Approach Spaces, Limit Tower Spaces, and Probabilistic Convergence Spaces

Applied Categorical Structures 5: 99–110, 1997. 99c© 1997 Kluwer Academic Publishers. Printed in the Netherlands.

Approach Spaces, Limit Tower Spaces, andProbabilistic Convergence Spaces

PAUL BROCK and D. C. KENTWashington State University, Department of Pure and Applied Mathematics, Pullman,WA 99164-3113, U.S.A.

(Received: 27 April 1995; accepted: 15 January 1996)

Abstract. The category LTS of limit tower spaces is defined and shown to be isomorphic to thecategory CAP of convergence approach spaces. The full subcategory of LTS determined by theobjects satisfying a diagonal axiom due to Cook and Fischer is shown to be isomorphic to thecategory AP of approach spaces. A family of isomorphisms is also obtained between LTS andcertain full subcategories of the category PCS of probabilistic convergence spaces.

Mathematics Subject Classifications (1991). 54B30, 18B30, 54A05, 54E70.

Key words: approach space, convergence approach limit, limit tower, isomorphism functor, prob-abilistic convergence space, t-norm.

1. Introduction

In 1989, R. Lowen [7], introduced the category AP of approach spaces whichcontains the categories TOP and MET as full subcategories. In [5], E. Lowenand R. Lowen embedded AP in the quasitopos CAP of convergence approachspaces, and in [6] the same authors constructed within CAP the smallest qua-sitopos PSAP containing AP .

Probabilistic convergence spaces were first defined in 1989 by Florescu [2], interms of net convergence. The filter formulation considered here was introducedin [11] by Richardson and Kent. Probabilistic convergence spaces are a naturalextension of probabilistic metric and probabilistic topological spaces which havebeen studied during the past forty years by Menger [9], Schweizer and Sklar [12,13], Frank [3], and others. The fundamental idea is to assign a probability to theconvergence of a given filter to a given point. Whereas approach spaces weremotivated in large measure by categorical considerations, probabilistic conver-gence spaces were designed to deal with certain convergence situations arisingnaturally in probability theory and analysis. Nonetheless, it can be shown that thecategory PCS of probabilistic convergence spaces and their natural morphismsis a quasitopos.

Our study of the relationship betweenAP and PCS led us to define a new cat-egory LTS of limit tower spaces which is isomorphic to CAP , but whose objectsresemble probabilistic convergence spaces. Indeed, the structures of objects in

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100 PAUL BROCK AND D. C. KENT

both LTS and PCS can be regarded as linearly ordered families of convergencestructures on a set X, indexed in the former case by [0,∞] and in the latter caseby [0, 1]. Every strictly decreasing map from [0, 1] onto [0,∞] defines a uniquet-norm and a corresponding isomorphism between LTS and a subcategory ofPCS determined by the t-norm.

Two diagonal axioms K and F introduced for convergence spaces by Kowal-sky [4] and Cook and Fischer [1], respectively, extend naturally to axioms [K]and [F ] in LTS, and we show that AP is isomorphic to the full subcategory ofLTS consisting of those objects which satisfy [F ]. The corresponding diagonalaxioms KT and FT in PCS depend on a t-norm T . The latter axioms, whichwere studied extensively in [11], provide convenient descriptions for a class ofsubcategories of PCS isomorphic to AP .

2. Convergence Spaces and Convergence Approach Spaces

Let X be a set, F(X) the set of all filters on X, and 2X the power set of X. Forx ∈ X, let x denote the fixed ultrafilter generated by {x}.

DEFINITION 1. A convergence structure q on a set X is a function q: F(X)→2X satisfying:

(C1): x ∈ q(x), for all x ∈ X;(C2): F ⊆ G ⇒ q(F) ⊆ q(G);(C3): x ∈ q(F)⇒ x ∈ q(F ∩ x).

The statement “x ∈ q(F)” will normally be written “Fq→ x” (F q-converges to

x). If q is a convergence structure on X, then (X, q) is a convergence space.Consider the additional axioms:

(C4): Fq→ x and G

q→ x implies F ∩ G

q→ x;

(C5): If each ultrafilter containing F q-converges to x, then Fq→ x.

A convergence structure is called a limit structure if it satisfies (C4) and a pseudo-topology if it satifies (C5). Limit space and pseudo-topological space are definedin the obvious way.

Let C(X) denote the complete lattice consisting of all convergence structureson X, partially ordered as follows: p 6 q iff F

q→ x implies F

p→ x; in this case

we say “q is finer than p” or “p is coarser than q.”If (X, q) is a convergence space such that

⋂i∈I Fi

q→ x whenever {Fi: i ∈

I} ⊆ F(X) and Fiq→ x for all i ∈ I, then q is called a pretopology and

(X, q) is a pretopological space. For any convergence space (X, q), the filterVq(x) =

⋂{F ∈ F(X): F

q→ x} is called the q-neighborhood filter at x. Clearly

q is a pretopology iff Vq(x)q→ x, for all x ∈ X. A topology on X can be regarded

as a special case of a pretopology, and hence of a convergence structure.

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APPROACH, LIMIT TOWER, AND PROBABILISTIC CONVERGENCE SPACES 101

Let CONV , LIM , PSTOP , PRTOP , and TOP be the categories whoseobjects are, respectively, the convergence spaces, limit spaces, pseudo-topologicalspaces, pretopological spaces, and topological spaces, and whose morphismsare continuous maps. All five categories are topological and the first three arequasitopoi (i.e. hereditary and cartesian closed). Each of the categories LIM ,PSTOP , PRTOP and TOP is a bireflective subcategory of every precedingcategory in the original list. The bireflection of (X, q) ∈ |CONV | in PRTOPis denoted by (X,πq); πq is called the pretopological modification of q.

We next recall two diagonal axioms K and F for a convergence space (X, q).Given arbitrary sets X and J , F ∈ F(J), and a function σ: J → F(X), let

κσF =⋃F∈F

⋂{σ(y): y ∈ F}.

Note that κσF is a filter on X, and if J = X and σ(x) 6 x for all x ∈ X, itfollows that κσF 6 F . In particular, if σ(x) = x for all x ∈ X, then κσF = F .

K: Let σ: X → F(X) be any function such that σ(y)q→ y, for all y ∈ X. If

Fq→ x, then κσF

q→ x.

F: Let J be any set, let ψ: J → X, and let σ: J → F(X) have the propertythat σ(y)

q→ ψ(y), for all y ∈ J . If F ∈ F(J) is such that ψ(F)

q→ x, then

κσFq→ x.

The axiom K was introduced by Kowalsky [4] in 1954, whereas F was intro-duced by Cook and Fischer [1] in 1967. Note that F reduces to K if J = X andψ is the identity map on X. The next proposition is proved in [11].

PROPOSITION 2. Let (X, q) be a convergence space.

(a) If (X, q) satisfies K, then (X, q) is a limit space and πq is a topology.(b) (X, q) satisfies F iff q is a topology.

We turn now to the category of convergence approach spaces introduced byE. Lowen and R. Lowen in [5].

DEFINITION 3. A convergence approach limit λ on a set X is a functionλ: F(X)→ [0,∞]X which satisfies the following conditions:

(CAL1): For any x ∈ X, λ(x)(x) = 0;(CAL2): If F ⊆ G, then λ(G) 6 λ(F);(CAL3): For all F , G ∈ F(X), λ(F ∩ G) = λ(F) ∨ λ(G).

The pair (X,λ) consisting of a set X and a convergence approach limit λ iscalled a convergence approach space. If (X,λ) and (X

′, λ′) are convergence

approach spaces, then a mapping f : (X,λ) → (X′, λ′) is called a contraction

if, for all F ∈ F(X),

λ′(f(F)) ◦ f 6 λ(F).

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The category CAP with convergence approach spaces as objects and contrac-tions as morphisms is proved in [6] to be a quasitopos. We give three additionalaxioms applicable to a convergence approach limit λ on a set X:

(PSAL): For each F ∈ F(X), λ(F) = sup{λ(U): U an ultrafilter, F ⊆ U};(PRAL): For any family (Fj)j∈J of filters on X, λ(

⋂j∈J Fj) = supj∈J λ(Fj);

(AL): Let σ: X → F(X) and F ∈ F(X). Then λ(κσF) 6 λ(F) +supy∈X λ(σ(y))(y).

It is easy to see that (PRAL) ⇒ (PSAL) ⇒ (CAL3).The axioms (PSAL) and (PRAL) define pseudo-approach limits and preap-

proach limits, respectively. The full subcategories of CAP determined by thepseudo-approach spaces and preapproach spaces are called PSAP and PRAP ,respectively. A convergence approach space satisfying (PRAL) and (AL) is calledan approach space, and the full subcategory of CAP whose objects are approachspaces is called AP .

Approach spaces were first introduced by R. Lowen [7] in 1989 as a categorycontaining both TOP and MET as full subcategories. In [7], six distinct butequivalent ways to axiomatize an approach space are given; the one most con-venient for our purposes is that based on “approach limits,” as defined above. Itis shown in [6] that PSAP is the quasitopos hull of AP .

3. Limit Tower Spaces

In this section we introduce the category LTS of limit tower spaces. The notionof “tower” was defined in [7] as one way to characterize approach spaces, and ourdefinition of “limit tower” generalizes “tower” in the same way that “convergenceapproach limit” generalizes “approach limit”. And indeed it turns out that LTSis isomorphic to CAP .

DEFINITION 4. A limit tower p on a set X is a family {pε: ε ∈ [0,∞]} of limitstructures on X satisfying the following conditions:

(LT1): ε 6 γ ⇒ pγ 6 pε;(LT2): p∞ is the indiscrete topology;(LT3): For each ε ∈ [0,∞], qε = sup{qγ : ε < γ}.

If p is a limit tower on X, then (X, p) is called a limit tower space. It will oftenbe convenient to write p = (pε), where the pε’s are called the component limitstructures of p. A partial order for limit towers on the same set X is defined asfollows:

p 6 p′ ⇔ pε 6 p′ε, for all ε ∈ [0,∞].

A limit tower p on X is said to be a pseudo-topological (respectively, pretopo-logical, topological) tower if pε is a pseudo-topology (respectively, pretopology,

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topology) for all ε ∈ [0,∞]. Furthermore, p is said to be a constant limit towerif pε = p0, for all ε ∈ [0,∞). The terms pseudo-topological tower space, pre-topological tower space, and topological tower space are defined in the obviousway.

A function f : (X, p) → (X ′, p′) between limit tower spaces is said to becontinuous if it is componentwise continuous (i.e., f : (X, pε) → (X ′, p′ε) iscontinuous for all ε ∈ [0,∞]). The category with limit tower spaces as objectsand continuous maps as morphisms is denoted by LTS. The pseudo-topological,pretopological, topological, and constant objects in LTS determine the full sub-categories PSTS, PRTS, TTS, and CLTS, respectively. It is clear that CLTSis isomorphic to the category LIM of limit spaces, and CLTS∩TTS is isomor-phic to the category TOP of topological spaces. We remark that the subcategoriesPSTS, PRTS, TTS, TOP , and LIM are all bireflectively embedded in LTS,while LIM is also bicoreflectively embedded in LTS.

We shall now proceed to establish an isomorphism between CAP and LTS.For (X,λ) ∈ CAP , let ηλ = {(ηλ)ε: ε > 0}, where for each ε ∈ [0,∞],

(ηλ)ε: F(X)→ 2X is defined by

x ∈ (ηλ)ε(F) ⇐⇒ λ(F)(x) 6 ε.

Using the notational convention of Definition 1, (ηλ)ε can be more intuitivelydescribed from a “convergence perspective” by:

F(ηλ)ε−→ x ⇐⇒ λ(F)(x) 6 ε.

PROPOSITION 5. If (X,λ) ∈ CAP , then (X, ηλ) ∈ LTS.Proof. To see that (ηλ)ε is a limit structure for each ε, note that (C1) and

(C2) of Definition 1 follow immediately from (CAL1) and (CAL2), respectively,

of Definition 3. If F(ηλ)ε−→ x and G

(ηλ)ε−→ x, then λ(F)(x) ∨ λ(G)(x) 6 ε, and by

(CAL3), λ(F ∩G)(x) 6 ε. Thus F ∩G(ηλ)ε−→ x, and so (ηλ)ε is a limit structure.

(LT1) If ε 6 γ and F(ηλ)ε−→ x, then λ(F)(x) 6 ε ⇒ λ(F)(x) 6 γ, which

implies F(ηλ)γ−→ x. Thus (ηλ)γ 6 (ηλ)ε.

(LT2) Since λ(F)(x) 6 ∞ holds for all F ∈ F(X) and x ∈ X, (ηλ)∞ isindiscrete.

(LT3) Let ε ∈ [0,∞], and let p = sup{(ηλ)γ : γ > ε}. p 6 (ηλ)ε follows

by (LT1). If Fp→ x then F

(ηλ)γ−→ x, for all γ > ε. Thus λ(F)(x) 6 ε, and so

F(ηλ)ε−→ x. 2

PROPOSITION 6. Let (X,λ) and (X ′, λ′) be convergence approach spaces ,and let f : (X,λ)→ (X ′, λ′) be a contraction. Then f : (X, ηλ) → (X ′, ηλ′) iscontinuous.

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Proof. Let ε ∈ [0,∞] and F(ηλ)ε−→ x. Since f is a contraction, λ′(f(F))(f(x))

6 λ(F)(x) 6 ε, and therefore f(F)(ηλ′)ε−→ f(x), establishing continuity of f :

(X, (ηλ)ε)→ (X ′, (ηλ′)ε). 2

The preceding propositions imply that η: CAP → LTS, defined for objectsby η(X,λ) = (X, ηλ) and for morphisms by η(f) = f is a functor.

For (X, p) ∈ LTS, let ρp: F(X) → [0,∞]X be defined as follows: ForF ∈ F(X) and x ∈ X,

(ρp)(F)(x) = inf{ε ∈ [0,∞]: Fpε→ x}.

PROPOSITION 7. If (X, p) ∈ LTS, then (X, ρp) ∈ CAP .Proof. (CAL1) and (CAL2) follow easily. To establish (CAL3), let F ,G ∈

F(X) and x ∈ X. It follows by (CAL2) that (ρp)(F) ∨ (ρp)(G) 6 (ρp)(F ∩G).To show the reverse inequality, let (ρp)(F)(x) ∨ (ρp)(G)(x) = ε. If ε < γ, then

Fpγ→ x and G

pγ→ x, so by (C4) of Definition 1.1, F ∩ G

pγ→ x. Thus

(ρp)(F ∩ G)(x) 6 inf{γ: F ∩ Gpγ→ x} 6 ε = (ρp)(F)(x) ∨ (ρp)(G)(x).

Since x is arbitrary, (CAL3) is obtained. 2

PROPOSITION 8. Let (X, p) and (X ′, p′) be limit tower spaces, and let f :(X, p)→ (X ′, p′) be continuous. Then f : (X, ρp)→ (X ′, ρp′) is a contraction.

Proof. Let F ∈ F(X) and x ∈ X, and let (ρp)(F)(x) = ε. Then Fpγ→ x, for

all γ > ε, which implies f(F)p′γ→ f(x), for all γ > ε. Thus (ρp′)(f(F))(f(x)) 6

inf{γ: γ > ε} = ε. 2

The two preceding propositions show that ρ: LTS → CAP , defined for objectsby ρ(X, p) = (X, ρp) and for morphisms by ρ(f) = f , is a functor.

THEOREM 9. η: CAP → LTS is an isomorphism.Proof. It remains to show: (a) for each (X,λ) ∈ |CAP |, (ρ◦η)(λ) = λ, and

(b) for each (X, p) ∈ |LTS|, (η ◦ ρ)(p) = p.(a) Let (X,λ) ∈ |CAP |. If F ∈ F(X) and x ∈ X, then

ρ(ηλ)(F)(x) = inf{ε: F(ηλ)ε−→ x} = inf{ε: λ(F)(x) 6 ε} = λ(F)(x).

Since F ∈ F(X) and x ∈ X are arbitrary, (ρ ◦ η)(λ) = λ.(b) Let (X, p) ∈ LTS, and let ε ∈ [0,∞]. For F ∈ F(X) and x ∈ X,

F → x relative to (η(ρp))ε ⇔ (ρp)(F)(x) 6 ε ⇔ inf{γ: Fpγ→ x} 6 ε. Let

δ = inf{γ: Fpγ→ x}. By (LT3) F

pδ→ x, and since δ 6 ε, F pε→ x by (LT1). Thus

F → x relative to (η(ρp))ε implies Fpε→ x. Conversely, it is clear that F

pε→ x

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implies inf{γ: Fpγ→ x} 6 ε, and so it follows by the above equivalences that

F → x relative to (η(ρp))ε. Thus we obtain (η ◦ ρ)(p) = p. 2

The next proposition states that η, when suitably restricted, yields an isomorphismbetween certain previously-mentioned subcategories of CAP and LTS. Theresults are predictable, and we omit the straightforward proofs.

PROPOSITION 10. If η: CAP → LTS is the isomorphism functor of Theo-rem 2.6, then η|PSAP : PSAP → PSTS and η|PRAP : PRAP → PRTS areboth isomorphism functors.

The axioms K and F defined for convergence spaces in Section 1 extend ina natural way to limit tower spaces. For (X, p) ∈ LTS, these axioms take thefollowing form:

[K]: Let ε, γ ∈ [0,∞] and let σ: X → F(X) be any function such that σ(y)pε→ y,

for all y ∈ X. If F ∈ F(X) and Fpγ→ x, then κσF

pε+γ→ x.

[F]: Let ε, γ ∈ [0,∞], and let J be a nonempty set. Let ψ: J → X andσ: J → F(X) be any functions which have the property that σ(y)

pε→ ψ(y),

for all y ∈ J . If F ∈ F(J) is such that ψ(F)pγ→ x, then κσF

pε+γ→ x.

It is obvious that [F] implies [K]. It is also obvious that under the naturalisomorphism between the full subcategory CLTS of constant limit tower spacesand LIM , [K] and [F] translate to K and F, respectively. Let KLTS and FLTSdenote the full subcategories of LTS whose objects satisfy the axioms [K] and[F], respectively.

PROPOSITION 11.

(a) TTS ⊆ FLTS ⊆ PRTS;(b) FLTS = KLTS ∩ PRTS.

Proof. (a) The first inclusion is an easy consequence of Proposition 2(b). Toprove the second, let (X, p) ∈ FLTS. For a given ε ∈ [0,∞], let {Hj : j ∈ J}be the set of all ultrafilters on X which pε-converge to x. Define ψ: J → X byψ(j) = x, and let σ(j) = Hj , for all j ∈ J . If F is the trivial filter {J}, then

ψF = xp0→ x, and therefore κσF

pε→ x (since 0 + ε = ε). But κσF = Vpε (x),and it follows that each pε is a pretopology.

(b) This assertion is proved in Proposition 3.10 [11], in the setting of proba-bilistic convergence spaces. The completion of this proof is therefore postponeduntil Section 3, where the formulation of [K] and [F] for probabilistic limit spaceswill be established. 2

There are examples which show that the categories KLTS, FLTS, PRTS, andTTS are mutually distinct.

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We next consider the translation of the diagonal axioms [K] and [F] underthe isomorphism between LTS and CAP . We first define the axiom (F) for(X,λ) ∈ CAP as follows:(F): If J is a non-empty set, ψ: J → X, σ: J → F(X) and F ∈ F(J), then:

λ(κσF) 6 λ(ψF) + supy∈J λ(σ(y))(ψ(y)).

PROPOSITION 12. Let (X,λ) ∈ |CAP |, and let η(X,λ) = (X, p) ∈ |LTS|where p = ηλ.

(a) (X,λ) satisfies (AL) iff (X, p) satisfies [K].(b) (X,λ) satisfies (F) iff (X, p) satisfies [F].Proof. We shall prove only (b); the proof of (a) is similar. First, assume that

(X,λ) satisfies (F). Let J be a nonempty set, ψ: J → X, and let σ: J → F(X)

have the property that σ(y)pε→ ψ(y), for all y ∈ J . Let γ ∈ [0,∞] and suppose

ψFpγ→ x for some x ∈ X. By (F) and the definition of p = ηλ, it follows that

λ(κσF) 6 λ(ψF) + ε 6 γ + ε. Therefore, λ(κσF)pε+γ−→ x, and so (X, p) ∈

|FLTS|.Conversely, assume that η(X,λ) = (X, p) satisfies [F]. Let J, ψ, σ, and F

be as in the statement of (F), and let x ∈ X. Choose ε ∈ [0,∞] such thatλ(ψF)(x) = ε, and let λ(σ(y))(ψ(y)) = γy , for all y ∈ X. Let γ = sup{γy: y ∈

J}, so that σ(y)pγ→ ψ(y), for all y ∈ J . Then supy∈J λ(σ(y))(ψ(y)) = sup{γy: y

∈ J} = γ. Since ψFpε→ x, and σ(y)

pγ→ ψ(y) for all y ∈ J , [F] implies that

κσFpε+γ−→ x, and so λ(κσF)(x) 6 ε+ γ. In other words,

λ(κσF)(x) 6 λ(ψF)(x) + supy∈J

λ(σ(y))(ψ(y))

holds for all x ∈ X. Therefore, (X,λ) satisfies (F). 2

The next theorem gives a simplified description of approach spaces in the cate-gory CAP .

THEOREM 13. Let (X,λ) ∈ |CAP |. Then (X,λ) ∈ |AP | iff (X,λ) satis-fies (F ).

Proof. From [5], we know that (X,λ) ∈ |AP | iff (X,λ) satisfies (PRAL)and (AL). Using Proposition 10 and 12(a), we obtain that (X,λ) ∈ |AP | iffη(X,λ) ∈ |PRTS| ∩ |KLTS|. Then applying Proposition 11 and 12(b), weobtain the desired conclusion. 2

COROLLARY 14. The categories AP and FLTS are isomorphic.

We conclude this section by pointing out the relationship between the notion of“tower” as defined in [7] and [8] and “limit tower” as introduced herein. A tower(τε)ε∈[0,∞] on X can be regarded as a family of closure operators on X satisfy-ing two additional axioms. Using the well known bijection between the closure

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operators and the pretopologies on a given set, we can interpret a “tower” in thesense of [7] to be a special case of a “pretopological tower” in our sense. Corol-lary 14 makes this relationship more precise; a “tower” [7] is a “pretopologicaltower” satisfying [K], or equivalently, a “limit tower” satisfying [F].

4. Probabilistic Convergence Spaces

Probabilistic convergence spaces have evolved from the study of probabilisticmetric spaces and their generalizations (see [2, 3, 9, 12, 13]). A filter-basedtheory for such spaces was introduced in [11].

Let I denote the unit interval [0, 1] in R.

DEFINITION 15. A probabilistic convergence structure q on X is a function

q: F(X) × I → 2X

satisfying:

(PCS1): For each µ ∈ I, q(F , µ) = qµ(F), where qµ ∈ C(X);(PCS2): If µ = 0, qµ is the indiscrete topology;(PCS3): If µ 6 ν ∈ I , then qµ 6 qν;(PCS4): For each µ ∈ I, qµ = sup{qν : ν < µ}.

We will generally write q = (qµ), where µ is assumed to range through I. If q isa probabilistic convergence structure on X, then (X,q) is called a probabilisticconvergence space. The probability that a filter F q-converge to x is defined tobe µ = sup{ν ∈ I: F

qν→ x}.

The condition (PCS4) of Definition 3.1 is called left-continuity. In [11], thiscondition was not required in the definition of “probabilistic convergence space”,but we have included it here, partly because the most significant examples areleft-continuous (for instance, see Examples 3.13 and 3.14 of [11]), but also tosimplify the relationship between probabilistic convergence spaces, convergenceapproach spaces, and limit tower spaces.

If (X,q) and (Y,p) are probabilistic convergence spaces and f : X → Y , thenf : (X,q)→ (Y,p) is said to be continuous if f : (X, qµ)→ (Y, pµ) is continu-ous for all µ ∈ I. The category with probabilistic convergence spaces as objectsand continuous functions as morphisms is denoted by PCS. If (X,q) ∈ |PCS|and each qµ is a limit structure (respectively, pseudotopology, pretopology, topol-ogy), then (X,q) is called a probabilistic limit space (respectively, probabilisticpseudo-topological space, probabilistic pretopological space, probabilistic topo-logical space), and the corresponding full subcategory of PCS is denoted byPLS (respectively, PPSS, PPRS, PTS).

We next define “t-norm,” a notion which is fundamental in the study of prob-abilistic metric spaces and their generalizations. For further information aboutthis topic and for justification of many of our unverified assertions, the reader isreferred to [13].

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DEFINITION 16. A t-norm is binary operation T : I2 → I which is associative,commutative, increasing in each variable, and satisfies: T (µ, 1) = µ, for allµ ∈ I.

Let T be the set of all t-norms. A partial order on T is defined as follows:

T 6 T ′ iff T (µ, ν) 6 T ′(µ, ν) for all (µ, ν) ∈ I2.

The smallest t-norm, T0 is defined by

T0(µ, ν) =

µ, if ν = 1ν, if µ = 10, otherwise.

The largest t-norm is T1, defined by

T1(µ, ν) = min{µ, ν}

for all (µ, ν) ∈ I2.A useful class of t-norms may be obtained as follows. Let S: I → [0,∞]

be any surjective, strictly decreasing function (in other words, an order-reversinghomeomorphism). Then TS : I2 → I, defined by TS(µ, ν) = S−1(S(µ) + S(ν))is a t-norm. Any t-norm T of the form T = TS for some order reversinghomeomorphism is called a strict t-norm. The subset of T consisting of strictt-norms is denoted by TS . Neither T0 nor T1 is in TS , but T0 is the infimum andT1 the supremum of TS in T.

Let S: I → [0,∞] be an order-reversing homeomorphism. If (X, p) is a limittower space, let φS p = q, where q = (qµ) and qµ = pS(µ), for all µ ∈ I. Acomparison of Definitions 4 and 15 reveals that φS p = q is a probabilistic limitstructure on X. On the other hand, if (X,q) ∈ |PLS| and we define ψSq = p,where pε = q

S−1(ε), then ψSq is a limit tower on X. Based on these simple

definitions, it is a trivial matter to verify that for every limit tower p on X,ψSφS p = p, and for every probabilistic limit structure q on X, φSψSq = q.

Let φS : LTS → PLS be defined for objects by φS(X, p) = (X,φS p) andfor morphisms by φ(f) = f . The preceding observations yield the followingtheorem.

THEOREM 17. Let S: I → [0,∞] be an order reversing homeomorphism. ThenφS : LTS → PLS is an isomorphism functor. Furthermore, the restrictionsof φS to the subcategories PSTS, PRTS, and TTS, respectively, establishisomorphisms between PSTS and PPSS, PRTS and PPRS, and TTS andPTS, respectively.

The importance of t-norms in the study of probabilistic convergence spacesbecomes apparent when we translate the axioms [K] and [F] from the setting

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of limit tower spaces under the isomorphism φS . Each such function S inducesa t-norm TS which becomes incorporated into the translated diagonal axiom.

For arbitrary T ∈ T, we formulate the following axioms for a probabilisticconvergence space (X,q).

KT: Let σ: X → F(X) be any function such that σ(y)qν→ y, for all y ∈ X,

where ν ∈ I. If µ ∈ I and Fqµ→ x, then κσF

qT (µ,ν)−→ x.

FT: Let J be any non-empty set, let ψ: J → X, and let σ: J → F(X) be suchthat σ(y)

qν→ ψ(y), for each y ∈ J , where ν ∈ I. If µ ∈ I, F ∈ F(J) and

ψFqµ→ x, then κσF

qT (µ,ν)−→ x.

PROPOSITION 18. Let S: I → [0,∞] be an order-reversing homeomorphism,φS : LTS → PLS the corresponding isomorphism from LTS to PLS, andT = TS the induced t-norm.

(a) (X, p) ∈ KLTS iff φS(X, p) satisfies KT.(b) (X, p) ∈ FLTS iff φS(X, p) satisfies FT.

Proof. Compare [K] with KT and [F] with FT, and recall that S(T (µ, ν)) =S(µ) + S(ν). Thus if S(µ) = ε and S(ν) = γ, then S(T (µ, ν)) = ε+ γ. Fromthe above observations, (a) and (b) follow immediately. 2

For arbitrary T ∈ T, let KTPLS and FTPLS be the full subcategories of PLSwhose objects satisfy KT and FT respectively.

COROLLARY 19. Let S: I → [0,∞] be an order-reversing homeomorphismwith induced t-norm TS = T . Then φS |KLTS: KLTS → KTPLS is an iso-morphism, and φS |FLTS: FLTS → FTPLS is an isomorphism. In particular,FTPLS is isomorphic to AP .

It is clear that the categories KTPLS and FTPLS depend on the t-norm T .Thus by varying the function S from which T is derived, we obtain differentsubcategories of PLS isomorphic to AP , and hence to one another.

The following proposition summarizes some results proved in [11].

PROPOSITION 20. Let T be an arbitrary t-norm.

(a) PPRS ⊆ FTPLS ⊆ PTS.(b) If T = T1 is the largest t-norm, FTPLS = PTS.(c) FTPLS = PPRS ∩KTPLS.

If the t-norm involved is strict, Proposition 18 and Proposition 20(c) supply theproof of Proposition 11(b).

The main results of this paper are summarized in the table below, in which allcategories at the same level are isomorphic to each other. We assume, of course,that T ∈ TS for KTPLS and FTPLS.

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Table I.

CAP LTS PLS

ALCAP KLTS KTPLS

PSAP PSTS PPSS

PRAP PRTS PPRS

AP FLTS FTPLS

TAP TTS PTS

By ALCAP , we designate the full subcategory of CAP whose objects satisfythe axiom (AL). TAP denotes the full subcategory of AP isomorphic to TTSunder the isomorphism η−1|TTS .

We further observe that

TTS ⊆ FLTS ⊆ PRTS ⊆ PSTS ⊆ LTS,

and

FLTS ⊆ KLTS ⊆ LTS.

Each of these categories is bireflectively embedded in the larger categories. LIMis bireflectively and bicoreflectively embedded in LTS, TOP is bireflectively andbicoreflectively embedded in TTS, and MET is embedded as a full subcategoryin FLTS.

Acknowledgement

The authors are grateful to the referee for suggested changes which significantlyimproved the paper.

References

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Approach Spaces, Limit Tower Spaces, and Probabilistic Convergence Spaces

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