FUZZY
TL-UNIFORM SPACES
KHALED A. HASHEM AND NEHAD N. MORSIReceived 12 September 2005; Revised 12 April 2006; Accepted 25 April 2006
The main purpose of this paper is to introduce a new structure that is a fuzzyTL-uniform space. We show that our structure generates a fuzzy topological space, precisely, a fuzzyT -locality space. Also, we deduce the concept of level uniformities of a fuzzyTL-uniformity. We connect the category of fuzzyTL-uniform spaces with the category of uniform spaces. We establish a necessary and sufficient condition, under which a fuzzyTL-uniformity is probabilistic pseudometrizable. Finally, we define a functor from the category of fuzzy
TL-uniform spaces into the category of fuzzyT-locality spaces and we show that it pre-serves optimal lifts.
Copyright © 2006 Hindawi Publishing Corporation. All rights reserved.
1. Introduction
In this paper, we introduce, for each continuous triangular normT, a new structure that is a fuzzyTL-uniform space, which generates a fuzzy topological space. We proceed as follows.
InSection 2, we present some basic definitions and ideas, also, we supply three lemmas on theα-cuts of fuzzy subsets on a universe setXfor allαin the interval [0, 1]. We need those lemmas for proofs scattered in subsequent sections.
InSection 3, we introduce our definition of fuzzyTL-uniform spaces, fuzzy uniform map, and the fuzzy topology associated with a fuzzyTL-uniform space.
InSection 4, we study the relationship between a fuzzyTL-uniformity and itsα-level uniformities. Also, we generate fuzzyTL-uniformities from classical uniformities.
InSection 5, we prove that each category of fuzzyTL-uniform spaces and uniform maps between them is a topological category, and we describe the fuzzyTL-uniform space of the optimal lift of a source in this category. Also, we define a functor from the category of fuzzyTL-uniform spaces into the category of fuzzyT-locality spaces, and we show that it preserves optimal lifts.
Hindawi Publishing Corporation
International Journal of Mathematics and Mathematical Sciences Volume 2006, Article ID 25094, Pages1–24
2. Prerequisites
A triangular norm (cf. [1,2]) is a binary operation on the unit intervalI=[0, 1] that is associative, symmetric, isotone in each argument and has neutral element 1. A continu-ous triangular normTis uniformly continuous, because its domainI×Iis compact. This means that for allε >0, there isθ=θT,ε>0 in such a way that for every (α,β)∈I×I, we
have
(αTβ)−ε≤(α−θ)T(β−θ)≤αTβ≤(α+θ)T(β+θ)≤(αTβ) +ε. (2.1)
Given a lower semicontinuous triangular normT, the following binary operation onI:
jT(α,γ)=sup{θ∈I:αTθ≤γ}, (α,γ)∈I×I, (2.2)
is called the residual implication of T [3], we often simplify jT to j. The best well-known triangular norms are the simplest three, namely Min (also denoted by∧),Tm(the Lukasiewicz conjunction), andπ(product), where for allα,β∈I,
αTmβ=(α+β)∧1, απβ=αβ, (2.3)
where the binary operation∧, above, is the truncated subtraction, defined on nonnega-tive real numbers byr∧s=max{r−s, 0},r,s≥0.
Proposition 2.1 [4]. TheT-residual implicationj, of a lower semicontinuous triangular normT, enjoys the following properties for allα,β,γ∈I:
(IT1)αTβ≤γif and only ifα≤j(β,γ), (IT2)αTβ > γif and only ifα > j(β,γ), (IT3)j(α,γ)=1 if and only ifα≤γ, (IT4)j(1,γ)=γ,
(IT5)j(αTβ,γ)=j(α,j(β,γ)), (IT6)αT j(α,γ)≤γ,
(IT7)j(α,β)T j(θ,γ)≤j(αTθ,βTγ),
(IT8)jis antimonotone in the left argument and monotone in the right argument. A fuzzy setλin a universeX, introduced by Zadeh in [5], is a functionλ:X→I. The collection of all fuzzy subsets ofXis denoted byIX. Ordinary subsets of a universeXwill frequently occur in what follows. We will often need to consider a subsetM⊆Xas a fuzzy subset ofX, said to be a crisp subset ofX, which we will denote by the symbol 1M. We do this by identifying 1M with its characteristic function. The collection of all crisp subsets ofXis denoted by 2X. The symbolαdenotes the constant fuzzy set in a universeXwith valueα∈I.
For the fuzzy setsμ,λ∈IX, the degree of containment ofμinλaccording toj=jT is the real number inI[6]:
jμ,λ =inf x∈Xj
Last, we denote by j|μ,λ|the following fuzzy subset ofX:
j|μ,λ|(x)=jμ(x),λ(x), x∈X. (2.5)
Given a fuzzy setλ∈IXand a real numberα∈I
1=[0, 1[, the strongα-cut ofλis the
following subset ofX:
λα=x∈X:λ(x)> α, (2.6)
and for a real numberα∈I, the weakα-cut ofλis the subset ofX:
λα∗=x∈X:λ(x)≥α, (2.7)
for both strongα-cuts and weakα-cuts ofλ, we identify crisp sets with their characteristic functions.
It is direct to verify that everyλ∈IXhas the following resolutions:
λ=
α∈I1
α∧λα= α∈I0
α∧λα∗
. (2.8)
We follow Lowen’s definition of a fuzzy interior operator on a setX[7].
This is an operatoro:IX→IXthat satisfiesμo≤μ, (μ∧λ)o=μo∧λofor allμ,λ∈IX, andαo=αfor allα∈I.
The pair (X,o) is called a fuzzy topological space (FTS).
The category of all fuzzy topological spaces and continuous functions between them (cf. [7]) is denoted by FTS.
Given a fuzzy topological space (X,o)=(X,τ), andα∈I1, the set
ια(τ)=λα⊆X:λ∈τ (2.9)
is a topology onX, said to be the α-level topology ofτ [8]. It is direct to see that its interior operator is given by
intια(τ)(M)= α∨1M oα
, M⊆X, (2.10)
and,ι(τ)=supα∈I1ια(τ) is called the topology modification ofτ[8].
Prefilters and prefilterbases were introduced by Lowen in [9]. A prefilter in a universe
Definition 2.2 [10]. TheT-saturation operator is the operator∼T which sends a pre-filterbaseᏮinXto the following subset ofIX:
Ꮾ∼T=
⎡ ⎣ ⎧ ⎨ ⎩
γ∈[0,1[
γTμγ:μγ∈Ꮾ∀γ∈[0, 1[
⎫ ⎬ ⎭ ⎤
⎦, (2.11)
said to be theT-saturation ofᏮ.
Theorem 2.3 [10]. TheT-saturation operator∼Tis isotone. Also, given a prefilterbaseᏮ inX,Ꮾ∼T satisfies
Ꮾ∼Tis a prefilter, ⊇[Ꮾ]⊇Ꮾ, sup
υ∈Ꮾjυ,λ =μ∈Ꮾsup∼Tjμ,λ , (2.12)
moreover,Ꮾ∼Tequals the following subset ofIX:
λ∈IX:jβ,λ∈[Ꮾ]∀β <1. (2.13)
Definition 2.4 [10]. A fuzzyT-locality space is a fuzzy topological space (X,o) whose fuzzy interior operator is induced by some indexed familyᏮ=(Ꮾ(x))x∈X, of prefilterbases in
IX, in the following manner:
μo(x)= sup υ∈Ꮾ(x)
jυ,μ , ∀(μ,x)∈IX×X. (2.14)
The familyᏮis said to be aT-locality basis for (X,o), the fuzzy topology of (X,o) will be denoted byτ(Ꮾ). The full subcategory, of FTS, of fuzzyT-locality spaces is denoted by
T-FLS.
Theorem 2.5 [10]. A family of prefilterbases inX,Ꮾ=(Ꮾ(x))x∈X, will be aT-locality base inXif and only if it satisfies the following two conditions for allx∈X.
(TLB1)υ(x)=1 for allυ∈Ꮾ(x).
(TLB2) Everyυ∈Ꮾ(x) has aT-kernel. This consists of two families (υγ∈Ꮾ(x))γ∈I1and (υyγθ∈Ꮾ(y))(y,γ,θ)∈X×I1×I0such that for all
(y,γ,θ)∈X×I1×I0,
γTυγ(y)∧θTυyγθ≤υ, whereI0=]0, 1]. (2.15)
Theorem 2.6 [10]. (i) A familyᏮ=(Ꮾ(x))x∈X will be aT-locality basis for aT-locality space (X,o) if and only ifᏮ∼Tis so. In this case,Ꮾ∼Tis the greatestT-locality basis for (X,o) (andᏮ∼T is called theT-locality system of (X,o)). It is the uniqueT-saturatedT-locality basis for (X,o). In this case,Ꮾis a basis for theT-locality systemᏮ∼T.
(ii) If Ꮾ=(Ꮾ(x))x∈X is a T-locality basis for a T-locality space (X,o), then Ꮾ∼T = ((Ꮾ(x))∼T)x∈Xis obtained also by
Ꮾ(x)∼T=υ∈IX:υo(x)=1, x∈X. (2.16)
A distance distribution function (DDF) [2] is a function from the setR+of positive
supremum 1. The set of all DDFs is denoted byD+. Every continuous triangular norm T induces onD+a binary operation⊕T by (η⊕Tζ)(s)=sup{η(b)Tζ(s−b) : 0< b < s}, s >0. The identity element of a semigroup (D+,⊕T) isε
0defined by
ε0(x)= ⎧ ⎨ ⎩
0, x≤0,
1, x >0. (2.17)
A probabilistic pseudometric on a setX (cf. [11]) is a functionΓ:X×X→D+that
satisfies for allx,y,zinXthe following properties: (PM1)Γ(x,x)=ε0,
(PM2)Γ(x,y)=Γ(y,x),
(PM3)Γ(x,y)⊕TΓ(y,z)≥Γ(x,z).
In [12], H¨ohle defines for everyψ,ϕ∈IX×X and λ∈IX, theT-section ofψ overλ byψλ T(x)=supy∈Xλ(y)Tψ(y,x),x∈X, theT-composition ofψ,ϕbyψoTϕ(x,y)= supz∈Xϕ(x,z)Tψ(z,y),x,y∈X, the symmetric ofψbysψ(x,y)=ψ(y,x),x,y∈X.
Now, we will use the following lemmas in the sequel.
Lemma 2.7. For allϕ,ψ∈IX×Xandα∈I1,
(i) (γTϕ)α=ϕj(γ,α)for allγ∈I,
(ii) (ϕoTψ)α=θTβ>α(ϕθ∗oψβ∗)=θTβ≤α(ϕθoψβ).
Proof. (i) For every (x,y)∈X×X, we get the equivalences
(x,y)∈ γTϕα
iff γTϕ(x,y)> α
iff ϕ(x,y)> j(γ,α), by (IT2),
iff (x,y)∈ϕj(γ,α).
(2.18)
This proves (i).
(ii) For the first equality, let (x,y)∈X×X, we get the equivalences
(x,y)∈ϕoTψα
iff ϕoTψ(x,y)> α
iff sup
z∈Xψ(x,z)Tϕ(z,y)> α
iff ∃z∈X, β,θ∈IowithβTθ > αsuch that
ψ(x,z)≥β,ϕ(z,y)≥θ
iff ∃z∈X, β,θ∈IowithβTθ > αsuch that
(x,z)∈ψβ∗, (z,y)∈ϕθ∗
iff (x,y)∈ θTβ>α
ϕθ∗oψβ∗.
(2.19)
For the second equality of (ii), we get also
(x,y)∈/ ϕoTψα
iff ϕoTψ(x,y)≤α
iff sup
z∈Xψ(x,z)Tϕ(z,y)≤α
iff ∀z∈X,∃β,θ∈I0withβTθ≤αsuch that
ψ(x,z)≤β,ϕ(z,y)≤θ
iff ∀z∈X,∃β,θ∈I0withβTθ≤αsuch that
(x,z)∈/ ψβ, (z,y)∈/ ϕθ
iff (x,y)∈/ θTβ≤α
ϕθoψβ.
(2.20)
This completes the proof.
Lemma 2.8 [13]. For allψ∈IX×Xandα∈I
1,
sψα=sψα. (2.21)
Lemma 2.9 [14]. Let (ψn)n∈Nbe a sequence of fuzzy sets onX×Xfor whichψn(x,x)=1 for allx∈X;ψn=s(ψn);ψn+1oTψn+1oTψn+1≤ψn. Then, there is a probabilistic pseudometric
ΓonXsatisfying
ψn+1(x,y)≤Γ(x,y)2−n≤ψn(x,y), ∀(x,y)∈X×X. (2.22)
3. FuzzyTL-uniform spaces
The fuzzyTL-uniform spaces are introduced in this section, and some of their properties are given. The uniform maps are defined. Also, we generate a fuzzy topology from a fuzzy
TL-uniform space.
Definition 3.1. (i) A fuzzyTL-uniform base on a setXis a subsetυ⊂IX×Xwhich fulfills the following properties.
(FLUB1)υis a prefilterbase.
(FLUB2) For allϕ∈υandx∈X,ϕ(x,x)=1.
(FLUB3) For allϕ∈υandγ∈I1, there isϕγ∈υwithγTϕγ≤sϕ.
(FLUB4) For allϕ∈υandγ∈I1, there isϕγ∈υwithγT(ϕγoTϕγ)≤ϕ.
Obviously, (FLUB3) and (FLUB4) can be replaced by the single condition (FLUB3\). For allϕ∈υandγ∈I1, there isϕγ∈υwithγT(ϕγoTϕγ)≤sϕ. (ii) A fuzzyTL-uniformity onXis aT-saturated fuzzyTL-uniform base onX. (iii) Ifμis a fuzzyTL-uniformity onX, thenυis a basis forμifυis a prefilterbase and
It follows that for a fuzzyTL-uniformityμon a setX, and allϕ∈μ,sϕ∈μ. The pair (X,μ) consisting of a setXand a fuzzyTL-uniformityμonXis called fuzzyTL-uniform space. The elements ofμare called fuzzy vicinities.
Proposition 3.2. Let υbe a fuzzy TL-uniform base on a setX, then the prefilter υ∼T determined byυis a fuzzyTL-uniformity onX.
Conversely, ifμis a fuzzyTL-uniformity onX, then conditions (FLUB2)–(FLUB4) are satisfied by every prefilterbaseυwhich determinedμ.
Proof. Supposeυis a fuzzyTL-uniform base onX. We show thatυ∼T is a fuzzyTL -uniformity.
Obviously,υ∼T isT-saturated prefilter.
Now, to prove (FLUB2), let ϕ∈υ∼T, then there exists (ψα∈υ)α∈I1 such that ϕ≥
α∈I1(αTψα). Hence for everyx∈X, we have
ϕ(x,x)≥ α∈I1
αTψα
(x,x)
=
α∈I1
(αT1) forψα∈υ
=
α∈I1 (α)=1.
(3.1)
Also, to prove (FLUB3\), letϕ∈υ∼T andγ∈I
1, then for allα∈I1, there existsψα∈υ such that
ϕ≥αTψα. (3.2)
But for eachβ∈I1, there isβψα∈υwith
βTβψαoTβψα≤sψα. (3.3)
By continuity ofT, we can chooseβ∈I1 such thatβTβ=γ, and by takingϕγ=βψβ∈
υ⊆υ∼T, we get
γTϕγoTϕγ=βTβTβψβoTβψβ
≤βTsψβ, by (3.3),
≤sϕ, directly from (3.2).
(3.4)
This proves thatυ∼Tis a fuzzyTL-uniformity.
Conversely, suppose thatμis a fuzzyTL-uniformity, and letυbe a prefilterbase which determinesμ.
To prove (FLUB4), letϕ∈υandγ∈I1, thenϕ∈μand hence, there is a family (ψγ∈
μ)γ∈I1such that
γTψγoTψγ≤ϕ. (3.5)
By continuity ofT, we can findα∈I1such thatγ≤αTαTα, and byTheorem 2.6, there
isϕγinυwithϕγ≤j|α,ψα|, consequently,
γTϕγoTϕγ≤αTαTαT jα,ψαoTjα,ψα
=αT αT jα,ψα
oT αT jα,ψα
, clear,
≤αTψαoTψα, by (IT6), ≤ϕ, by (3.5),
(3.6)
which proves thatυsatisfies (FLUB4), and completes the proof.
Lemma 3.3. If{ψα:α∈I1} ⊆IX×Xandψ=α∈I1(αTψα)∈IX×X, then for everyM∈2X,
ψ1M T=α∈I1[αTψα1M T]∈IX. Proof. For everyx∈X,
ψ1MT(x)=sup
y∈X1M(y)Tψ(y,x)=supy∈Mψ(y,x)
=sup y∈M
⎧ ⎨ ⎩
α∈I1
αTψα(y,x)
⎫ ⎬ ⎭
=
α∈I1
αT sup y∈Mψα(y,x)
!"
, by isotonicity ofT,
=
α∈I1
αTψα1MT
(x).
(3.7)
Henceψ1M T=α∈I1[αTψα1M T].
Lemma 3.4. For a prefilterbase#inX×XandM∈2X,
ψ1MT:ψ∈
∼T
$"
=
ψ1MT:ψ∈$
"∼T
. (3.8)
Proof. First, we show that{ψ1M T:ψ∈#∼T}isT-saturated prefilter. Letψ,ϕ∈#∼T and letμ∈IXbe such thatμ≥ψ1M T∧ϕ1M T. Then we defineφ∈IX×Xby
φ(x,y)=
⎧ ⎨ ⎩
μ(x) ifx∈M,
It is easy to see thatμ=φ1M T andφ≥ψ∧ϕand henceφ∈#∼T. This proves that
μ∈ {ψ1M T:ψ∈#∼T}, demonstrates that{ψ1M T:ψ∈#∼T}is a prefilter. Now, given a subfamily{ψγ:γ∈I1}of
#∼T
. Since#∼TisT-saturated,α∈I1(αTψα)∈
#∼T, so,
α∈I1
αTψα1MT
=
α∈I1
αTψα1MT, byLemma 3.3,
∈
ψ1MT:ψ∈
∼T
$"
.
(3.10)
This proves our assertion. Finally, we have
ψ1MT:ψ∈$⊆
ψ1MT:ψ∈
∼T
$"
=ψ1MT:∀γ∈I1∃ψγ∈ $
withγTψγ≤ψ
⊆ψ1MT:∀γ∈I1∃ψγ∈ $
withγTψγ1MT≤ψ1MT
=ψ1MT:ψ∈$∼T,
(3.11)
since the prefilter{ψ1M T:ψ∈#∼T}isT-saturated, then
ψ1MT:ψ∈$∼T⊆
ψ1MT:ψ∈
∼T
$"
⊆ψ1MT :ψ∈$∼T.
(3.12)
So, equality holds, and winds up the proof.
Theorem 3.5. Ifυis a fuzzyTL-uniform base on a setX, then the indexed familyᏮ= (Ꮾ(x))x∈X, given by
Ꮾ(x)=ϕ1xT:ϕ∈υ, is aT-locality base onX. (3.13)
Proof. Givenx∈Xandυ∈Ꮾ(x), then there isϕ∈υsuch thatυ=ϕ1x T, hence
υ(x)=ϕ1xT(x)=sup y∈X
1x(y)Tϕ(y,x)=1x(x)Tϕ(x,x)=1. (3.14)
Also, for everyυ∈Ꮾ(x),γ∈I1, andz∈X, we have
υ(z)=ϕ1xT(z) for someϕ∈υ
=ϕ(x,z)≥γTϕγoTϕγ(x,z), by (FLUB4)
=sup y∈X
γTϕ
γ(x,y)Tϕγ(y,z)
≥sup y∈X
γTϕγ(x,y)TδTϕγδoTϕγδ(y,z), by (FLUB4) again
=sup y∈X
γTϕγ(x,y)TδTsup s∈X
ϕγδ(y,s)Tϕγδ(s,z)
≥γTϕγ(x,y)TδTϕγδ(y,z)Tϕγδ(z,z)
=γTϕγ1xT(y)Tδ
Tϕγδ1yT(z)
.
(3.15)
Now, for everyθ∈I0, we can get (by continuity ofT)δ=δθ∈I1for which [γTϕγ1x T (y)Tδ]≥[(γTϕγ1x T(y))∧θ], hence takeυxγ=ϕγ1x T∈Ꮾ(x) andυyγθ=ϕγδ1y T∈ Ꮾ(y) which satisfyυ(z)≥[(γTυxγ(y))∧θ]Tυyγθ(z).
This shows thatᏮsatisfies (TLB2), which completes the proof.
Proposition 3.6. Ifμis a fuzzyTL-uniformity on a setX, then the indexed familyᏮ= (Ꮾ(x))x∈X, given by
Ꮾ(x)=ϕ1xT:ϕ∈μ, is aT-locality system onX. (3.16)
The proof follows at once fromTheorem 3.5andLemma 3.4.
We will see how a fuzzyTL-uniform base can generate a fuzzy topology.
Definition 3.7. The fuzzyT-locality space (X,τT(υ)) induced by a fuzzyTL-uniform base
υonXis theT-FLS defined by means of the fuzzyT-locality base of the preceding theo-rem, for which its fuzzy interior operatorois given by
λo(x)=sup ϕ∈υj
%
ϕ1xT,λ&, (λ,x)∈IX×X. (3.17)
The fuzzy topologyτT(υ) of that space is also denoted simply byτ(υ). Notice that due to Theorem 1.4, a fuzzyTL-uniform baseυand its fuzzyTL-uniformityυ∼Tinduce the same fuzzyT-locality space, that is,τ(υ)=τ(υ∼T).
Uniform-type continuity:
Definition 3.8. Let (X,μ) and (X\,μ\) be fuzzyTL-uniform spaces and f :X→X\. f is a fuzzy uniform map (or fuzzy uniformly continuous) if any of the following equivalent conditions hold.
(i) For eachψ∈μ\, (f×f)−1(ψ)∈μ.
The composite of two uniformly continuous functions f : (X,μ)→(Y,U) and g: (Y,U)→(Z,℘) is again uniformly continuous, since (g f ×g f)−1(ψ)=(f ×f)−1{(g× g)−1(ψ)} ∈μ, for allψ∈℘.
Proposition 3.9. If (X,μ) and (X\,μ\) are fuzzyTL-uniform spaces, with basesυandυ\, respectively, and f :X→X\, thenf is uniformly continuous if and only if for allϕ∈υ\and allγ∈I1, there isϕ∈υsuch that
γTϕ≤(f ×f)−1(ϕ). (3.18)
Proof. Let f : (X,μ)→(X\,μ\) be uniformly continuous andϕ∈υ\,γ∈I
1. Thenϕ∈ υ\∼T=μ\, sincef is uniformly continuous, we get (f ×f)−1(ϕ)∈μ=υ∼T, hence there isϕ∈υsuch thatγTϕ≤(f ×f)−1(ϕ).
Conversely, suppose that the stated condition holds. Letψ∈μ\, sinceμ\=υ\∼T, then for allγ∈I
1, there existsϕγ∈υ\ such thatψ≥
γTϕ γ, hence
(f×f)−1(ψ)≥(f×f)−1 γTϕγ
=γT(f ×f)−1ϕγ
≥γTθTϕγθ for someθ∈I1,ϕγθ∈υ, by hypothesis. (3.19)
SinceTis continuous, then for allα∈I1, there existsγ=γα∈I1such thatα=γTγ, by
takingφα=ϕγγ∈υ, we get
(f×f)−1(ψ)≥γTγTϕ
γγ=αTφα, (3.20)
which implies that (f ×f)−1(ψ)∈μ. Hence f is uniformly continuous, and winds up
the proof.
A function f : (X,o)=(X,τ)→(Y,o)=(Y,τ\), between two fuzzy topological spaces, is said to be continuous [7]; if f−1(μ)∈τ for all μ∈τ\, μ∈IY. Equivalently, if
f−1(λo)≤[f−1(λ)]ofor allλ∈IY.
Theorem 3.10. Let (X,μ) and (Y,℘) be fuzzyTL-uniform spaces. Iff :X→Yis uniformly continuous, then it is continuous with respect to the fuzzy topologies generated byμand℘, respectively.
Proof. Letλ∈IYandx∈X, we denote byo
1ando2the fuzzy interior operators ofτ(μ)
andτ(℘), respectively. Then, we have
f−1λo2(x)=λo2f(x)=sup ψ∈℘j
%
ψ1f(x)
T,λ
&
=sup ψ∈℘yinf∈Yj ψ
1f(x)
T(y),λ(y)
≤sup ψ∈℘zinf∈Xj
=sup ψ∈℘zinf∈Xj
(f×f)−1(ψ)(x,z),f−1(λ)(z)
≤sup ϕ∈μinfz∈Xj
ϕ(x,z),f−1(λ)(z), by uniformly continuous of f and (IT8),
=sup ϕ∈μinfz∈Xj ϕ
1xT(z),f−1(λ)(z)
=f−1(λ)o1(x),
(3.21)
which proves the continuity of f : (X,τ(μ))→(Y,τ(℘)) at each pointxinX.
4. The operatorsιu,αandωu
Now, we introduce the concept ofα-level uniformities for a given fuzzyTL-uniformity and we study the relationships between them.
For a fuzzyTL-uniformityμon a setXandα∈I1, we define
ιu,α(μ)=
ψj(β,α)⊆X×X:ψ∈μ,β∈]α, 1], (4.1)
we next show thatιu,α(μ) is a uniformity onXwheneverμis a fuzzyTL-uniformity, called
theα-level uniformity ofμ.
In the following two propositions, letυbe a basis for a fuzzyTL-uniformityμon a nonempty setX.
Proposition 4.1. ιu,α(υ) is a filterbasis for the filterιu,α(μ).
Proof. We first show thatιu,α(υ) is indeed a filterbase.
(a)∅∈/ ιu,α(υ), because every member inιu,α(υ) contains the diagonal ofX(D(X)).
(b) The intersection of two members of ιu,α(υ) contains a member: given U,V ∈ ιu,α(υ), then there areψ,ϕ∈υandβ1,β2∈]α, 1] such thatU=ϕj(β1,α),V=ψj(β2,α),
with-out loss of generality,β1≤β2. So
U∩V=ϕj(β1,α)∩ψj(β2,α)
⊇ϕj(β1,α)∩ψj(β1,α), becausejβ 2,α
≤jβ1,α
=(ϕ∧ψ)j(β1,α).
(4.2)
But, there isζ∈υwithζ≤ϕ∧ψ (becauseυis a prefilterbase). ThusU∩V contains a memberζj(β1,α)ofιu
,α(υ).
To prove thatιu,α(μ) is a filter, it now suffices to show thatιu,α(μ) is closed under
su-persets: givenU∈ιu,α(μ), letV⊇U. Then
If we takeϕ=Vj(β,α)∈IX×X, we getϕ≥ψj(β,α)j(β,α)≥ψtherefore,ϕis also inμ
and
V= V∨j(β,α)j(β,α)=ϕj(β,α)∈ι
u,α(μ), (4.4)
which proves our assertion.
Finally, we show thatιu,α(μ) is generated byιu,α(υ): letU∈ιu,α(μ), thenU=ψj(β,α)for
someψ∈μandβ∈]α, 1]. Sinceυ⊆υ∼T⊂μ, then for allγ∈I
1, there existsϕγ∈υwith
γTϕγ≤ψ. (4.5)
By continuity ofT, we can chooseγo∈I1such thatγoTβ∈]α, 1]. Then ϕ
γoj(γoTβ,α)=ϕγoj(γo,j(β,α)), by (IT5),
=γoTϕγoj(β,α), byLemma 2.7(i), ⊆ψj(β,α), by (4.5),
=U.
(4.6)
Thus there is a member (ϕγo)j(γoTβ,α) ofι
u,α(υ) contained inU. Since alsoιu,α(υ) is
in-cluded inιu,α(μ).
This completes the proof that it is a filterbasis forιu,α(μ).
Proposition 4.2. ιu,α(υ) is a uniform basis for the uniformityιu,α(μ).
Proof. ιu,α(υ) is a uniform base.
(UB1)ιu,α(υ) is a filterbase (fromProposition 4.1).
(UB2) Each member ofιu,α(υ) evidently contains the diagonalD(X): sinceϕ(x,x)=
1> j(β,α) for all (ϕ,x,β)∈υ×X×]α, 1], then (x,x)∈ϕj(β,α)for all (ϕj(β,α),x)∈(ιu ,α(υ))
×X.
(UB3) For every U∈ιu,α(υ), its symmetricsU contains a member of ιu,α(υ): given U∈ιu,α(υ), then there existϕ∈υandβ∈]α, 1], such thatU=ϕj(β,α), hence for allγ∈I1,
there existsϕγ∈υwith
γTϕγ≤sϕ. (4.7)
By continuity ofT, we can getγo∈I1such thatγoTβ∈]α, 1]. Then
ϕγoj(γoTβ,α)=ϕγoj(γo,j(β,α)), by (IT5), =γoTϕγoj(β,α), byLemma 2.7(i), ⊆(sϕ)j(β,α), by (4.7),
=sϕj(β,α), byLemma 2.8, =sU.
(4.8)
(UB4) For everyU∈ιu,α(υ), there existsV∈ιu,α(υ) such thatVoV ⊂U: givenU∈ ιu,α(υ), there areϕ∈υandβ∈]α, 1], such thatU=ϕj(β,α). Also, for allγ∈I1, there exists ϕγ∈υwith
γTϕγoTϕγ≤ϕ. (4.9)
By continuity ofT, we can chooseγo∈I1such thatγo≥βandγoTβ∈]α, 1]. Then
U=ϕj(β,α)⊇γ
oT
ϕγooTϕγo j(β,α)
=ϕγooTϕγoj(γo,j(β,α)), byLemma 2.7(i), =ϕγooTϕγoj(γoTβ,α), by (IT5),
=
θTδ≤j(γoTβ,α)
ϕ
γoθoϕγoδ
, byLemma 2.7(ii),
⊇
θTδ≤j(γoTβ,αTα)
ϕ
γoθoϕγoδ
, by (IT8),
⊇ϕγoj(γo,α)oϕγoj(β,α), by (IT7),
⊇ϕγoj(β,α)oϕγoj(β,α), by (IT8) forγo≥β.
(4.10)
TakingV=(ϕγo)j(β,α), which is inι
u,α(υ), proves our assertion. This completes the proof
thatιu,α(υ) is a uniform base.
Consequently, byProposition 4.1,ιu,α(μ) is uniformity withιu,α(υ) a basis.
Corollary 4.3. Let (X,μ) be a fuzzyTL-uniform space, in the casesT=Tm, min, the level uniformities of (X,μ) form an ascending chain, while in the caseT=π, the level uniformities form an antichain.
Proof. Let 0≤α1< α2<1 andV∈ιu,α1(μ).
Then there areϕ∈μandβ∈]α1, 1] such thatV=ϕj(β,α1). First, wheneverT=Tm, we
have
jβ,α1
=min1−β+α1, 1
, by (2.2), =min1−β+α1−α2+α2, 1
=jα2+β−α1,α2
,
(4.11)
hence,V =ϕj(β,α1)=ϕj(α2+β−α1,α2)∈ι
u,α2(μ), because it is easy to see thatα2+β−α1∈
]α2, 1], sinceβ > α1, which proves thatιu,α1(μ)⊆ιu,α2(μ).
Second, wheneverT=Min, we have
Hence
V=ϕj(β,α1)=ϕα1 ⊇ϕα2, becauseα
1< α2,
=ϕj(γ,α2) for anyγ∈α 2, 1
.
(4.13)
But for everyγ∈]α2, 1],ϕj(γ,α2)∈ιu,α2(μ),V ∈ιu,α2(μ), becauseιu,α2(μ) is a filter. Hence ιu,α1(μ)⊆ιu,α2(μ).
Finally, wheneverT=π, we have
j(β,α)=αβ ∀β > α, by (2.2). (4.14)
LetW∈ιu,α2(μ). ThenW=ψj(βo,α2)for someψ∈μandβo∈]α2, 1]. For this numberβo,
takeγ0=α1(βo/α2), we get
W=ψj(βo,α2)=ψ(α2/βo)=ψ(α1/γ0)=ψj(γo,α1)∈ι
u,α1(μ) (4.15)
because, obviously, we can see thatγo∈]α1, 1], which proves thatιu,α2(μ)⊆ιu,α1(μ).
Proposition 4.4. If (X,μ) and (Y,℘) are fuzzyTL-uniform spaces, and f :X→Y is a uniform map, then it is also a uniform map when considered as a function between the uniform spaces (X,ιu,α(μ)) and (Y,ιu,α(℘)) for allα∈I1.
The proof immediately follows from the definition ofα-level uniformities and the fact that (f ×f)−1(ψβ)=((f ×f)−1(ψ))βfor allψ∈℘,β∈I
1.
For every fixedα∈I1, define a functionι∼u,αfrom the category of fuzzyTL-uniform spaces and fuzzy uniform maps to the category of uniform spaces and uniform maps by
on objects :ι∼u,α isιu,α,
on morphisms :ι∼u,α is the identity function.
(4.16)
Then an obvious conclusion from the above proposition is that theseι∼u,αare well-defined
functors.
Now, for a uniformity u on a setX, we define
ωu(u)=ψ∈IX×X:∀γ∈I
1,ψγ∈u
, (4.17)
we supply the proof thatωu(u) is a fuzzyTL-uniformity onX.
Proposition 4.5. Ifuis a uniformity on a setX, thenωu(u) is a fuzzyTL-uniformity with u as a basis.
Proof. First, it is easy to see thatωu(u) is a fuzzyTL-uniform base. To show thatωu(u) is
T-saturated, letψ∈(ωu(u))∼T. Then
For everyα∈I1, chooseγ=γα> αinI1for which
ψα⊇ γTϕ γ
α
=ϕγj(γ,α), (4.19)
since (ϕγ)j(γ,α)∈u, because j(γ,α)∈I
1, thenψα∈u because u is a filter, henceψ∈ ωu(u). This shows that (ωu(u))∼T=ωu(u).
Second, to show that u is a basis forωu(u), we prove that (u)∼T=ωu(u): letψ∈(u)∼T. Then there is a family (Wβ∈u)β∈I1such that
ψ≥βT1Wβ. (4.20)
Now, for everyα∈I1, we chooseγ > αinI1for which we get
ψα⊇ γT1Wγα
= γ∧1Wγα for 1Wγis crisp =1Wγ=Wγ∈u,
(4.21)
thusψα∈u, because u is filter, consequently,ψ∈ωu(u) renders (u)∼T⊆ωu(u). On the other hand, ifϕ∈ωu(u), then
ϕ=
α∈I1
α∧ϕα≥ α∈I1
αTϕα. (4.22)
Sinceϕα∈u, for allα∈I1, we getϕ∈(u)∼T, therefore,ωu(u)⊆(u)∼T, and hence
equal-ity holds.
Proposition 4.6. If (X, u) and (Y,ᐃ) are uniform spaces, and f :X→Y is uniformly continuous in the usual sense, then it is uniformly continuous when considered as a function between the fuzzyTL-uniform spaces (X,ωu(u)) and (Y,ωu(ᐃ)).
Now, if we denote by TL-FUS (US) the category of fuzzy TL-uniform spaces (the category of uniform spaces) together with the uniformly continuous functions between these spaces, and define the map ω∼: US→TL-FUS by settingω∼(X,u)=(X,ωu(u)) andω∼(f)=f, we get thatω∼is a well-defined functor.
In the following, we denote the fuzzy topology associated with a fuzzyTL-uniformity
μbyτ(μ) and the topology associated with a uniformityuby T(u). Now, we introduce some compatibility between the above notions.
Theorem 4.7. (i) Theα-level topologyια(τ(μ)) of the fuzzyT-locality space (X,τ(μ)), co-incides with the topology T(ιu,α(μ)) onX, induced by theα-level uniformityιu,α(μ) of a fuzzy TL-uniform space (X,μ).
(ii) For a uniform space (X,u), the fuzzy topologyτ(ωu(u)) induced by the above fuzzy
Proof. (i) Leto be the fuzzy interior operator of the T-FLS (X,τ(μ)). Then for every nonemptyM∈2X,
intια(τ(μ))M=
α∨Moα, by (2.10),
=x: α∨Mo(x)> α
=
'
x: sup ψ∈μyinf∈Xj ψ
1xT(y), α∨M(y)> α
(
, byDefinition 2.4,
=x:∃ψ∈μ,β∈]α, 1[ s.t.∀y /∈M, jψ(x,y),α≥β
=x:∃ψ∈μ,β∈]α, 1[ s.t.∀y /∈M,βTψ(x,y)≤α, by (IT1),
=x:∃ψ∈μ,β∈]α, 1[ s.t.∀y /∈M, (x,y)∈/ βTψα
=x:∃ψ∈μ,β∈]α, 1[ s.t.∀y /∈M, (x,y)∈/ ψj(β,α)
=x:∃V∈ιu,α(μ) s.t.∀y /∈M, (x,y)∈/ V
=x:∃V∈ιu,α(μ) s.t.∀y /∈M, y /∈Vx
=x:Vx ⊆Mfor someV∈ιu,α(μ)
=intT(ιu,α(μ))M.
(4.23)
This demonstrates thatια(τ(μ)))=T(ιu,α(μ)), which renders (i).
(ii) Let the fuzzy setλbeτ(ω(u))-open.
Ifx /∈(intT(u)λε) for somex∈Xandε∈I1, we getVx ⊂λε, for allV∈u, thus, there
existsz∈Xsuch thatz∈Vx for allV∈u withz /∈λε, that is, there existsz∈Xsuch thatz∈Vx for allV∈uwithλ(z)≤ε, hence
λ(x)=intτ(ω(u))λ
(x), by hypothesis
= sup ψ∈ω(u)
inf y∈Xj ψ
1xT(y),λ(y)
=sup V∈u
inf y∈Xj 1V
1xT(y),λ(y), byProposition 4.4,
≤sup V∈uj
1V1xT(z),λ(z)
=j1,λ(z) =λ(z), by (IT4), ≤ε.
(4.24)
That is,x /∈λε, which shows thatλε⊆intT(u
)λε,ε∈I1. Consequently,λε=intT(u)λε, that
On the other hand, letλbeω(T(u))-open.
Givenx∈X, then for allε∈I1withε < λ(x), we havex∈λε, and hence, by hypothesis x∈intT(u)λε, thus there existsW∈usuch thatWx ⊂λε, that is, there existsW∈u
such that for allz∈Wx , we getz∈λε, that is, there existsW∈usuch that for all
z∈Wx , we getλ(z)> ε. Hence,
intτ(ω(u))λ
(x)= sup ψ∈ω(u)
inf y∈Xj ψ
1xT(y),λ(y)
=sup V∈u
inf y∈Xj 1V
1xT(y),λ(y)
≥inf y∈Xj 1W
1xT(y),λ(y)≥ε.
(4.25)
This shows thatλ≤intτ(ω(u))λ, which proves thatλisτ(ωu(u))-open, and winds up the
proof of (ii).
(iii) Follows immediately from the definitions.
Example 4.8. Let (X,<) be an ordered set. For each pointxo∈X, letVxo= {(x,y) :x,y >
xo}and defineUxo=D(X)∪Vxo. Then the reader can easily check that the collection u∗= {Ux:x∈X}is a basis for a uniformity u onX. Hence by the above notions, we have
ωu(u)=ψ∈IX×X:∀γ∈I
1,ψγ∈u
(4.26)
is a fuzzyTL-uniformity onXwith u as a basis. Moreover,
ιu,αωu(u)=
ψj(β,α)⊆X×X:ω∈ωu(u),β∈]α, 1]=u. (4.27)
Definition 4.9. A fuzzy topological space (X,τ) is called a fuzzyTL-uniformizable if there is a fuzzyTL-uniformityμonXsuch thatτ=τ(μ).
Corollary 4.10. A topological space (X, T) is uniformizable if and only if (X,ω(T)) is fuzzyTL-uniformizable.
Proof. If T=T(u), then
ω(T)=ωT(u)=τω(u). (4.28)
Conversely, letω(T)=τ(μ), then
T=ιω(T)=sup α∈I1ια
ω
(T), by definition of modification topology,
=sup α∈I1ια
τ(μ), by hypothesis,
=sup α∈I1T
ιu,α(μ)
, byTheorem 4.7(i),
=T
)
sup α∈I1ιu,α(μ)
*
.
(4.29)
Definition 4.11. AT-fls (X,τ(Ꮾ)) is said to be the following.
(i)L-T0, if for everyx=yinX, there isυ∈Ꮾ(x)∪Ꮾ(y) such that
υ(x)∧υ(y)<1. (4.30)
(ii)L-regular, if for every (M,x,ε)∈2X×X×I0, there isυ∈Ꮾ(x) with supy∈X(υ ∧1M)(y)< ε, then there are an open setμandρ∈Ꮾ(x) such that 1M ≤μand supy∈X(μTρ)(y)< ε.
Theorem 4.12. TL-uniformizability⇒L-regularity. Proof. Suppose (X,μ) is a fuzzyTL-uniform space.
LetM∈2X,x∈X, andε∈I0are such that there isυ∈Ꮾ(x) with
sup y∈X
υ
∧1M(y)< ε, (4.31)
consequently, we can findε0very small such that
sup y∈X
υ
∧1M(y) +ε0< ε. (4.32)
Since,υ∈Ꮾ(x), then for all γ∈I1 there areψ,ψγ∈μsuch thatυ=ψ1x T andψ≥ γT(ψγoTψγ), (by using (FLUB4)). Hence
ε >sup z∈X ψ
1xT∧1M(z) +ε0
≥sup z∈M
γTωγoTψγ1xT
(z) +ε0
=sup z∈M
γTψ
γoTψγ(x,y)+ε0
=
'
γTsup z∈Msupy∈X
ψγ(x,y)Tψγ(y,z)(+ε0
=
'
γTsup y∈Xzsup∈M
ψ
γ(x,y)Tsψγ(z,y)(+ε0
≥(γ+θ)Tsup y∈X
⎡
⎣ψγ1xT(y)T z∈M
sψγ1zT(y)
⎤
⎦, θ=θT,ε0>0 as in (2.1).
Choosingγ0∈I1for which (γ0+θ)=1, and takingρ=sψγ01xT,μ=(z∈M sψγ01z T)o, we getρ∈Ꮾ(x), andμis open, withε >supy∈X[ρ(y)Tμ(y)], alsoμ≥1M, because for ev-eryx∈M, we have
μ(x)=
⎛
⎝
z∈M
sψγ01zT
⎞ ⎠
o
(x)
≥ sψγ01xT
o (x)
=υxo(x) forsψγ0∈μ
=1, byTheorem 2.3.
(4.34)
This proves theL-regularity of (X,τ(μ)).
Theorem 4.13. If (X,μ) is a fuzzyTL-uniform space, then (X,τ(μ)) isL-T0if and only if
(infψ∈μψ)1∗=D(X).
Proof. AT-FLS (X,τ(μ)) isL-T0
iff ∀x=yinX,∃υ∈Ꮾ(x)∪Ꮾ(y) such thatυ(x)∧υ(y)<1
iff ∀x=yinX,∃ψ∈μsuch thatψ1xT(x)∧ψ1xT(y)<1
iff ∀x=yinX,∃ψ∈μsuch thatψ(x,y)=ψ1xT(y)<1 iff ∀x=yinX,∃ψ∈μsuch that (x,y)∈/ ψ1∗
iff ψinf ∈μ
1∗=
/
ψ∈μ
ψ1∗=D(X).
(4.35)
Theorem 4.14. LetΓbe a probabilistic pseudometric on a setX. Then there is a fuzzyTL -uniformityμ=μ(Γ) given by
μ(Γ)=ψ∈IX×X:∀γ∈I
1∃n∈NwithγTΓ(x,y)
2−n≤ψ(x,y). (4.36)
Proof. Obviously,μ(Γ) isT-saturated prefilter.
Now, if we putψn(x,y)=Γ(x,y)(2−n), we get a sequence (ψn)n∈Nsatisfiesψn(x,x)=1,
ψn=s(ψn), andψn+1oTψn+1≤ψn, which shows thatμ(Γ) is a fuzzyTL-uniformity.
Remark 4.15. Letμbe a fuzzyTL-uniformity onX, a nonempty subsetᏮofμis said to be a base ofμif for eachψ∈μand eachn∈N, there isϕn∈Ꮾsuch that (1−1/n)Tϕn≤ψ.
Theorem 4.16. A fuzzyTL-uniformityμon a setX is probabilistic pseudo-metrizable if and only if it has a countable base.
Proof. Letμbe probabilistic pseudo-metrizable. Then by using (PM1) andTheorem 4.14, we get the necessary condition. For the sufficiency of the condition, suppose thatμhas a countable baseᏮ= {ψ1,ψ2,...,ψn,...}.
For a fixedγinI1, apply (FLUB4) twice for the memberϕ1∧ψ2which is inμ, then
there is a symmetric member sayϕ2inμsuch that
γTϕ2oTϕ2oTϕ2
≤ϕ1∧ψ2. (4.37)
Next, considerϕ2∧ψ3and get a symmetricϕ3inμsuch that
γTϕ3oTϕ3oTϕ3≤ϕ2∧ψ3. (4.38)
In this manner, we proceed by induction a sequence (ϕn)n∈Nof fuzzy vicinities for which
ϕn=sϕn, γTϕn+1oTϕn+1oTϕn+1
≤ϕn∧ψn+1≤ϕn, (4.39)
by usingLemma 2.9, there is a probabilistic pseudometricΓonX, with
γTϕn+1(x,y)≤Γ(x,y)
2−n≤ϕn(x,y); (4.40)
that is, the fuzzyTL-uniformityμis generated byΓ, which proves our assertion.
5. Optimal lifts in fuzzyTL-uniform spaces
We prove that the category of fuzzyTL-uniform spaces and uniform maps between them is a topological category, by constructing optimal lifts of sources in it. For expansion on the categorical notions mentioned below, see [15].
The category FTS of fuzzy topological spaces and their continuous functions is a topo-logical category. This means that its objects are sets with structures (here, fuzzy topolo-gies), its morphisms are admissible functions between those sets (here, the continuous functions), morphism composition and the identity morphisms are the usual ones for functions, and the following three conditions are satisfied for all nonempty setsX.
(1) Every source (fj:X→(Xj,σj)j∈J) of functions in the category FTS has a unique optimal lift (also called initial lift), namely, the coarsest structure onX making each function fj a morphism. Specifically, the optimal lift of that source is the fuzzy topologyτonXwith subbasis [16]:
τ∗=f−1
j (λ)∈IX:j∈J,λ∈σj
, (5.1)
that is,τis the smallest fuzzy topology onXthat containsτ∗.
(2) The class FTS(X), of all FTS-structures (i.e., fuzzy topologies) on X, is a set (smallness condition).
(3) IfXis a singleton, then FTS(X) is a singleton.
An important construction is that of initial fuzzyTL-uniformities. Given a family (fj:
If (Y,℘) is a fuzzyTL-uniform space andh:Y→X, then ifXcarries the initial fuzzy
TL-uniformity,his uniformly continuous if and only if each fjohis uniformly continu-ous.
Lemma 5.1. For a fuzzyTL-uniform baseυon a setYand a function f :X→Y,
(f ×f)−1υ∼T⊆(f×f)−1(υ)∼T=(f ×f)−1(ϕ) :ϕ∈υ∼T. (5.2)
Proof. Letψ∈(f×f)−1(υ∼T) andγ∈I
1. Then there areϕ∈υ∼T andϕγ∈υ, withψ=
(f ×f)−1(ϕ) andγTϕγ≤ϕ.
By takingψγ=(f ×f)−1(ϕγ)∈(f ×f)−1(υ), we have
γTψγ=γT(f ×f)−1ϕγ=(f ×f)−1 γTϕγ
≤(f ×f)−1(ϕ)=ψ. (5.3)
Therefore,ψ∈[(f×f)−1(υ)]∼T, which completes the proof of the lemma.
Proposition 5.2. Let f :X→Y be a function. Ifυis a fuzzyTL-uniform base onY, then (f ×f)−1(υ) is a fuzzyTL-uniform base onX.
Proof. (FLUB1) (f ×f)−1(υ) is indeed a prefilterbase.
(FLUB2) For allψ∈(f×f)−1(υ) andx∈X, we have
ψ(x,x)=(f×f)−1(ϕ)(x,x) for someϕ∈υ
=ϕf(x),f(x) =1.
(5.4)
(FLUB3\) Letψ∈(f ×f)−1(υ) andγ∈I
1, we getϕandϕγinυsuch that
ψ=(f×f)−1(ϕ), γTϕ
γoTϕγ≤sϕ. (5.5)
By takingψγ=(f ×f)−1(ϕγ)∈(f ×f)−1(υ), we have
γTψγoTψγ=γT(f ×f)−1ϕγoT(f ×f)−1ϕγ
=γT(f ×f)−1ϕ
γoTϕγ
≤(f ×f)−1sϕ =sψ,
(5.6)
which completes the proof that (f×f)−1(υ) is a fuzzyTL-uniform base onX.
Proposition 5.3. Let fj:X→Yjbe a function,j∈J. Ifυjis a fuzzyTL-uniform base on
Yjfor every j∈J, thenj∈J(f ×f)−1(υj) is a fuzzyTL-uniform base onX, where
j∈J
(f ×f)−1υj=
⎧ ⎨ ⎩
0
j∈J1
(f ×f)−1ϕj:J1is a finite subset ofJ,ϕj∈υj∀j
⎫ ⎬ ⎭.
(5.7)
Theorem 5.4. A source inTL-FUS (fj:X→(Yj,μj))j∈J has an optimal lift inTL-FUS, with fuzzyTL-uniform base
υ=
j∈J
fj×fj−1μj. (5.8)
Proof. Since (fj×fj)−1(μj)⊆j∈J(fj×fj)−1(μj)=υ⊆υ∼T, then byProposition 3.6, all
fjare uniformly continuous: (X,υ∼T)→(Yj,μj).
Let (Z,℘) be inTL-FUS, and let a functionh:Z→X be such that fjoh: (Z,℘)→ (Yj,μj) are uniformly continuous for all j∈J. Then byProposition 3.6again,
℘⊇fjoh×fjoh−1μj
=fj×fjo(h×h)−1μj
∀j∈J, that is,
℘⊇ j∈J
(h×h)−1fj×fj−1μj
=(h×h)−1 ⎡
⎣
j∈J
fj×fj−1μj
⎤ ⎦
=(h×h)−1(υ).
(5.9)
Referring toProposition 3.6, once more, we find thathis uniformly continuous: (Z,℘)→ (X,υ∼T). This establishes that (X,υ∼T) is the optimal lift inTL-FUS of the given source. Proposition 5.5. TL-FUS is a topological category.
Definition 5.6. The functor-t∼:TL-FUS→T-FLS leaves the functions unaltered, and is defined on objects (X,μ) inTL-FUS by-t∼(X,μ)=(X,τ(μ)).
Theorem 5.7. The above-t∼is a well-defined functor. The proof follows immediately from Theorem 3.10.
Theorem 5.8. The functor-t∼preserves optimal lifts. The proof can be analogously as [17, Theorem 7.3].
Theorem 5.9. FuzzyTL-uniformizability is an initial property.
Acknowledgments
The authors express their sincere thanks to the referees and the Area Editor for their thorough analysis of this manuscript and their most valuable suggestions which helped improve this manuscript greatly.
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Khaled A. Hashem: Department of Mathematics, Faculty of Science, Benha University, Benha, Egypt E-mail address:khaledahashem@yahoo.com
Nehad N. Morsi: Basic & Applied Science Department, Arab Academy for Sciences & Technology and Maritime Transport, P.O. Box 2033, Al-Horraya, Heliopolis, Cairo, Egypt
