converges to x. To prove that xnM̂-converges to x, let t>0 and δ∈(0,1). Choose r∈(0,1) such that (1-r)*(1-r)≥1-δ. Since {xn} is M̂-Cauchy sequence, there is n0∈ℕ such that M̂(xm,xn,t/2)>1-r for all m,n≥n0. Since xnkM̂-converges to x, there is k1∈ℕ such that k1>n0,M̂(xk1,x,t/2)>1-r. This implies that if n≥n0, then M̂(xn,x,t)≥M̂(xn,xk1,t/2)*M̂(xk1,x,t/2>(1-r)*(1-r)≥1-δ. Therefore, xnM̂-converges to x and hence (X,M,*) is M̂-complete.Letd be a quasimetric on X, Md the corresponding fuzzy quasimetric, and d̂=max{d(p,q),d(q,p)} for each p,q∈X. Since a sequence {xn} is an M̂d-Cauchy sequence in X if and only if {xn} is a d̂-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 d̂-complete if and only if (X,Md) is M̂d-complete.Definition 2.10.
Let(X,M,*)be a fuzzy quasimetric space. A collection{Fn}n∈ℕis said to have fuzzy diameter zero if for eachr∈(0,1)and eacht>0, there existsn0∈ℕsuch thatM(x,y,t)>1-rfor allx,y∈Fn0.It is clear that a nonempty subsetF 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 M̂-complete if and only if every nested sequence {Fn}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 M̂-complete. Let {xn} be a M̂-Cauchy sequence in X. Set Bn={xk:k⩾n} and Fn=Bn¯, then we claim that {Fn} 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 {xn} is M̂-Cauchy sequence, there exists n0∈ℕ such that M̂(xn,xm,(1/3)t)>1-r for all m,n⩾n0. Therefore, M̂(x,y,(1/3)t)>1-r for all x,y∈Bn0. Let x,y∈Fn0.  Then there exist sequences {xn′} and {yn′} in Bn0 such that xn′→x and yn′→y. Hence xn′∈B(x,r,t/3) and yn′∈B(y,r,t/3) for sufficiently large n. Now we have M̂(x,y,t)⩾M̂(x,xn′,t/3)*M̂(xn′,yn′,t/3)*M̂(yn′,y,t/3)>(1-r)*(1-r)*(1-r)>1-s. Therefore, M̂(x,y,t)>1-s for all x,y∈Fn0. Thus {Fn} has fuzzy diameter zero and hence by hypothesis ⋂n∈ℕFn is nonempty.
Takex∈⋂n∈ℕFn. We show that xn→x. Then, for r∈(0,1) and t>0, there exists n1∈ℕ such that M̂(xn,x,t)>1-r for all n⩾n1. Therefore, for each t>0, M̂(xn,x,t)→1 as n→∞ and hence xn→x. Therefore, (X,M,*) is M̂-complete.
Conversely, suppose that(X,M,*) is M̂-complete and {Fn}n∈ℕ is nested sequence of non empty closed sets with fuzzy diameter zero. For each n∈ℕ, choose a point xn∈Fn. We claim that {xn} is a M̂-Cauchy sequence. Since {Fn} has fuzzy diameter zero, for t>0 and r∈(0,1), there exists n0∈ℕ such that M̂(x,y,t)>1-r for all x,y∈Fn0. Since {Fn} is a nested sequence, M̂(xn,xm,t)>l-r for all n,m⩾n0. Hence {xn} is a M̂-Cauchy sequence. Since (X,M,*) is M̂-complete, xn→x for some x∈X. Therefore, x∈Fn¯=Fn for every n, and hence x∈⋂n∈ℕFn. This completes the proof.Remark 2.12.
The elementx∈⋂n∈ℕFn is unique. For if there are two elements x,y∈⋂n∈ℕFn since {Fn} 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 d̂=max{d(p,q),d(q,p)} for each p,q∈X. Then (X,d) is d̂-complete if and only if every nested sequence {Fn}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={xn:n∈ℕ} be a countable dense subset of X. Consider the family ℬ={B(xj,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 xj∈A such that xj∈B(x,1/m,1/m). Now, if y∈B(xj,1/m,1/m), then M(x,y,t)⩾M(x,xj,t/2)*M(y,xj,t/2)⩾M(x,xj,1/m)*M(y,xj,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,…,xn∈X and i∈{1,2,3,…,n} such that M̂(x,xi,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,τM) is a second countable if and only if it is totally bounded.A fuzzy quasimetricM 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, BδM(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,τM) is metrizable.Definition 2.19.
LetX be any nonempty set and (Y,M,*) a fuzzy quasimetric space. Then a sequence {fn} 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(fn(x),f(x),t)>1-r for all n⩾n0 and for all x∈X.Theorem 2.20 (Uniform Limit Theorem).
Letfn:X→Y be a sequence of continuous functions from a topological space X to a fuzzy quasimetric space (Y,M,*). If {fn} converges uniformly to f: X→Y, then f is continuous.Proof.
LetV 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 {fn} converges uniformly to f, given t>0 and s∈(0,1), there exists n0∈ℕ such that M(fn(x),f(x),t/3)>1-s for all n⩾n0 and for all x∈X. Since fn is continuous for all n∈ℕ, there exists a neighborhood U of x0 such that fn(U)⊂B(fn(x0),s,t/3). Hence M(fn(x),fn(x0),t/3)>1-s for all x∈U. Now M(f(x),f(x0),t)⩾M(f(x),fn(x),t/3)*M(fn(x),fn(x0),t/3)*M(fn(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 subsetA 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.
LetU 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, supposeA is not nowhere dense. Then int(A̅)≠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{Un}n∈ℕ is a sequence of dense open subsets of a  M̂-complete fuzzy quasimetric space  (X,M,*). Then  ⋂n∈ℕUn is also dense in  X.Proof.
LetV 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 r1′<r1 and t1′=min(t1,1) such that B(x1,r1′,t1′)⊂V∩U1. Since U2 is dense in X,B(x1,r1′,t1′)∩U2≠ϕ. Let x2∈B(x1,r1′,t1′)∩U2. Since B(x1,r1′,t1′)∩U2 is open, there exist r2∈(0,1) and t2>0 such that B(x2,r2,t2)¯⊂B(x1,r1′,t1′)∩U2. Choose r2′<r2 and t2′=min{t2,1/2} such that B(x2,r2′,t2′)⊂B(x1,r1′,t1′)∩U2. Continuing in this manner, we obtain a sequence {xn} in X and a sequence {tn′} such that 0<tn′<1/n and B(xn,rn′,tn′)¯⊂B(xn-1,rn-1′,tn-1′)∩Un.
Now we claim that{xn} is a M̂-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, M̂(xn,xm,t)⩾M̂(xn,xm,1/n)⩾1-1/n>1-ϵ.
Therefore,{xn} is a M̂-Cauchy sequence. Since X is M̂-complete, there exists x∈X such that xn→x. Since xk∈B(xn,rn′,tn′) for k⩾n, we obtain x∈B(xn,rn′,tn′)¯. Hence x∈B(xn,rn′,tn′)¯⊂B(xn-1,rn-1′,tn-1′)∩Un for all n. Therefore, V∩(⋂n∈ℕUn)≠ϕ.
Hence⋂n∈ℕUn is dense in X.


## 3. Compactness Results

Theorem 3.1.
EveryT1 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 aT1 fuzzy quasimetric space  (X,M,*) is  IF-bounded.Proof.
LetA 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,…,xn∈A such that A⊆⋃i=1nB(xi,r,t). Let x,y∈A.
Thenx∈B(xi,r,t) and y∈B(xj,r,t) for some i,j. Thus we have M(x,xi,t)>1-r,M(y,xj,t)>1-r. Now, let α=min{M(xi,xj,t):1⩽i,j⩽n}. Then α>0. Now, we have M(x,y,3t)⩾M(x,xj,t)*M(xi,xj,t)*M(xj,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 Remark3.3 and Theorems 3.1 and 3.4, we have the following corollary.Corollary 3.5.
In aT1 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 M̂-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.



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