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TRANSITIVITY IN UNIFORM APPROACH THEORY
Y. J. LEE and B. WINDELS Received 26 February 2002
We introduce a notion of transitivity for approach uniformities and approach uniform convergence spaces, yielding reflective subconstructs of AUnif and AUCS. Further, we investigate how these new categories are related touACHY,uACHYU, anduMET, and we show that these relationships are similar to those in the classical case.
2000 Mathematics Subject Classification: 18B30, 54E15.
1. Introduction. Since the first considerations on zero-dimensional spaces, by F.
Hausdorff, and the original study of non-Archimedean metric spaces, by A. F. Monna, the amount of literature on transitive structures has become extensive. Transitivity turned out to be interesting in a wide range of fields (functional analysis, Boolean algebra, valuation rings, domain theory, and many others) which proves the great importance of the concept. Therefore, an investigation of this topic in the setting of uniform approach structures is inevitable.
This paper presents a transitivity condition for two important quantified uniform structures: one for approach uniformities (introduced in Lowen and Windels [5] as a quantification of Unif) and a related concept for approach uniform convergence spaces (introduced in Windels [7] as a quantification of UCS). These definitions in turn yield different transitivity concepts in the setting of approach Cauchy spaces (introduced in Lowen and Lee [4] as a quantification ofCHY).
Although the categoriesUCSandCHYare well known to be Cartesian closed (see Lee [3] and Bentley et al. [1], respectively), the associated quantified structures yield cate- gories which do not share this property; the triangle inequality-like axiom turned out to be the essential problem. One possible solution, which is discussed in [7], is to drop this particular axiom. Alternatively, we can demand a stronger (non-Archimedean) tri- angle inequality to be fulfilled: in the case of Cauchy spaces, this approach leads to the Cartesian closed categoryuACHY(see [4]). In this paper, we will pursue the same method for uniform convergence spaces.
For any setX, we denote the set of all filters onXbyᏲ(X). The filter generated by a filter basisᏮis denoted by[Ꮾ]. In particular, the point filter generated by the set {x}is denoted by ˙x. If Ᏺ,Ᏻ∈Ᏺ(X), then Ᏺ×Ᏻ=[{F×G:F ∈Ᏺ, G∈Ᏻ}]. If f:X→Y is a map andᏲ∈Ᏺ(X), thenf (Ᏺ)=[{f (F ):F∈Ᏺ}]. IfΦ∈Ᏺ(X×X), then Φ−1= {U−1:U∈Φ}, whereU−1= {(y, x)∈X×X:(x, y)∈U}. IfΦ,Ψ∈Ᏺ(X×X), thenΦ◦Ψ=[{U◦V:U∈Φ, V∈Ψ}], provided that everyU◦V= {(x, y)∈X×X: there existsz∈Xsuch that(x, z)∈Uand(z, y)∈V}is not empty; whenever this notation is used, we will tacitly assume this condition to be fulfilled.
Recall from [8] that asemi-uniform convergence structureLon a setXis a collection of filters onX×Xsuch that
(UCS1) ˙x×x˙∈Lfor allx∈X, (UCS2) ifΦ∈LandΦ⊂Ψ, thenΨ∈L, (UCS3) ifΦ,Ψ∈L, thenΦ∩Ψ∈L, (UCS4) ifΦ∈L, thenΦ−1∈L.
The collectionL is called auniform convergence structureif it also satisfies the supplementary condition
(UCS5) ifΦ,Ψ∈L, thenΦ◦Ψ∈L.
The pair(X,L)is called auniform convergence space.
For any semi-uniform convergence spaces(X,L)and (Y ,K), a mapf :X→Y is calleduniformly continuousifΦ∈Limplies(f×f )(Φ)∈K. LetUCSdenote the cate- gory of uniform convergence spaces and uniformly continuous maps.
2. The category uAUCS. In this section, we introduce a notion of transitivity for approach uniform convergence structures. Recall from Windels [7] that anapproach uniform convergence structureon a setXis a mapη:Ᏺ(X×X)→[0,∞]satisfying the following conditions: for allx∈Xand allΦ,Ψ∈Ᏺ(X×X),
(AUCS1) η(˙x×x)˙ =0,
(AUCS2) Φ⊂Ψimpliesη(Φ)≥η(Ψ), (AUCS3) η(Φ∩Ψ)=η(Φ)∨η(Ψ), (AUCS4) η(Φ−1)=η(Φ),
(AUCS5) η(Φ◦Ψ)≤η(Φ)+η(Ψ).
Alternatively, such a structure can be described by auniform convergence tower (Lε)ε∈R+(or(Lε)ε), that is a collection of semi-uniform convergence structuresLεon Xsuch that
(UCT1) ifε, ε∈R+andΦ∈Lε,Ψ∈Lε, thenΦ◦Ψ∈Lε+ε, (UCT2) for anyε∈R+,Lε=
α>εLα.
The equivalence is shown by consideringLε= {Ᏺ∈Ᏺ(X×X):η(Ᏺ)≤ε}andη(Ᏺ)= min{ε∈R+:Ᏺ∈Lε}. The pair(X, η)(or, equivalently, the pair(X, (Lε)ε∈R+)) is called anapproach uniform convergence space(AUC-space for short).
Given AUC-spaces(X, η)and (Y , η)with uniform convergence towers(Lε)ε and (Kε)ε, respectively, a map f :X →Y is called auniform contraction if one of the following equivalent conditions is satisfied:
(1) η((f×f )(Φ))≤η(Φ)for allΦ∈Ᏺ(X×X),
(2) for eachε∈R+,f:(X,Lε)→(Y ,Kε)is uniformly continuous. LetAUCSdenote the category of AUC-spaces and uniform contractions. For details, the reader is referred to [7].
Definition2.1. LetXbe a set. An AUC-structureη:Ᏺ(X×X)→[0,∞]is called anultra approach uniform convergence structureif it satisfies instead of (AUCS5) the stronger condition: (uAUCS5) ifΦ,Ψ∈Ᏺ(X×X), thenη(Φ◦Ψ)≤η(Φ)∨η(Ψ).
The pair(X, η)is called anultra approach uniform convergence space(uAUC-space for short).
uAUC-spaces can be described by uniform convergence towers too.
Proposition2.2. Let(X, η)be an AUC-space, and let(Lε)ε∈R+ denote its uniform convergence tower. Then, the following are equivalent:
(1) (X, η)is an ultra approach uniform convergence space, (2) for everyε∈R+,Lεis a uniform convergence structure.
LetuAUCSdenote the full subcategory ofAUCSconsisting of all uAUC-spaces.
Theorem2.3. The categoryuAUCSis a bireflective subcategory of AUCS.
Proof. For a family((Xj, ηj))j∈J of uAUC-spaces and a source(Xf→j (Xj, ηj))j∈J inAUCS, the initial approach uniform convergence structureη:Ᏺ(X×X)→[0,∞]on Xdefined by
Φ→η(Φ)=sup
j∈J
ηj fj×fj
(Φ)
(2.1)
satisfies (uAUCS5). For this, letΦ,Ψ∈Ᏺ(X×X)be such that there existsΦ◦Ψ, then for eachj∈J,(fj×fj)(Φ)◦(fj×fj)(Ψ)exists and
ηj fj×fj
(Φ)◦ fj×fj
(Ψ)
≤ηj fj×fj
(Φ)
∨ηj fj×fj
(Ψ)
. (2.2) SouAUCSis initially closed inAUCSand sinceuAUCScontains all indiscrete objects, this proves the claim.
Theorem2.4. The categoryuAUCSis a topological construct.
Proof. This is an immediate consequence ofTheorem 2.3and [2, Theorem A.10].
Initial sources can be described by means of towers as well.
Proposition2.5. Let(X, η) and((Xj, ηj))j∈J be uAUC-spaces, and let(Lε)ε and (Ljε)εdenote the respective towers. Then, the following are equivalent:
(1) ((X, η)f→i (Xj, ηj))j∈Jis initial (inAUCS),
(2) ∀ε∈R+:((X,Lε)f→j(Xj,Ljε))j∈Jis initial (inUCS).
Proof. For everyε∈R+, letKεbe the initial uniform convergence structure for the source(Xf→j(Xj,Ljε))j∈J. Then, for anyΦ∈Ᏺ(X×X),η(Φ)=supj∈Jηj((fj×fj)(Φ)) and thus we have
Lε=
Φ∈Ᏺ(X×X):η(Φ)≤ε
=
Φ∈Ᏺ(X×X):ηj
fj×fj
(Φ)
≤ε∀j∈J
=
Φ∈Ᏺ(X×X): fj×fj
(Φ)∈Ljε∀j∈J
=Kε,
(2.3)
which proves the claim.
For any uAUC-spaces(X, η)and(Y , η), letC(X, Y )be the set of all uniform con- tractions fromXtoY. Then, for anyΦ∈Ᏺ(X×X)andΘ∈Ᏺ(C(X, Y )×C(X, Y )), the set{H(A):A∈Φ, H∈Θ}, whereH(A)= {(h(a), k(b)):(a, b)∈A, (h, k)∈H}for
eachA∈ΦandH∈Θ, forms a filter basis onY×Y. LetΘ(Φ)be the filter onY×Y generated by this basis and define a mapη∗:Ᏺ(C(X, Y )×C(X, Y ))→[0,∞]by
Θ→η∗(Θ)=inf
α:α∈L(Θ)
, (2.4)
where
L(Θ)= α:η
Θ(Φ)
≤η(Φ)∨α∀Φ∈Ᏺ(X×X)
. (2.5)
Proposition2.6. The mapη∗yields the coarsest uAUC-structure onC(X, Y )with respect to which the evaluation mapev :X×C(X, Y )→Y defined by(x, f )f (x)is a uniform contraction.
Proof. Clearly,η∗is well defined. (AUCS1) follows from the inequality η
(f˙×f )(˙ Φ)
=η
(f×f )(Φ)
≤η(Φ) (2.6)
for allf∈C(X, Y ),Φ∈Ᏺ(X×X)and (AUCS2) is trivial sinceΘ⊂ΘinᏲ(C(X, Y )× C(X, Y )) impliesΘ(Φ)≤Θ(Φ)in Ᏺ(Y×Y ) for all Φ ∈Ᏺ(X×X). For (AUCS3), let Θ,Θ∈Ᏺ(C(X, Y )×C(X, Y )). Then
η∗(Θ∩Θ)=inf α:η
(Θ∩Θ)(Φ)
≤η(Φ)∨α∀Ᏺ∈Ᏺ(X) , η(Θ∩Θ)(Φ)=η
Θ(Φ)∩Θ(Φ)
=η Θ(Φ)
∨η Θ(Φ)
. (2.7)
Soη∗(Θ∩Θ)≤η∗(Θ)∨η∗(Θ)and the converse follows from (AUCS2). Since for anyΘ∈Ᏺ(C(X, Y )×C(X, Y ))andΦ∈Ᏺ(X×X)it holds thatΘ−1(Φ)=(Θ(Φ−1))−1, (AUCS4) is immediate. Finally, letΘ,Θ∈Ᏺ(C(X, Y )×C(X, Y )) be such that there existΘ◦ΘandΦ∈Ᏺ(X×X). Then for anyH∈Θ,K∈Θ, andA∈Φ, it holds that (H◦K)(A)⊆H(A)◦K(A−1◦A)and hence
Θ(Φ)◦Θ Φ−1◦Φ
⊂(Θ◦Θ)(Φ). (2.8)
So (uAUCS5) is fulfilled, consequentlyη∗is an uAUC-structure onC(X, Y ). Since for anyΨ∈Ᏺ((X×C(X, Y ))×(X×C(X, Y ))),
η
(ev×ev)(Ψ)
≤η
(ev×ev)
π1×π1 (Ψ)×
π2×π2 (Ψ)
=η
π2×π2 (Ψ)
π1×π1 (Ψ)
≤η π1×π1
(Ψ)
∨η∗ π2×π2
(Ψ)
= η×η∗
(Ψ),
(2.9)
whereπ1andπ2are the canonical projection maps fromX×C(X, Y )toXandC(X, Y ), respectively, the map ev :X×C(X, Y )→Y is a uniform contraction with respect toη∗. Letη∗ be another uAUC-structure onC(X, Y )with respect to which ev is a uniform contraction. Then for allΦ∈Ᏺ(X×X)andΘ∈Ᏺ(C(X, Y )×C(X, Y )), we have
η
(ev×ev)(Φ×Θ)
=η Θ(Φ)
≤η(Φ)∨η∗(Θ), (2.10) consequently,η∗(Θ)∈L(Θ)for allΘ∈Ᏺ(C(X, Y )×C(X, Y )). Soη∗(Θ)≤η∗(Θ)for allΘ∈Ᏺ(C(X, Y )×C(X, Y ))and hence we have the result.
Proposition2.7. Let(X, η),(Y , η), and(Z, η)be uAUC-spaces and letf:X×Z→ Y be a uniform contraction. Then there exists a unique uniform contractionfˆ:Z→ C(X, Y )such thatev◦(1X×f )ˆ =f.
Proof. Define a map ˆf:Z→C(X, Y )by z→f (z)ˆ :X →Y ,
x→f (z)(x)ˆ =f (x, z). (2.11)
Then for eachz∈Z, ˆf (z)=f◦(1X×[z]), where [z]:X→Z is a map defined by xzfor allx∈X. Since the identity map, the constant map, and the composition of uniform contractions are uniform contractions, ˆfis a uniform contraction and hence the map ˆf is well defined. Furthermore, for anyΦ∈Ᏺ(X×X)andΨ∈Ᏺ(Z×Z), we have
η
(fˆ×f )(ˆ Ψ) (Φ)
=η
(f×f )(Φ×Ψ)
≤(η×η)(Φ×Ψ)
=η(Φ)∨η(Ψ).
(2.12)
Soη∗((fˆ×f )(ˆ Ψ))≤η(Ψ)for allΨ∈Ᏺ(Z×Z)and hence ˆfis a uniform contraction.
Clearly, ev◦(1X×f )ˆ =f and such an ˆfis unique.
Combining Propositions2.6and2.7, we have the following theorem.
Theorem2.8. The categoryuAUCSis Cartesian closed.
For any uniform convergence space(X,L), the mapηL:Ᏺ(X×X)→[0,∞]defined by
Φ→ηL(Φ)=
0 forΦ∈L,
∞ forΦ∈L (2.13)
is clearly an uAUC-structure onX. Furthermore, for any uniform convergence spaces (X,L)and (Y ,K), a mapf :(X,L)→(Y ,K) is uniformly continuous if and only if f:(X, ηL)→(Y , ηK)is a uniform contraction.
SoUCSis embedded as a full subcategory inuAUCSby the functor UCS →uAUCS,
(X,L)→ X, ηL
, f→f ,
(2.14)
and analogously to [7, Proposition 11], we have the following proposition.
Proposition2.9. An uAUC-space(X, η)is a uniform convergence space if and only ifη(Ᏺ(X×X))⊆ {0,∞}.
Theorem2.10. The categoryUCSis a bicoreflective subcategory of uAUCS.
Theorem2.11. The categoryUCSis a bireflective subcategory of uAUCS.
3. The category AUnifU. In this section and inSection 4we discuss two different notions of transitivity for approach uniformities. Recall from Lowen and Windels [5]
that an approach uniformity on a setX, is an idealᐁof functions fromX×X into [0,∞], satisfying the following conditions:
(AU1) for allu∈ᐁ, for allx∈X:u(x, x)=0,
(AU2) for allv∈[0,∞]X×X:(∀ε >0,∀N <∞:∃uNε ∈ᐁs.t.v∧N≤uNε +ε)⇒v∈ ᐁ,
(AU3) for allu∈ᐁ, for allN <∞, there existuN∈ᐁs.t.∀x, y, z∈X:u(x, z)∧N≤ uN(x, y)+uN(y, z),
(AU4) for allu∈ᐁ:us∈ᐁ.
Equivalently, an approach uniformity can be described with auniform tower, that is, a family of semi-uniformities(ᐁε)ε∈R+(or(ᐁε)ε) onX, such that
(UT1) for allε, ε∈R+:ᐁε◦ᐁε⊃ᐁε+ε, (UT2) for allε∈R+:ᐁε= α>εᐁα.
The equivalence is shown by consideringᐁε= {{u < α}:α > ε, u∈ᐁ}. The pair (X,ᐁ)(or, equivalently, the pair(X, (ᐁε)ε∈R+)) is called anapproach uniform space.
The functionf:(X,ᐁ)→(Y ,ᐁ)is called auniform contractionif and only ifu∈ ᐁ impliesu◦(f×f )∈ᐁ. The category of approach uniform spaces and uniform contractions is denoted byAUnif. For details, the reader is referred to [5].
Definition3.1. An approach uniform space(X, (ᐁε)ε∈R+)satisfying the supple- mentary condition that everyᐁεis a uniformity, is calledlevel-uniform.
This definition establishes a notion of transitivity in the sense of previous section.
To be precise, if(X, (ᐁε)ε∈R+)is an approach uniform space, then the AUC-structure ηdefined byη(Φ)=min{ε∈R+:Φ⊃ᐁε}is an uAUC-structure if and only if(ᐁε)εis level-uniform.
LetAUnifUdenote the full subcategory ofAUnifconsisting of all level-uniform ap- proach uniform spaces (for short, AUnifU-spaces). Level-uniform spaces can be char- acterized nicely by ideals of functions too.
Proposition3.2. Let(X,ᐁ)be an approach uniform space. Then the following are equivalent:
(1) (X,ᐁ)is level-uniform,
(2) ᐁhas a basisᏮsuch that for allu∈Ꮾ, for allN <∞, there existuN∈Ꮾ, for all x, y, z∈X:u(x, z)∧N≤uN(x, y)∨uN(y, z),
(3) for allu∈ᐁ, for allε >0, for allN <∞, there existuNε ∈ᐁ, for allx, y, z∈X: u(x, z)∧N≤uNε(x, y)∨uNε(y, z)+ε.
Proof. In order to prove (1)⇒(2), consider anyu=infni=1(αi−1+θUi), which form a basis forᐁ(see [5]). ChooseVi∈ᐁαisuch thatVi◦Vi⊂ᐁi(i=1, . . . , n)andV1⊂V2⊂
··· ⊂Vn. Letv=infni=1(αi−1+θVi). Now supposev(x, y)=αi−1andv(y, z)=αj−1. Then(x, y)∈Vi⊂Vi∨jand(y, z)∈Vj⊂Vi∨j, consequently,(x, z)∈Ui∨j. Therefore u(x, z)≤αi∨j−1=αi−1∨αj−1=v(x, y)∨v(y, z). The fact that (2)⇒(3) is immediate.
To prove that (3)⇒(1), letu∈ᐁbounded,ε∈R+, andα > ε. By(3), there is some
v∈ᐁsuch that for allx, y, z∈X:u(x, z)≤v(x, y)∨v(y, z)+(α−ε)/2. Then
v <α+ε 2
◦
v <α+ε 2
⊂ {u < α}. (3.1)
With every level-uniform approach uniform space(X, (ᐁε)ε), we can associate an uAUC-structureηonXdefined byη(Φ)=min{ε∈R+:Φ⊃ᐁε}. This procedure yields an embedding ofAUnifUintouAUCS.
Theorem3.3. The categoryAUnifUis a bireflective subcategory of uAUCS.
Proof. Let((X, η)f→j (Xj, ηj))j∈Jbe an initial source inuAUCS, and suppose that every(Xj, ηj)is level-uniform. If(Lε)εis the tower ofηand for allj∈J:(Ljε)εis the tower ofηj, then, byProposition 2.5, for allε∈R+:((X,Lε)f→j(Xj,Ljε))j∈J is initial, and sinceUnifis a reflective subcategory ofUCS, every(X,Lε)is level-uniform. Conse- quently,(X, η)is level-uniform. ThusAUnifUis initially closed inAUCS. Furthermore, sinceAUnifUcontains all indiscrete objects, we have the result.
Proposition3.4. The categoryAUnifUis a topological construct.
Proof. This is an immediate consequence ofTheorem 3.3and [2, Theorem A.10].
The categoryAUnifUis not Cartesian closed, since it containsUnifboth reflectively and coreflectively.
4. The category tAUnif. SinceAUnifcontains both the category of uniform spaces and the category of pseudo-metric spaces, it is natural to seek a subcategory ofAUnif that generalizes the notions of transitive uniform spaces and ultra-metric spaces.
Recall that a uniform space(X,ᐁ)is calledtransitiveifᐁhas a basis of entourages Uwith the property thatU◦U=U. A pseudo-metricdonXis called anultra-pseudo- metric(ornon-Archimedean pseudo-metric) ifdsatisfies the strong triangle inequality d(x, z)≤d(x, y)∨d(y, z)for everyx, y, z∈X.
Every approach uniformity induced by a transitive uniformity or by an ultra-metric is level-uniform, but not vice versa. In fact,everyuniformly generated approach uni- formity is level-uniform. This section establishes a stronger notion of transitivity for approach uniformities, in order to eliminate this disadvantage. Since every approach uniformity has a basis of pseudo-metrics, it seems natural to adopt the following definition.
Definition4.1. An approach uniform space(X,ᐁ)is calledtransitiveifᐁhas a basis consisting of ultra-pseudo-metrics.
Transitive approach uniformities can be described nicely in terms of uniform towers too.
Proposition4.2. Let(X,ᐁ)be an approach uniform space with a uniform tower (ᐁε)ε∈R+. Then the following are equivalent:
(1) (X,ᐁ)is transitive,
(2) for everyε∈R+,ᐁεis a transitive uniformity.
Proof. To see that (1)⇒(2), notice that
{d < α}:α > ε, d∈ᐁ, dultra-pseudo-metric
(4.1) is a transitive basis forᐁε. Conversely, if everyᐁεis a transitive uniformity, then (by [5, Lemma 2.7]) we know that
Ꮾ= n
infi=1
αi−1+θUi
|
α0, . . . , αn
δ-net on 0, αn
,
∀i∈ {1, . . . , n}:Ui∈ᐁαiandUi◦Ui=Ui
(4.2)
is a basis forᐁ. Now supposeu∈Ꮾ andu(x, y)=αi−1 andu(y, z)=αj−1. Then (x, y)∈ Ui and (y, z) ∈ Uj and consequently (x, z) ∈ Ui∨j. Therefore u(x, z)≤ αi∨j−1=αi−1∨αj−1=u(x, y)∨u(y, z). ThusᏮis a basis consisting of ultra-pseudo- metrics.
LettAUnifdenote the full subcategory ofAUnifconsisting of all transitive approach uniformities.
Theorem4.3. The category tAUnifis a reflective subcategory of AUnifU. Conse- quently,tAUnifis a topological construct.
Proof. SinceAUnifUis a reflective subcategory ofAUCS, initial structures in both categories are the same. Therefore the same argument as for Theorems 3.3 and Proposition 3.4can be used.
Transitive approach uniformities generalize the notions of transitive uniformity and ultra-pseudo-metric.
Proposition4.4. Let(X,ᐁ)be a principal approach uniform space, that is, for all ε∈R+,ᐁε=ᐁ0. Then the following are equivalent:
(1) (X,ᐁ)is transitive, (2) ᐁ0is transitive.
Proof. By virtue ofProposition 4.2, this is evident.
Proposition4.5. Let(X,ᐁ)be a metric approach uniform space, that is,ᐁ= {u: u≤d}for some pseudo-metricdonX. Then the following are equivalent:
(1) (X,ᐁ)is transitive,
(2) dis an ultra-pseudo-metric.
Proof. To see that (1)⇒(2), suppose that Ꮾis a basis for ᐁconsisting of ultra- pseudo-metrics. Thend=supu∈Ꮾu, and therefore dis an ultra-pseudo-metric too.
The converse is trivial, since{d}is a basis forᐁ.
The categories of ultra-pseudo-metric spaces and of transitive uniform spaces are nicely embedded intAUnif, analogously to the classical case.
Theorem4.6. The categorytUnifis a bireflective and bicoreflective subcategory of tAUnif. The categoryuMETis a bicoreflective subcategory of tAUnif.
Therefore, we have the following diagram:
tAUnif
r
upMET
c
r
AUnif tUnif
c r
r
pMET
c
Unif
c r
(4.3)
The categorytAUnifis not Cartesian closed, since it containstUnifboth reflectively and coreflectively, andtUnifis not Cartesian closed (in fact, any reflective subcategory ofUnifcontaining a nondiscrete object is not Cartesian closed).
5. Embedding uACHY in uAUCS. Recall from Lee and Lowen [4] that a function γ :Ᏺ(X)→[0,∞]is called an ultra approach Cauchy structure (for short, uACHY- structure) onXif it satisfies the following conditions:
(AF1) γ(˙x)=0 for allx∈X,
(AF2) ifᏲ,Ᏻ∈Ᏺ(X)andᏲ⊂Ᏻ, thenγ(Ᏺ)≥γ(Ᏻ),
(uACHY) ifᏲ,Ᏻ∈Ᏺ(X)and∃Ᏺ∨Ᏻ, thenγ(Ᏺ∩Ᏻ)≤γ(Ᏺ)∨γ(Ᏻ).
The pair(X, γ)is called anultra approach Cauchy space(for short, uACHY-space).
For any set X and Φ∈Ᏺ(X×X), let β(Φ)be the collection of all finite families (Ᏺj)nj=1⊆Ᏺ(X)such thatn
j=1(Ᏺj×Ᏺj)⊂Φ.
For any uACHY-space(X, γ), define a mapηγ:Ᏺ(X×X)→[0,∞]by Φ→ηγ(Φ)=inf
n sup
j=1
γ Ᏺj
: Ᏺjn
j=1∈β(Φ)
. (5.1)
Proposition5.1. For any uAUCHY-space(X, γ), the mapηγis an uAUC-structure onX.
Proof. (AUCS1)–(AUCS4) are routine. To show thatηγfulfills (uAUCS5), letΦ,Ψ∈ Ᏺ(X×X)be such that there existΦ◦Ψand take any(Ᏺi)ni=1∈β(Φ),(Ᏻj)mj=1∈β(Ψ).
SinceΦ◦Ψexists, then there exists at least one pair of indices(i0, j0)such thatᏲi0∨Ᏻj0
exists. Take all the pairs(ik, jk)such thatᏲik∨Ᏻjkexists.
Then
Φ◦Ψ>
n
i=1
Ᏺi×Ᏺi
◦
m
j=1
Ᏻj×Ᏻj
=
i=1,...,n j=1,...,m
Ᏺi×Ᏺi
◦
Ᏻj×Ᏻj
=
k
Ᏺik×Ᏺik
◦
Ᏻjk×Ᏻjk
=
k
Ᏺik×Ᏻjk
>
k
Ᏺik∩Ᏻjk
×
Ᏺik∩Ᏻjk
.
(5.2)
So(Ᏺik∩Ᏻjk)k∈β(Φ◦Ψ)and sinceγ(Ᏺik∩Ᏻjk)=γ(Ᏺik)∨γ(Ᏻjk)for each pair(ik, jk), we have supkγ(Ᏺik∩Ᏻjk)≤supni=1γ(Ᏺi)∨supmj=1γ(Ᏻj)and consequentlyηγ(Φ◦Ψ)≤ ηγ(Φ)∨ηγ(Ψ).
Proposition5.2. For any uACHY-spaces(X, γ)and(Y , γ), if a mapf:(X, γ)→ (Y , γ)is a contraction, thenf:(X, ηγ)→(Y , ηγ)is a uniform contraction.
Proof. For anyΦ∈Ᏺ(X×X)and(Ᏺj)nj=1∈β(Φ), we have(f (Ᏺj))nj=1∈β((f× f )(Φ))and supnj=1γ(f (Ᏺj))supnj=1γ(Ᏺj).
Therefore, we have a functor
uACHY →uAUCS, (X, γ)→
X, ηγ
, f→f .
(5.3)
For any uAUC-space(X, η), letγη:Ᏺ(X)→[0,∞]be the map defined by
Ᏺ→γη(Ᏺ)=η(Ᏺ×Ᏺ). (5.4)
Proposition5.3. For any uAUC-space(X, η), the pair(X, γη)is an ultra approach Cauchy space.
Proof. (AF1) and (AF2) are immediate. For (uACHY) note that for anyᏲ,Ᏻ∈Ᏺ(X), it holds that(Ᏺ∩Ᏻ)×(Ᏺ∩Ᏻ)=(Ᏺ×Ᏺ)∩(Ᏺ×Ᏻ)∩(Ᏻ×Ᏺ)∩(Ᏻ×Ᏻ)and ifᏲ∨Ᏻexists, then(Ᏺ×Ᏺ)◦(Ᏻ×Ᏻ)=Ᏺ×Ᏻ. So for anyᏲ,Ᏻ∈Ᏺ(X)such thatᏲ∨Ᏻexists, we have γη(Ᏺ∩Ᏻ)=γη(Ᏺ)∨γη(Ᏻ)by (AUCS3), (AUCS4), and (uAUCS5).
Proposition5.4. For any uAUC-spaces(X, η)and(Y , η), iff:(X, η)→(Y , η)is a uniform contraction, thenf:(X, γη)→(Y , γη)is a contraction.
So there exists a functor
uAUCS→uACHY, (X, η)→
X, γη
, f→f .
(5.5)
Proposition5.5. (1)For any uACHY-structureγon a setX,γ=γηγ. (2)For any uAUC-structureηonX,η≤ηγη.
Proof. (1) For anyᏲ∈Ᏺ(X), take any(Ᏺj)nj=1∈β(Ᏺ×Ᏺ). Without loss of gener- ality, we may assumeᏲi∨Ᏺjdoes not exist fori=j and hence we can take(Aj)nj=1 such thatAj∈Ᏺjfor eachj=1, . . . , nandAi∩Aj= ∅fori=j. Then there exists F ∈Ᏺsuch thatF×F ⊆ nj=1(Aj×Aj)and soF ⊆Ak for somek∈ {1, . . . , n}. Since Aj∩Ak= ∅forj=k, we getᏲk⊂Ᏺand consequentlyγ(Ᏺk)≥γ(Ᏺ). Thusγ≤γηγ
and the converse is obvious.
(2) For anyΦ∈Ᏺ(X×X)and(Ᏺj)nj=1∈β(Φ), we have supn
j=1
γη
Ᏺj
=supn
j=1
η Ᏺj×Ᏺj
=η n
j=1
Ᏺj×Ᏺj
≥η(Φ) (5.6)
and hence we have the result.
Theorem5.6. The categoryuACHYis a bicoreflective subcategory of uAUCS.
Proof. By [6, Theorem 2.2.10], for any uAUC-space(X, η) 1X:
X, ηγη
→(X, η) (5.7)
is theuACHY-bicoreflection.
6. The categories uACHYUand uACHYtU. Throughout this section(X,ᐁ)will be a level-uniform approach uniform space, and(ᐁε)εwill denote its uniform tower. Then the mapγᐁ:Ᏺ(X)→[0,∞]defined by
Ᏺ→γᐁ(Ᏺ)=inf
ε∈R+:Ᏺis aᐁε-Cauchy filter
(6.1) is an ultra approach Cauchy structure onX. Conditions (AF1) and (AF2) are obvious and (uACHY) is immediate from (UT4) and the fact that eachᐁεis a uniform structure onX. We say that γᐁ and ᐁ are compatible andγᐁ is called the uACHY-structure induced by ᐁ. Given a set X, an uACHY-structure γ on X is said to beapproach uniformizableif there exists a compatible AUnifU-structureᐁonX, that is,γ=γᐁfor some AUnifU-structureᐁonX.
LetuACHYUbe the full subcategory ofuACHYconsisting of all approach uniformiz- able uACHY-spaces (for short, uACHYU-spaces).
For any AUnifU-space (X,ᐁ), the pair (X, γᐁ) is a uACHYU-space and if a map f :(X,ᐁ)→(Y ,ᐁ) is a uniform contraction between AUnifU-spaces, then for any ε∈R+andᏲ∈Ᏺ(X)such thatᏲis aᐁε-Cauchy filter,f (Ᏺ)is aᐁε-Cauchy filter and soγᐁ(f (Ᏺ))≤γᐁ(Ᏺ). Thus the mapf :(X, γᐁ)→(Y , γᐁ)is a contraction between uACHYU-spaces. Thus it defines a functor
AUnifU →uACHYU, (X,ᐁ)→
X, γᐁ , f→f .
(6.2)
Proposition6.1. Ifᐁis the initial approach uniformity for a source
X fj→
Xj,ᐁj
j∈J (6.3)
inAUnifUandγis the initial uACHY-structure for the induced source X fj→
Xj, γᐁj
j∈J (6.4)
inuACHY, thenγ=γᐁ.
Proof. Denote for everyj∈Jthe uniform tower ofᐁjby(ᐁjε)ε. For anyj∈J, the mapfj:(X,ᐁ)→(Xj,ᐁj)is a uniform contraction and hence the induced map fj:(X, γᐁ)→(Xj, γᐁj)is a contraction. So the map 1X:(X, γᐁ)→(X, γ)is a contrac- tion by the initiality ofγand we haveγ≤γᐁ. For the converse, note thatγ:Ᏺ(X)→ [0,∞]is a map defined by
Ᏺ→γ(Ᏺ)=sup
j∈J
γᐁj
fj(Ᏺ). (6.5)
Letε∈R+andᏲ∈Ᏺ(X)be such thatγ(Ᏺ)≤ε. Then for eachj∈J,γᐁj(fj(Ᏺ))≤ε and fj(Ᏺ) is a ᐁjε-Cauchy filter on Xj. Thus Ᏺ is a ᐁε-Cauchy filter on X and so γᐁ(Ᏺ)≤ε. Thereforeγ≥γᐁ, which proves the claim.
For any uACHYU-space (X, γ), let ᐁ(γ)be the class of all AUnifU-uniform towers inducingγ.
Theorem6.2. The categoryuACHYUis a bireflective subcategory of uACHY.
Proof. For any family((Xj, γj))j∈Jof uACHYU-spaces, sayγj=γᐁj, and any source X fj→
Xj, γj
j∈J (6.6)
inuACHY, letγbe the initial uACHY-structure onXand letᐁbe the initial AUnifU- uniform tower onXfor the source
X fj→ Xj,ᐁj
j∈J. (6.7)
ThenᐁinducesγbyProposition 6.1and henceuACHYUis initially closed inuACHY.
Furthermore, sinceuACHYUcontains all indiscrete objects, we have the result.
For any uACHYU-space(X, γ), letᐁγbe the initial AUnifU-structure with respect to the source(X1X→(X,ᐁ))ᐁ∈ᐁ(γ). thenᐁγ is the finest AUnifU-structure onXinducing γbyProposition 6.1.
Proposition6.3. For any uACHYU-spaces(X, γ)and(Y , γ), iff:(X, γ)→(Y , γ) is a contraction, thenf:(X,ᐁγ)→(Y ,ᐁγ)is a uniform contraction between AUnifU- spaces.
Proof. For any uACHYU-space(X, γ), letC(X, γ)be the collection of all contrac- tions from(X, γ)to uACHYU-spaces and letᐁbe the initial AUnifU-structure onXfor the source
X g→ Z,ᐁγ
g∈C(X,γ) (6.8)
inAUnifU. Thenγᐁ is the initial uACHYU-structure onXfor the source Xg→(Z, γ)
g∈C(X,γ) (6.9)
inuACHYUand hence 1X:(X, γ)→(X, γᐁ)is a contraction. For the converse, note that if a mapf:(X, γ)→(Y , γ)is a contraction, thenf:(X, γᐁ)→(Y , γ)is a contraction.
So by the fact that the identity map is a contraction, we get 1X:(X, γᐁ)→(X, γ)is a contraction. Thereforeᐁ∈ᐁ(γ)and so 1X:(X,ᐁγ)→(X,ᐁ)is a uniform contraction.
Thus, iff:(X, γ)→(Y , γ)is a contraction, thenf :(X,ᐁ)→(Y ,ᐁγ)is a uniform contraction and hencef:(X,ᐁγ)→(Y ,ᐁγ)is a uniform contraction.
Hence there is a functor
uACHYU →AUnifU, (X, γ)→
X,ᐁγ
, f→f .
(6.10)
Theorem6.4. The categoryuACHYUis a bicoreflective subcategory of AUnifU. Proof. For any uACHYU-structure γ on a set X, we have γᐁγ =γ and for any AUnifU-structureᐁ on X, ᐁγᐁ is finer than ᐁ. So by [6, Theorem 2.2.10], for any AUnifU-space(X,ᐁ)
1X: X,ᐁγᐁ
→(X,ᐁ) (6.11)
is theuACHYU-bicoreflection.
For any setX, a uACHYU-structureγ onXis said to betransitively approach uni- formizableifγis compatible with some transitive approach uniformityᐁonX.
LetuACHYtUbe the full subcategory ofuACHYconsisting of all transitive approach uniformizable uACHY-spaces (for short, uACHYtU-spaces).
SincetAUnifis initially closed inAUnifU, we have the following theorem.
Theorem6.5. The categoryuACHYtUis a bireflective subcategory of uACHYU. For any uACHYtU-space(X, γ), letᐁtγ be the initial transitive approach uniformity with respect to the source(X1X→(X,ᐁ))ᐁ∈ᐁt(γ), whereᐁt(γ)is the class of all tran- sitive approach uniformities inducingγ. Thenᐁtγ is the finest transitive approach uniformity onXinducingγ. Clearly the restriction of the above two functors are well defined and we get the analogous result.
Theorem6.6. The categoryuACHYtUis a bicoreflective subcategory of tAUnif.
7. Categorical overview. Summarizing the results in foregoing sections, we obtain the following diagram:
uAUCS
UCS
r c
AUnifU r
uACHY
c
Unif
r c r
tAUnif
r
CHY
r c
c
uACHYU r
c
tUnif
r c r
CHYU r r c
c
uACHYtU r
c
CHYtU
r c r
c
(7.1)
The categoriestUnifandCHYtUare the full subcategories ofUnifandCHYUwhose objects are transitive uniform spaces and transitive uniformizable Cauchy spaces, respectively. These categories form a similar diagram asUnifandCHYU. At the end ofSection 2, we showed thatUCSis both reflectively and coreflectively embedded in uAUCS. The argument is representative for all upward arrows in the diagram.
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Y. J. Lee: Department of Mathematics, Yonsei University, Seoul120-749, Korea B. Windels: Department of Mathematics and Computer Science, RUCA, University of Antwerp, Groenenborgerlaan171,2020Antwerp, Belgium
