Completely regular fuzzifying topological spaces

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TOPOLOGICAL SPACES

A. K. KATSARAS

Received 24 March 2005 and in revised form 22 September 2005

Some of the properties of the completely regular fuzzifying topological spaces are inves-tigated. It is shown that a fuzzifying topologyτis completely regular if and only if it is induced by some fuzzy uniformity or equivalently by some fuzzifying proximity. Also,τis completely regular if and only if it is generated by a family of probabilistic pseudometrics.

1. Introduction

The concept of a fuzzifying topology was given in [1] under the nameL-fuzzy topology. Ying studied in [9,10,11] the fuzzifying topologies in the case ofL=[0, 1]. A classical topology is a special case of a fuzzifying topology. In a fuzzifying topologyτon a setX, every subsetAof X has a degreeτ(A) of belonging toτ, 0≤τ(A)≤1. In [4], we de-fined the degrees of compactness, of local compactness, Hausdorﬀnes, and so forth in a fuzzifying topological space (X,τ). We also introduced the fuzzifying proximities. Every fuzzifying proximityδinduces a fuzzifying topologyτδ. In [6], we studied the level classi-cal topologiesτθ_{, 0}_≤_{θ <}_{1, corresponding to a fuzzifying topology}_τ_{. In the same paper,} we studied connectedness and local connectedness in fuzzifying topological spaces as well as the so-called sequential fuzzifying topologies. In [5], we introduced the fuzzifying syn-topogenous structures. We also proved that every fuzzy uniformityᐁ, as it is defined by Lowen in [7], induces a fuzzifying proximityδᐁ, and that for every fuzzifying proximity δ, there exists at least one fuzzy uniformityᐁwithδ=δᐁ. Some of the results contained

in papers [4,6] are closely related to those which appeared in the papers [12,13]. In this paper, we continue with the investigation of fuzzifying topologies. In particular, we study the completely regular fuzzifying topologies, that is, those fuzzifying topologies

τfor which each level topologyτθ_{is completely regular. As in the classical case, we prove} that for a fuzzifying topologyτ onX, the following properties are equivalent: (1) τ is completely regular; (2)τis uniformizable, that is, it is induced by some fuzzy uniformity; (3)τis proximizable, that is, it is induced by some fuzzifying proximity; and (4)τis gen-erated by a family of so-called probabilistic pseudometrics onX. We also give a charac-terization of completely regular fuzzifying spaces in terms of continuous functions. Many Theorems on classical topologies follow as special cases of results obtained in the paper.

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2. Preliminaries

A fuzzifying topology on a setX(see [1,9,10,11]) is a mapτ: 2X_→_{[0, 1] (where 2}X _is the power set ofX) satisfying the following conditions:

(FT1)τ(X)=τ(∅)=1;

(FT2)τ(A1∩A2)≥τ(A1)∧τ(A2); (FT3)τ(Ai)≥infiτ(Ai).

Ifτis a fuzzifying topology onXandx∈X, then theτ-neighborhood system ofxis the function

Nx=Nxτ: 2X−→[0, 1], Nx(A)=sup

τ(B) :x∈B⊂A. (2.1)

By [9, Theorem 3.2], we have thatτ(A)=infx∈ANx(A). The following theorem is contained in [9] (see also [3,13]).

Theorem2.1. Ifτis a fuzzifying topology on a setX, then the mapx→Nx=Nxτ, fromX to the fuzzy power setᏲ(2X₎_of₂X_{, has the following properties:}

(FN1)Nx(X)=1andNx(A)=0ifx /∈A; (FN2)Nx(A1∩A2)=Nx(A1)∧Nx(A2); (FN3)Nx(A)≤sup_x_∈_D_⊂_Ainfy∈DNy(D).

Conversely, if a mapx→Nx, fromXtoᏲ(2X_{), satisfies}_{(FN1)–(FN3), then the map}

τ: 2X_−→_{[0, 1],} _τ₍_A₎₌_inf

x∈ANx(A), (2.2)

is a fuzzifying topology andNx=Nxτfor everyx∈X.

Let now (X,τ) be a fuzzifying topological space. To every subsetAofXcorrersponds a fuzzy subset ¯A=_A¯τ _of _X _{defined by ¯}_A₍_x₎₌₁₋_N_x(_Ac_{) (see [}₁₃_{, Remark 3.16]). A} function f, from a fuzzifying topological space (X,τ1) to another one (Y,τ2), is said to be continuous at somex∈X(see [4,13]) ifNx(f−1₍_A₎₎_≥_N

f(x)(A) for every subsetA ofY. If f is continuous at every point ofX, then it is said that (τ1,τ2)-continuous. As it is shown in [4], f is continuous if and only if τ2(A)≤τ1(f−1₍_A_{)) for every subset}

AofY. For f :X→Y a function andτ a fuzzifying topology onY, f−1₍_τ_{) is defined} to be the weakest fuzzifying topology onX for which f is continuous. By [4], f−1₍_τ₎ is given by the neighborhood structureNx(A)=Nf(x)(Y\ f(Ac)). If (τi)i∈I is a family of fuzzifying topologies onX, we will denote byi∈Iτi, or by supτi, the weakest of all fuzzifying topologies onX which are finer than eachτi. As it is proved in [4],

i∈Iτi is given by the neighborhood structure

Nx(A)=sup

inf i∈JN

τi

x

Ai

:x∈ i∈J

Ai⊂A , (2.3)

where the infimum is taken over the family of all finite subsetsJ ofI and allAi⊂X,

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fuzzifying topologyτ=τi on X=

Xi is the weakest fuzzifying topology on X for which each projectionπi:X→Xiis continuous. Thus,τ=

iπi−1(τi) and it is given by the neighborhood structure

Nx(A)=sup

inf i∈JNxi

Ai

:x∈ i∈J

πi−1

Ai

⊂A , (2.4)

where the supremum is taken over the family of all finite subsetsJ ofI andAi⊂Xi, for

i∈J(see [4]).

The degree of convergence to anx∈X, of a net (xδ) in a fuzzifying topological space (X,τ), is the numberc(xδ→x)=cτ(xδ→x) defined by

cxδ−→x

=inf1−Nx(A) :A⊂X,xδ

frequently inAc_. _(2.5)

As it is shown in [6], forA⊂Xandx∈X, we have

¯

A(x)=maxcxδ−→x:xδnet inA. (2.6)

The degree of Hausdorﬀness ofX(see [4]) is defined by

T2(X)=1−sup x =y

supcxδ−→x

∧cxδ−→y

:xδ

net inX. (2.7)

Also, the degree ofXbeingT1is defined by

T1(X)=inf x infy =xsup

Nx(B) :y /∈B. (2.8)

Let now (X,τ) be a fuzzifying topological space. For each 0≤θ <1, the familyB_θτ= {A⊂

X:τ(A)> θ}is a base for a classical topologyτθ _on_X _{(see [}₅_{]). It is easy to see that} a subsetBofXis aτθ_{-neighborhood of}_x_{if and only if}_N

x(B)> θ. By [6],T2(X) (resp.,

T1(X)) is the supremum of all 0≤θ <1 for whichτθ_is_T

2(resp.,T1). Also, forτ= ∨τi, we have thatτθ₌_sup

iτiθ(see [6, Theorem 3.5]). Ifτ=

τiis a product fuzzifying topology, thenτθ₌_τθ

i (see [6, Theorem 3.5]). If Y is a subspace of (X,τ) andτ1=τ|Y, then

τ1θ=τθ|Y. By [6, Theorem 3.10], for a fuzzifying topological space (X,τ), co(X) coincides with the supremum of all 0< θ <1 for whichτ1−θ_{is compact.}

Next, we will recall the notion of a fuzzifying proximity given in [4]. A fuzzifying proximity on a setXis a mapδ: 2X_×₂X_→_{[0, 1] satisfying the following conditions:}

(FP1)δ(A,B)=1 if theA,Bare not disjoint; (FP2)δ(A,B)=δ(B,A);

(FP3)δ(∅,B)=0;

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Every fuzzifying proximityδinduces a fuzzifying topologyτδgiven by the neighborhood structureNx(A)=1−δ(x,Ac). A fuzzifying proximityδ1is said to be finer than another oneδ2ifδ1(A,B)≤δ2(A,B) for all subsetsA,BofX. For f :X→Y a function andδa fuzzifying proximity onY, the function

f−1₍_δ_{) : 2}X_×₂X_−→_{[0, 1],} _f−1₍_δ₎₍_A_,_B₎₌_δ_f₍_A_),_f₍_B₎_, _(2.9)

is a fuzzifying proximity onX (see [4]) and it is the weakest of all fuzzifying proximi-tiesδ1onX for which f is (δ1,δ)-proximally continuous, that is, it satisfiesδ1(A,B)≤

δ(f(A),f(B)) for all subsetsA,BofX. As it is shown in [4],τf−1_(δ)=f−1(τ_δ).

Let now (δλ)λ∈Λbe a family of fuzzifying proximities on a setX. We will denote by

δ=λδλ, or by supδλ, the weakest fuzzifying proximity onXwhich is finer than eachδλ. By [4, Theorem 8.10],δis given by

δ(A,B)=inf

sup i,j

inf λ∈Λδλ

Ai,Bj

, (2.10)

where the infimum is taken over all finite collections (Ai), (Bj) of subsets ofXwithA=

Ai,B=

Bj. Moreover,τδ=

τδλ(see [4]).

Finally, we will recall the definition of a fuzzy uniformity introduced by Lowen in [7]. For a set X, let ΩX be the collection of all functions α:X×X→[0, 1] such that

α(x,x)=1 for allx∈X. Forα,β∈ΩX, theα∧β,α◦βandα−1are defined byα∧β(x,y)=

α(x,y)∧β(x,y),α◦β(x,y)=sup_zβ(x,z)∧α(z,y),α−1₍_x_,_y₎₌_α₍_y_,_x_{). If}_α₌_α−1_{, then}_α is called symmetric. A fuzzy uniformity onXis a nonempty subsetᐁofΩXsatisfying the following conditions.

(FU1) Ifα,β∈ᐁ, thenα∧β∈ᐁ.

(FU2) Ifα∈ᐁis such that, for every>0, there exists aβ∈ᐁwithβ≤α+, then

α∈ᐁ.

(FU3) For eachα∈ᐁand each>0, there exists aβ∈ᐁwithβ◦β≤α+. (FU4) Ifα∈ᐁ, thenα−1_∈_ᐁ_.

A subsetᏮ, of a fuzzy uniformityᐁ, is a base forᐁif for each α∈ᐁand each>0, there existsβ∈Ꮾwithβ≤α+. It is easy to see that for a subsetᏮofΩX, the following are equivalent.

(1)Ꮾis a base for a fuzzy uniformity onX.

(2) (a) Ifα,β∈Ꮾand>0, then there existsγ∈Ꮾwithγ≤α∧β+. (b) For eachα∈Ꮾand each>0, there existsβ∈Ꮾwithβ◦β≤α+.

(c) For eachα∈Ꮾand each>0, there existsβ∈Ꮾwithβ≤α−1₊_.

In case (2) is satisfied, the fuzzy uniformityᐁfor whichᏮis a base consists of allα∈ΩX such that for each>0, there exists aβ∈Ꮾwithβ≤α+.

By [5], every fuzzy uniformityᐁonXinduces a fuzzifying proximityδᐁdefined by

δᐁ(A,B)= inf

α∈ᐁx∈supA,y∈B

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In caseᏮis a base forᐁ, then

δᐁ(A,B)= inf

α∈Ꮾx∈supA,y∈B

α(x,y). (2.12)

Every fuzzy uniformityᐁinduces a fuzzifying topologyτᐁgiven by the neighborhood

structure

Nx(A)=1−δᐁx,Ac=1− inf

α∈ᐁsupy /∈A

α(x,y). (2.13)

For every fuzzifying proximityδ, there exists at least one compatible fuzzy uniformity, that is, a fuzzy uniformityᐁwithδᐁ=δ(see [5, Theorem 11.4]).

3. Probabilistic pseudometrics

A fuzzy real number is a fuzzy subsetuof the real numbersRwhich is increasing, left continuous, and such that limt→+∞u(t)=1, limt→−∞u(t)=0. A fuzzy real numberuis said to be nonnegative if u(t)=0 ift≤0. We will denote by R+

φ the collection of all nonnegative fuzzy real numbers. To every real numberrcorresponds a fuzzy real number ¯

r, where ¯r(t)=0 ift≤rand ¯r(t)=1 ift > r. Foru,v∈R+

φ, we defineuvif and only ifv(t)≤u(t) for allt∈R. If Ꮽis a nonempty subset ofR+

φ and ifuo∈R+φ is defined byuo(t)=supv∈Ꮽv(t), thenuois the biggest of allu∈R+φ withuvfor allv∈Ꮽ. We will denoteuoby infᏭor by

Ꮽ. Foru1,u2∈R+φ, we defineu=u1⊕u2∈R+φbyu(t)= sup{u1(t1)∧u2(t2) :t=t1+t2}. Also, foru∈R+_φ andλ >0, we defineλuby (λu)(t)=

u(λ−1_t_{). It is easy to see that for}_u_∈_R+

φandλ >0, we have (¯λ⊕u)(t)=u(t−λ).

Definition 3.1. A probabilistic pseudometric on a setX(see [2]) is a mappingF:X×X→

R+

φsuch that for allx,y,z∈X,

F(x,x)=¯0, F(x,y)=F(y,x), F(x,z)F(x,y)⊕F(y,z). (3.1)

If in additionF(x,y)(0+)=0 whenx =y, thenFis called a probabilistic metric.

Ifr1, r2 are nonnegative real numbers, thenr1r2 if and only ifr1≤r2. Also, for

r= |r1−r2|, we have that

r= ∧u∈R+

φ: ¯r2u⊕r1,r1u⊕r¯2

. (3.2)

In fact, letuo= ∧{u∈R+φ: ¯r2u⊕r1andr1u⊕r2}and assume that (say)r1≥r2. Letu∈᏾+

φbe such that ¯r2u⊕r1,r1u⊕r2. Thenr1(t)≥(u⊕r2)(t)=u(t−r2) for all

t. Ifs < r1, then 0=r1(s)≥u(s−r2) and sou(r1−r2)=sup_s<r₁u(s−r2)=0 which implies thatru. Thusruo. On the other hand, we haver⊕r2=r1andr⊕r1=2r1−r2. Since

r22r1−r2, it follows thatuor, and hencer=uo. Motivated by the above, we define the following distance function on᏾+_φ:

D:᏾+_φ×᏾+

φ−→᏾+φ, D

u1,u2

= ∧u∈᏾+

φ:u1u2⊕u,u2u⊕u1

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Then D is a probabilistic pseudometric on᏾+φ. In fact, it is clear thatD(u1,u2)=

D(u2,u1). Also, since u=u⊕¯0, when u∈᏾+

φ, we have that D(u,u)=¯0. Finally, let

D(u1,u2)(t1)∧D(u2,u3)(t2)> θ >0. There arev1,v2∈᏾+φ withu1v1⊕u2,u2v1⊕

u1,u3v2⊕u2,u2v2⊕u3,v1(t1)> θ,v2(t2)> θ. Nowu1v1⊕u2v1⊕(v2⊕u3)= (v1⊕v2)⊕u3 andu3v2⊕u2v2⊕(v1⊕u1)=(v1⊕v2)⊕u1. Thus,D(u1,u3)v1⊕

v2 and D(u1,u3)(t1+t2)≥v1(t1)∧v2(t2)> θ. This proves thatD(u1,u3)D(u1,u2)⊕

D(u2,u3) and the claim follows. We will refer toDas the usual probabilistic pseudomet-ric on᏾+

φ.

Let nowFbe a probabilistic pseudometric onX. Fort >0, letuF,tbe defined onX2by

uF,t(x,y)=F(x,y)(t). The familyᏮF= {uF,t:t >0}is a base for a fuzzy uniformityᐁF onX. LetτFbe the fuzzifying topology induced byᐁF.

In the rest of the paper, we will consider on᏾+_φthe fuzzifying topology induced by the usual probabilistic pseudometricD.

Theorem3.2. A probabilistic pseudometricF, on a fuzzifying topological space(X,τ), is

τ×τcontinuous if and only ifτF≤τ.

Proof. Assume thatτF≤τand letGbe a subset ofR+φandu=F(xo,yo) withNu(G)> θ > 0. There exists at >0 such that 1−sup_{v /}_∈_GD(v,u)(t)> θ. Forx,y∈X, we have

F(x,y)Fx,xo⊕xo,yo⊕Fyo,y=Fx,xo⊕Fy,yo⊕Fxo,yo. (3.4)

Similarly,F(xo,yo)[F(x,xo)⊕F(y,yo)]⊕F(x,y). Thus,

DF(x,y),Fxo,yo

Fx,xo

⊕Fy,yo

. (3.5)

Let

A1=

x∈X:Fx,xot 2

≥1−θ

, A2=

x∈X:Fy,yot 2

≥1−θ

. (3.6)

Ifx∈A1,y∈A2, then

DF(x,y),Fxo,yo

(t)≥Fx,xo

t

2

∧Fy,yo

t

2

≥1−θ, (3.7)

and soF(x,y)∈G. Also,Nτ

xo(A1)≥N

τF

xo(A1)≥1−supx /∈A1F(x,xo)(t/2)≥θandN τ yo(A2) ≥θ. Therefore,

Nτ×τ (xo,yo)

F−1₍_G₎_≥_Nτ xo

A1

∧Nτ yo

A1

≥θ, (3.8)

which proves that N_(xτ×_o_,yτ_o)(F−1(G))≥NF(xo,yo)(G) and so F isτ×τ continuous.

Con-versely, assume that F isτ×τ continuous and letNτF

xo(A)> θ >0. Choose >0 such

thatNτF

xo(A)> θ+. There exists at >0 such that 1−supx /∈AF(x,xo)(t)> θ+. If

Z=u∈R+

φ:D(u, ¯0)(t)=u(t)>1−θ−

, (3.9)

then

N¯0(Z)≥1−sup u /∈Z

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Since F isτ×τ continuous and F(xo,xo)=¯0, there exists a subset A1 of X contain-ingxosuch thatA1×A1⊂F−1(Z) andNxo(A1)> θ. Ifx∈A1, thenF(x,xo)∈Zand so F(x,xo)(t)>1−θ−, which implies thatx∈A. Thus,A1⊂Aand soNxo(A)≥NxτFo(A)

for every subsetAofXand everyxo∈X. Hence,τF≤τand the result follows.

Theorem3.3. LetFbe a probabilistic pseudometric on a setX,τ=τF,(xδ)δ∈∆a net inX, andx∈X. Then

cxδ−→x

=inf

t>0lim infδ F

xδ,x

(t). (3.11)

Proof. Letd=inft>0lim infδF(xδ,x)(t) and assume thatd < θ <1. There exists a t >0 such that lim infδF(xδ,x)(t)< θ. LetA= {y:F(y,x)(t)> θ}. Then (xδ) is not eventually in A, and soc(xδ→x)≤1−Nx(A)≤supy /∈AF(y,x)(t)≤θ, which proves thatc(xδ→

x)≤d. On the other hand, letc(xδ→x)< r <1. There exists a subsetBofXsuch that (xδ) is not eventually inBand 1−Nx(B)< r. Lets >0 be such that 1−sup_{y /}_∈_BF(y,x)(s)>1−

r. For eachδ∈∆, there existsδ≥δwithxδ∈/ B, and soF(x_δ,x)(s)≤sup_{y /}_∈_BF(y,x)(s).

Thus,d≤lim infδF(xδ,x)(s)< r, which proves thatd≤c(xδ→x) and the result follows.

Theorem3.4. LetF1,F2,. . .,Fnbe probabilistic pseudometrics onXand defineFby

F(x,y)(t)= min

1≤k≤nFk(x,y)(t). (3.12)

ThenFis a probabilistic pseudometric andτF=nk+1τFk.

Proof. Using induction onn, it suﬃces to prove the result in the case ofn=2. It follows easily thatFis a probabilistic pseudometric. SinceF1,F2F, it follows thatτF1,τF2≤τF and soτo=τF1∨τF2≤τF. On the other hand, letNxτF(A)> θ >0. There exists at >0 such that 1−sup_{y /}_∈_AF(y,x)(t)> θ. LetBi= {y∈Ac:Fi(y,x)(t)<1−θ},i=1, 2. ThenAc=

B1∪B2and soA=A1∩A2,Ai=Bic. MoreoverN τ_Fi

x (Ai)≥1−sup_y_∈_B_iFi(y,x)(t)≥θ, and thus

Nτo

x (A)≥Nxτo

A1

Nτo

x

A2

≥NτF1 x A1

NτF2 x A2

≥θ. (3.13)

This proves thatNτo

x (A)≥NτF

x (A) and the result follows.

ForᏲa family of probabilistic pseudometrics on a set X, we will denote byτᏲ the

supremum of the fuzzifying topologiesτF,F∈Ᏺ, that is,τᏲ=F∈ᏲτF.

Theorem3.5. Ifτ=τᏲ, whereᏲis a family of probabilistic pseudometrics on a setX, then

T2(X)=T1(X)=1−sup_y₌_xinfF∈ᏲF(x,y)(0+).

Proof. Letd=1−sup_y₌_xinfF∈ᏲF(x,y)(0+). It is always true thatT2(X)≤T1(X).

Sup-pose thatT1(X)> r >0 and letx =y. Sinceτr _is_T_{1, there exists a}_τr_{-neighborhood}_A ofxnot containing y. NowNx(A)> r, and hence there are subsetsA1,. . .,AnofX and

F1,. . .,Fn∈Ᏺsuch that

Ak⊂A,N τ_Fk

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y /∈Ak. Lett >0 be such that

1−sup z /∈Ak

Fk(z,x)(t)> rand so inf

F∈ᏲF(x,y)(t)(0+)≤Fk(x,y)(t)<1−r, (3.14)

which proves thatd≥r. Thusd≥T1(X). On the other hand, assume thatd > θ >0 and letx =y. Choose>0 such thatd > θ+. There existsF∈ᏲwithF(x,y)(0+)<1−θ− , and henceF(x,y)(t)<1−θ−for somet >0. Let

A=

z:F(z,x)

_t

2

>1−θ−

, B=

z:F(z,y)

_t

2

>1−θ−

. (3.15)

Clearlyx∈A,y∈B. Ifz∈A∩B, then

F(x,y)(t)≥F(x,z)

_t

2

∧F(z,y)

_t

2

>1−θ−, (3.16)

a contradiction. ThusA∩B= ∅. Moreover

Nx(A)≥NxτF(A)≥1−sup z /∈A

F(x,z)

t

2

≥θ+> θ, Ny(A)> θ. (3.17)

It follows thatT2(X)≥dand the proof is complete.

Let us say that a fuzzifying topologyτon a setXis pseudometrizable if there exists a probabilistic pseudometricFonXwithτ=τF.

Theorem3.6. A fuzzifying topologyτonX is pseudometrizable if and only if each level topologyτθ_,₀_≤_{θ <}_{1, is pseudometrizable.}

Proof. Assume thatτ=τF for some probabilistic pseudometricFand let 0≤θ <1. For each positive integern, withn >1/(1−θ), let

An=

(x,y)∈X2:F(x,y)

₁

n

>1−θ−1_n

. (3.18)

ThenAn+1⊂Anand the familyᏰ= {An:n∈N,n >1/(1−θ)}is a base for a uniformity

ᐁonX. The topologyσθinduced byᐁis pseudometrizable sinceᏰis countable. More-overσθ=τθ. Indeed, letAbe aσθ-neighborhood ofx. There existsn∈N,n >1/(1−θ), such thatB= {y:F(x,y)(1/n)>1−θ−1/n} ⊂A. Now

Nτ

x(A)≥Nxτ(B)≥1−sup y /∈B

F(x,y)

1

n

≥θ+1

n> θ, (3.19)

and soAis aτθ_{-neighborhood of}_x_{. Conversely, assume that}_A_{is a}_τθ_{-neighborhood of}

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that 1−sup_{y /}_∈_AF(x,y)(1/n)> θ+ 1/n. Hence

y:F(x,y)

1

n

>1−θ−1

n

⊂A, (3.20)

which implies thatA is aσθ-neighborhood ofx. Thus τθ=σθ, and therefore eachτθ is pseudometrizable. Conversely, suppose that eachτθ_{is pseudometrizable. By an} argu-ment analogous to the one used in the proof of [6, Theorem 3.3], we show that there exists a family{dθ: 0≤θ <1}of pseudometrics onXsuch thatdθ=supθ1>θdθ1, for each 0≤θ <1, and τθ _{coincides with the topology induced by the pseudometric} _d_{θ. Now,} forx,y∈X, defineF(x,y) :R→[0, 1] byF(x,y)(t)=0 ift≤0 andF(x,y)(t)=sup{θ: 0< θ≤1,d1−θ(x,y)< t}ift >0. It is clear thatF(x,y) is increasing and left continuous. For 0< r <1 andt > d1−r(x,y), we have thatF(x,y)(t)≥r, and so limt→∞F(x,y)(t)=1. AlsoF(x,x)(t)=1 for everyxand everyt >0. To show thatFis a probabilistic pseudo-metric onX, we must prove that it satisfies the triangle inequality. So, letF(x,y)(t1)∧

F(y,z)(t2)> θ >0. Thend1−θ(x,y)< t1,d1−θ(y,z)< t2, and sod1−θ(x,z)< t1+t2, which implies thatF(x,z)(t1+t2)≥θ. Thus the triangle inequality is satisfied andFis a proba-bilistic pseudometric. We will finish the proof by showing thatτF=τ. So letNxτF> θ >0 and chooset >0 such that 1−sup_{y /}_∈_AF(y,x)(t)> θ. If nowdθ(x,y)< t, thenF(x,y)(t)≥ 1−θ, and thusy∈A, which proves thatAis aσθ=τθneighborhood ofx. Henceτ≥τF. On the other hand, letBbe aτθ_{-neighborhood of}_x_{. There exists}_θ

1> θsuch thatNx(B)>

θ1. NowBis aτθ1-neighborhood ofx, and so there existst >0 such that{y:dθ1(x,y)<

t} ⊂B. IfF(x,y)(t)>1−θ1, then there existsα >1−θ1such thatd1−α(x,y)< tand so

dθ1(x,y)< t. Thus{y:F(x,y)(t)>1−θ1} ⊂B, and therefore

NτF

x (B)≥1−sup y /∈B

F(x,y)(t)≥θ1> θ. (3.21)

Thus,τF≥τand the result follows.

Theorem3.7. Let(X,F)be a probabilistic pseudometric space,A⊂X, andx∈X. Let

α=sup

inf

t>0lim infn F

xn,x(t) :xnsequence inA

,

β=sup

lim inf n F

xn,xtn:tn−→0+, xnsequence inA

,

γ=sup

lim inf n F

xn,x(1/n) :xn

sequence inA

.

(3.22)

Thenα=β=γ=_A¯₍_x_). Proof. If (xn)⊂A, then ¯

A(x)≥cxn−→x=inf

t>0lim infn F

xn,x(t), (3.23)

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and choose k such that tn< t when n≥k. For m≥k, we have infn>mF(xn,x)(t)≥ infn≥mF(xn,x)(tn)> θ. Thus lim infnF(xn,x)(t)> θ for eacht >0 and so α≥θ, which proves that α≥β. Clearly β≥γ. Finally, Nx(Ac₎_≥₁₋_sup

y∈AF(y,x)(1/n), and so sup_y_∈_AF(y,x)(1/n)≥1−Nx(Ac₎₌_A¯₍_x₎_>_A¯₍_x₎₋₁_/n_{. Hence, for each}_n_∈_N_{, there} ex-istsxn∈AwithF(xn,x)(1/n)>A¯(x)−1/n. Consequently,

γ≥lim inf n F

xn,x1 n

≥lim inf n

¯

A(x)−1

n

=_A¯₍_x_), _(3.24)

and soγ≥_A¯₍_x₎_≥_α_≥_β_≥_γ_{, which completes the proof.}

In view of [6, Theorem 4.14], we have the following corollary.

Corollary3.8. Every pseudometrizable fuzzifying topological space isN-sequential and hence sequential.

Theorem3.9. If(Fn)is a sequence of probabilistic pseudometrics on a setX, then there exists a probabilistic pseudometricFsuch thatτF=

nτFn.

Proof. If F is a probabilistic pseudometric on X and if ¯F is defined by ¯F(x,y)(t)=

F(x,y)(t) ift≤1 and ¯F(x,y)(t)=1 ift >1, then ¯Fis a probabilistic pseudometric onX

andτ_F¯=τF. Hence, we may assume thatFn(x,y)(t)=1, for alln, ift >1. Forx,y∈X, de-fineF(x,y) onRbyF(x,y)(t)=0 ift≤0 andF(x,y)(t)=infn[(1/n)Fn(x,y)](t) ift >0. Clearly,F(x,y) is increasing andF(x,y)(t)=1 ift >1. Also, F(x,y) is left continuous. In fact, letF(x,y)(t)> θ >0 and choosensuch that (n+ 1)t >1. There exists 0< s1< t such thatFk(x,y)(ks1)> θfork=1,. . .,n. Chooses1< s < tsuch that (n+ 1)s >1. Now,

Fm(x,y)(ms)=1 ifm > n. Thus

F(x,y)(s)= min 1≤k≤n

1

kFk(x,y)

(s)> θ, (3.25)

which proves thatF(x,y) is inR+φ. It is clear thatF(x,x)=¯0. We need to prove thatF satisfies the triangle inequality. So assume thatF(x,y)(t1)∧F(y,z)(t2)> θ >0. Ifm is such that (m+ 1)(t1+t2)>1, then

F(x,z)t1+t2

= min

1≤k≤mFk(x,z)

kt1+t2

. (3.26)

Since

Fk(x,z)kt1+t2

≥Fk(x,y)kt1

∧Fk(y,z)kt2

> θ, (3.27)

it follows thatF(x,z)(t1+t2)> θ, and soFsatisfies the triangle inequality. We will fin-ish the proof by showing thatτF=

τFn. To see this, we first observe that (1/n)FnF,

which implies that τFn=τ(1/n)Fn≤τF, and soτo=

nτ(1/n)Fn≤τF. On the other hand,

let NτF

x (A)> θ and choose >0 such thatNxτF(A)> θ+. Lett >0 be such that 1− sup_{y /}_∈_AF(y,x)(t)> θ+. If (m+ 1)t >1, then

F(y,z)(t)= min

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LetAk= {y:Fk(y,x)(kt)≥1−θ−}. Then

Nτo

x

Ak≥N τ_Fk

x Ak≥1−sup z /∈Ak

Fk(z,x)(kt)≥θ+> θ (3.29)

andmk=1Ak⊂A. Hence,Nxτo(A)≥min1≤k≤mNxτo(Ak)> θ. This proves thatτF≤τoand

the result follows.

Theorem3.10. Let f :X→Y be a function and letFbe a probabilistic pseudometric onY. Then the function

f−1₍_F_{) :}_X2_−→_R+1

φ , f−1(F)(x,y)=F

f(x),f(y), (3.30)

is a probabilistic pseudometric onXandτf−1_(F)=f−1(τF).

Proof. It follows easily that f1₍_F_{) is a probabilistic pseudometric on} _X_{. Let}_x_∈_X _and

B⊂X. IfD=Y\f(Bc_{), then}

Nxτf−1 (F)(B)=inf t>0

1−sup y /∈B

Ff(y),f(x)(t)

=inf t>0

1−sup z∈Dc

Fz,f(x)(t)

=NτF

f(x)(D)=N f−1_(τ

F)

x (B),

(3.31)

which clearly completes the proof.

Corollary3.11. IfFis a probabilistic pseudometric on a setXandY⊂X, thenτF|Y is induced by the probabilistic pseudometricG=F|Y×Y,G(x,y)=F(x,y).

Corollary3.12. If(Xn,τn)is a sequence of pseudometrizable fuzzifying topological spaces, then the Cartesian product(X,τ)=(Xn,τn)is pseudometrizable.

Proof. LetFnbe a probabilistic pseudometric onXninducingτn. IfGn=πn−1(Fn), then

τGn=π−

1

n (τn), and soτ=

nπn−1(τn) is pseudometrizable.

4. Level proximities

Letδbe a fuzzifying proximity on a setX. For each 0< d≤1, letδd_{be the binary relation} on 2X _{defined by}_Aδd_B_{if and only if}_δ₍_A_,_B₎_≥_d_{. It is easy to see that}_δd _{is a classical} proximity onX. We will show that the classical topology σd induced byδd coincides withτ1−d_{. In fact, let}_x_∈_A_∈_σ_{d. Then,}_x _{is not in the}_σ_{d-closure of}_Ac_{, which implies} thatx δd_Ac_{, that is,}_δ₍_x_,_Ac₎_{< d}_{, and so}_Nτ

x(A)=1−δ(x,Ac)>1−d. This proves that

A∈τ1−d_{. Conversely, if}_x_∈_B_∈_τ1−d_{, then}_Nτ

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Theorem4.1. Ifδis a fuzzifying proximity on a setXand0< d≤1, then

δd₌

0<θ<d

δθ_. _(4.1)

Proof. If 0< θ < d, thenδθ_{is coarser than}_δd_{, and so}_δ

o=0<θ<dδθ is coarser thanδd. On the other hand, letAδoB. Sinceδois finer thanδθ (for 0< θ < d), we have thatAδθB and soδ(A,B)≥θ, for each 0< θ < d, which implies thatδ(A,B)≥d, that is,Aδd_B_{. So}_δ

o

is finer thanδd_{and the result follows.}

Theorem4.2. For a family{γd: 0< d≤1}of classical proximities on a setX, the following are equivalent.

(1)There exists a fuzzifying proximityδonXsuch thatδd₌_γ

dfor alld. (2)γd=0<θ<dγθfor each0< d≤1.

Proof. In view of the preceding theorem, (1) implies (2). Assume now that (2) is satis-fied and defineδon 2X_×₂X _by_δ₍_A_,_B₎₌_sup_{_d_:_Aγ

dB}(the supremum over the empty family is taken to be zero). It is clear thatδ(A,B)=1 if theA,Bare not disjoint. Also,

δ(A,B)=δ(A,B) andδ(A,B≥δ(A1,B1) ifA1⊂A,B1⊂B. Let nowδ(A,B)< d <1. Then

A γdB, and so there exists a subsetDofXsuch thatA γdDandDc γdB. SinceA γdD, we have thatδ(A,D)≤d. Similarlyδ(Dc_,_B₎_≤_d_{, and so inf}_{_δ₍_A_,_D₎_∧_δ₍_Dc_,_B₎_{} ≤}_δ₍_A_,_B_). On the other hand, ifδ(A,D)∧δ(Dc_,_B₎_{< θ <}_{1, then}_A_⊂_Dc_{, and so}_δ₍_A_,_B₎_≤_δ₍_Dc_,_B₎_<

θ. This proves thatδis a fuzzifying proximity onX. We will finish the proof by showing that δd₌_γ

d for all d. Indeed, ifAγdB, thenδ(A,B)≥d, that is, AδdB. On the other hand, letAδd_B_{and let (}_A_{i), (}_B_{j) be finite families of subsets of}_X_with_A₌

i,B=

Bj. Sinceδ(A,B)=i,jδ(Ai,Bj)≥d, there exists a pair (i,j) such thatδ(Ai,Bj)≥d. If now 0< θ < d, then there existsr > θwithAiγrBj, and soAiγθBj. This proves thatAγdBsince

γd=

0<θ<dγθ. This completes the proof.

Theorem 4.3. Let(X,δ1),(Y δ2)be fuzzifying proximity spaces and let f :X→Y be a function. Then f is proximally continuous if and only if f : (X,δ1d)→(Y,δ2d)is proximally continuous for each0< d≤1.

Proof. It follows immediately from the definitions.

Theorem4.4. Let (Xλ,δλ)λ∈Λ be a family of fuzzifying proximity spaces and let(X,δ)= (Xλ,δλ)be the product fuzzifying proximity space. Thenδd₌_δd

λ for all0< d≤1. Proof. Since each projectionπλ: (X,δd)→(Xλ,δλd) is proximally continuous, it follows thatδd_{is finer than}_σ₌_δd

λ. On the other hand, letAσB. We need to show thatδ(A,B)≥

d. In fact, let (Ai), (Bj) be finite families of subsets ofX such thatA=Ai,B=Bj. SinceAσBand σ=λπλ−1(δdλ), there exists a pair (i,j) such thatAiπλ−1(δd)Bj, that is,

δλ(πλ(Ai),πλ(Bj))≥d. In view of [4, Theorem 8.9], we conclude thatδ(A,B)≥d. Hence,

σ=δd_{and the proof is complete.}

We have the following easily established theorem.

Theorem4.5. Let(Y,δ)be a fuzzifying proximity space and let f :X→Y. Thenf−1₍_δ₎d₌

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Theorem4.6. Let(δλ)λ∈Λ be a family of fuzzifying proximities on a setXandδ= ∨λδλ. Thenδd₌

λδλdfor each0< d≤1.

Proof. Letσ=λδλd. Sinceδis finer than eachδλ, it follows thatδd is finer than eachδλd, and soδd _{is finer than}_σ_{. On the other hand, let}_AσB_{and let (}_A_{i), (}_B_{j) be finite families} of subsets ofX such thatA=Ai,B=Bj. There exists a pair (i,j) such that AiσBj. Sinceσ is finer than eachδd

λ, we have thatAiδdλBj, that is,δλ(Ai,Bj)≥d. In view of [4, Theorem 8.10], we get thatδ(A,B)≥d, that is,Aδd_B_{. So}_σ_{is finer than}_δd_{and the proof}

is complete.

5. Completely regular fuzzifying spaces

Definition 5.1. A fuzzifying topological space (X,τ) is called completely regular if each of the classical level topologiesτd_{, 0}_≤_{d <}_{1, is completely regular.}

Definition 5.2. A fuzzifying proximityδon a setXis said to be compatible with a fuzzi-fying topologyτifτcoincides with the fuzzifying topologyτδinduced byδ.

We have the following easily established theorem.

Theorem5.3. Subspaces and Cartesian products of completely regular fuzzifying spaces are completely regular.

Theorem 5.4. Let(X,τ)be a completely regular fuzzifying topological space and define

δ=δ(τ) : 2X_×₂X_→_{[0, 1]}_by

δ(A,B)=1−supd: 0≤d <1,∃f :X,τd_−→_{[0, 1]}_continuous _f₍_A₎₌_0, _f₍_B₎₌₁_. (5.1)

Then,(1)δis a fuzzifying proximity onXcompatible withτ;

(2)ifδ1is any fuzzifying proximity onXcompatible withτ, thenδis finer thanδ1. Proof. It is easy to see thatδsatisfies (FP1), (FP2), (FP3), and (FP5). We will prove that

δsatisfies (FP4). Let

α=infδ(A,D)∨δDc,B:D⊂X. (5.2)

Ifδ(A,D)∨δ(Dc_,_B₎_{< θ}_{, then}_A_⊂_Dc_{, and so}_δ₍_A_,_B₎_≤_δ₍_Dc_,_B₎_{< θ}_{, which proves that}

δ(A,B)≤α. On the other hand, assume thatδ(A,B)< r <1. There exist ad, 1−r < d <1, and f :X→[0, 1]τd_{-continuous such that} _f₍_A₎₌_0, _f₍_B₎₌_{1. Let}_D_{= {}_x_∈_X_{: 1}_/₂_≤

f(x)≤1}and defineh1,h2: [0, 1]→[0, 1],h1(t)=2t,h2(t)=0 if 0≤t≤1/2 andh1(t)= 1,h2(t)=2t−1 if 1/2< t≤1. Ifgi=hi◦f,i=1, 2, theng1(A)=0,g1(D)=1,g2(Dc)=0,

g2(B)=1. Thus,δ(A,D)≤1−d < r,δ(Dc_,_B₎_{< r}_{, which proves that}_α_≤_δ₍_A_,_B_{). Hence,}

δis a fuzzifying proximity onX. We need to show thatτ=τδ. So, letτ(A)> θ >0. Since

τθ_{is completely regular, given}_x_∈_A_{, there exist} _f

x:X→[0, 1],τθ-continuous, fx(x)= 0, fx(Ac₎₌_{1. Thus}_δ₍_x_,_Ac₎_≤₁₋_θ_{, and so} _Nτδ

x (A)=1−δ(x,Ac)≥θ. It follows that

τδ(A)=infx∈ANxτδ(A)≥θ, which proves thatτδ≥τ. On the other hand, assume that

τδ(A)> r >0. Ifx∈A, thenδ(x,Ac₎₌₁₋_Nτδ

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G= {y:f(y)<1/2}is inτd_and_x_∈_G_⊂_A_{. Thus,}

Nτ

x(A)≥Nxτ(G)≥d > r. (5.3)

This proves thatτ(A)≥rand soτ≥τδ, which completes the proof of (1).

Letδ1be a fuzzifying proximity onX compatible withτand letA,Bbe subsets ofX withδ1(A,B)< θ <1. Ifd=1−θ, thenδ1θis compatible withτd. SinceA δθ1B, there exists (by [8, Remark 3.15]) an f :X→[0, 1]τd-continuous, with f(A)=0, f(B)=1, and so

δ(A,B)≤1−d=θ, which proves thatδ(A,B)≤δ1(A,B), and thereforeδis finer thanδ1.

This completes the proof.

Theorem5.5. For a fuzzifying topological space(X,τ), the following are equivalent. (1) (X,τ)is completely regular.

(2)There exists a fuzzifying proximityδonXcompatible withτ.

(3) (X,τ)is fuzzy uniformizable, that is, there exists a fuzzy uniformityᐁonXsuch that

τcoincides with the fuzzifying topologyτᐁinduced byᐁ.

Proof. By [5], (2) is equivalent to (3). Also (1) implies (2) in view of the preceding the-orem. Assume now thatτ=τδfor some fuzzifying proximityδ. For each 0< d≤1,δd is a classical proximity compatible withτ1−d_{, and so}_τ1−d_{is completely regular. This}

com-pletes the proof.

Theorem5.6. Every pseudometrizable fuzzy topological space(X,τ)is completely regular. Proof. Ifτis pseudometrizable, then eachτd_{, 0}_≤_{d <}_{1, is pseudometrizable, and hence}

τd_{is completely regular.}

Theorem5.7. For a fuzzifying topological space(X,τ), the following are equivalent. (1) (X,τ)is completely regular.

(2)IfᏲ=Ᏺτ is the family of all probabilistic pseudometrics onXwhich areτ×τ con-tinuous as functions fromX2_to_R+

φ, thenτ=τᏲτ.

(3)There exists a familyᏲof probabilistic pseudometrics onXsuch thatτ=τᏲ.

Proof. (1)⇒(2). For eachF∈Ᏺτ, we have thatτF≤τ(byTheorem 3.2), and soτᏲτ ≤τ. Let nowA⊂Xandxo∈XwithNxτo(A)> θ >0. Sinceτ

θ_{is completely regular, there exists}

aτθ_{-continuous function} _f _from_X_{to [0,1] such that} _f₍_x

o)=0, f(Ac)=1. Forx,y∈X, defineF(x,y) onRby

F(x,y)(t)=

        

0 ift≤0,

1−θ iff(x)−f(y)≥t >0, 1 iff(x)−f(y)< t.

(5.4)

Clearly,F(x,y)=F(y,x)∈R+

φ andF(x,x)=¯0. We will prove thatF satisfies the trian-gle inequality. So, assume thatF(x,y)(t1)∧F(y,z)(t2)> F(x,z)(t1+t2). Then,t1,t2>0,

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thatNτF

x (B)> θ1. Chooset >0 such that 1−supy /∈BF(x,y)(t)> θ1, and soF(x,y)(t)= 1−θand|f(x)−f(y)| ≥tify /∈B. Thus,{y:|f(x)−f(y)|< t} ⊂B. This shows thatB

is aτθ_{-neighborhood of}_x_{. As}_{r < θ}_,_B_{is a}_τr_{-neighborhood of}_x_{, that is,}_Nτ

x(B)> r, and soτF≤τ. Finally ify /∈A, then|f(y)−f(xo)| =1, and soF(y,xo)(1/2)=1−θ, which implies that

NτᏲ

xo(A)≥N

τF

xo(A)≥1−sup

y /∈A

Fy,xo

1 2

≥θ. (5.5)

This shows thatNτᏲ

xo ≥Nxτo, and soτ≤τᏲ, which completes the proof of the

implica-tion (1)⇒(2).

(3)⇒(1). Assume thatτ=τᏲ for some familyᏲof probabilistic pseudometrics on X. For eachF∈Ᏺ,τFis completely regular and soτᏲ is completely regular sinceτᏲd =

F∈ᏲτFdfor each 0≤d <1. Hence the result follows.

We will denote by [0, 1]φthe subspace ofR+

φ consisting of allu∈R+φ withu(t)=1 if

t >1.

Theorem5.8. A fuzzifying topological space(X,τ)is completely regular if and only if the following condition is satisfied. IfNxo(A)> θ >0, then there existsf :X→[0, 1]φcontinuous

such that f(xo)=¯0and f(y)(t)=1−θify /∈Aand0< t <1.

Proof. Assume that (X,τ) is completely regular and letNxo(A)> θ >0. Sinceτθ is

com-pletely regular, there existsh: (X,τθ₎_→_{[0, 1] continuous,}_h₍_x_o)₌_0,_h₍_y₎₌_{1 if} _{y /}_∈_A_. Forx,y∈X, defineF(x,y) onRby

F(x,y)(t)=

        

0 ift≤0,

1−θ ifh(x)−hxo≥t >0, 1 ifh(x)−hxo< t.

(5.6)

F(x,z)(t1+t2)=1, a contradiction. SoFis a probabilistic pseudometric. Moreover,F is

τ×τcontinuous, or equivalentlyτF≤τ. In fact, letNxτF(B)> r >0. There exists at >0 such that 1−sup_{z /}_∈_BF(z,x)(t)> r. Ifz /∈B, thenF(z,x)(t)<1−r <1, and soF(z,x)(t)= 1−θ <1−r, that is,r < θ, and|h(z)−h(x)| ≥t. Hence

M=z:h(z)−h(x)< t⊂B. (5.7)

The setM is aτθ_{-neighborhood of}_x_{, and hence a}_τr_{-neighborhood, that is,}_Nτ x(B)> r. Thus τ≥τF. Finally, define f :X→[0, 1]φ, f(y)=F(y,xo). Then f is τ-continuous,

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exists aθ1> θsuch thatNxτo(A)> θ1. By our hypothesis, there exists f :X→[0, 1]φ

con-tinuous such that f(xo)=¯0 and f(y)(t)=1−θ1if y /∈Aand 0< t <1. DefineF(x,y) =D(f(x),f(y)). ThenFisτ×τcontinuous and

NτᏲ

xo(A)≥N

τF

xo(A)≥1−sup

y /∈A

Fxo,y(1)

=1−sup y /∈A

D¯0,f(y)(1)

=1−sup y /∈A

f(y)(1)≥θ1> θ.

(5.8)

ThusNτᏲ

xo(A)≥Nxτo(A), for every subsetAofX, and soτ≤τᏲ. Therefore,τ=τᏲ, and

soτis completely regular.

For a fuzzifying topological spaceX, we will denote byC(X, [0, 1]φ) the family of all continuous functions fromXto [0, 1]φ.

Theorem5.9. A fuzzifying topological space(X,τ)is completely regular if and only ifτ co-incides with the weakest of all fuzzifying topologiesτ1onXfor which each f ∈C(X, [0, 1]φ) is continuous.

Proof. Assume that (X,τ) is completely regular and letτ1be the weakest of all fuzzify-ing topologies onX for which each f ∈C(X, [0, 1]φ) is continuous. Clearlyτ1≤τ. On the other hand, letτ2be a fuzzifying topology onX for which each f ∈C(X, [0, 1]φ) is continuous. LetNτ

x(A)> θ >0. In view of the preceding theorem, there exists an f ∈

C(X, [0, 1]φ) such that f(x)=¯0, f(y)(t)=1−θify /∈Aand 0< t <1. Let

G=

u∈R+ φ:D

f(x),u1

2

=u

1 2

>1−θ

. (5.9)

Then

N¯0(G)≥1−sup u /∈G

Df(x),u1

2

≥θ. (5.10)

Since f isτ2-continuous, we have thatNτ2

x (f−1(G))≥θ. Butf−1(G)⊂Asince, fory /∈A, we have that f(y)(1/2)=1−θ. ThusNτ2

x (A)≥θ. This proves thatNxτ2(A)≥Nxτ(A), for every subsetAofX, and soτ2≥τ. This clearly proves thatτ1=τ. Conversely, assume thatτ1=τ. Ifσis the usual fuzzifying topology ofR+_φ, then

τ=τ1=

f∈C(X,[0,1]φ)

f−1(σ). (5.11)

Sinceσ is completely regular, each f−1₍_σ_{) is completely regular, and so}_τ_{is completely}

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A. K. Katsaras: Department of Mathematics, School of Natural Sciences, University of Ioannina, 45110 Ioannina, Greece