@ Applied General Topology c© Universidad Politécnica de Valencia Volume 4, No. 2, 2003 pp. 475–486 A note on separation in AP R. Lowen and M. Sioen Dedicated to Professor S. Naimpally on the occasion of his 70th birthday. Abstract. It is our aim in this note to take a closer look at some separation axioms in the construct AP of approach spaces and con- tractions. Whereas lower separation axioms seem to be qualitative, the higher ones seem to have a quantitative nature. Also some characteri- zations for the corresponding epireflectors will be given. 2000 AMS Classification: 54A05, 54D10, 54D15, 54B30. Keywords: approach space, separation property, reflection. 1. Introduction and preliminaries. As one readily knows, metric structures behave badly with respect to the formation of initial structures or in particular, of products, since e.g. for an infinite family of metrics, their pMET∞-product is not compatible with the topological product of the associated underlying topologies. As a remedy to this defect, the common supercategory AP (the objects of which are called approach spaces) of TOP and pqMET∞ was introduced in [7], where e.g. for a considered (infinite) family of metric spaces, their AP-product carries precisely that part of the numerical information present, which can be retained if one demands compatibility with the topological product of the family of underlying metric topologies. The basic difference in nature between approach and metric spaces consists in the fact that in an approach space, one specifies all the point-set distances (subsequent to some axioms), where such a point-set distance does not have to equal the infimum over the considered set of all the point-point distances, like in the metric case. We now recall the definition of an approach space. In the sequel, X stands for an arbitrary set and 2X stands for its powerset. Definition 1.1. A map δ : X × 2X −→ [0,∞] is called a distance on X if it satisfies the following conditions: (D1) ∀x ∈ X : δ(x,{x}) = 0, 476 R. Lowen and M. Sioen (D2) ∀x ∈ X : δ(x,∅) = ∞, (D3) ∀x ∈ X,∀A,B ∈ 2X : δ(x,A∪B) = δ(x,A) ∧δ(x,B), (D4) ∀x ∈ X,∀A ∈ 2X,∀ε ≥ 0 : δ(x,A) ≤ δ(x,A(ε)) + ε, with A(ε) := {z ∈ X | δ(z,A) ≤ ε}. The pair (X,δ) is called an approach space. The morphisms to go along with approach spaces are the so-called contrac- tions: if (X,δ), (X′,δ′) are approach spaces, then a map f : X −→ X′ is called a contraction if ∀x ∈ X,∀A ∈ 2X : δ′(f(x),f(A)) ≤ δ(x,A). It was shown in [7] that approach spaces and contractions constitute a topolog- ical construct, denoted by AP, into which TOP (resp. p(q)MET∞ (via the usual definition of point-set distances in the metric case)) can be concretely embedded as a full concretely bireflective and concretely bicoreflective (resp. concretely bicoreflective) subconstruct. Given an approach space, the topo- logical and ∞p(q)−metric coreflections have to be interpreted as the topology, resp. the ∞p(q)-metric “underlying” the given approach space, in the same sense as we think of the induced topology underlying a given metric. We will write Tδ for the topological coreflection of a distance δ. For any background material of categorical nature, we refer to [1] and for detailed information on approach theory, we refer to [7] and [8]. Let us only mention some alternative axiomizations for approach spaces which are described in [7], [8] and which will be used throughout the paper: approach systems (parallelling neighbourhood systems in topology), regular function frames (parallelling closed subsets in topology), hulls (parallelling topological closures) and approach limits (paral- lelling the description of a topology via convergence of filters). For an approach space (X,δ), the corresponding approach system, resp. regular function frame, hull and approach limit will be denoted by A := (A(x))x∈X, resp. R,h and λ and for explicit definitions and transition formulas between these equivalent characterizations, we again refer to [8]. If confusion might arise, we use a no- tation like Aδ to denote the approach system corresponding to the distance δ. We will make no distinction between a distance and its associated approach system, regular function frame, hull and approach limit. To finish, we agree upon some notations. If C is a full subconstruct of AP, then EAP(C) stands for its epireflective hull in AP, being the full subconstruct of AP consisting of all subspaces of products of C-objects. The metric approach space correspond- ing to the real line equipped with the Euclidean metric will be denoted by R. Also the following particular approach spaces will play a role in the sequel: the indiscrete 2-space I2 := ({0, 1},δi), where δi(x,∅) := ∞ and δi(x,A) := 0 if A 6= ∅, and P := ([0,∞],δP) with δP(x,∅) := ∞ and δP(x,A) := (x−sup A)∨0 for all x and A 6= ∅. Separation axioms, measuring to which extent different points, or points not belonging to a closed set, resp. disjoint closed sets can be recognized as A note on separation in AP 477 such by a given space play an important role in topology. In particular, many well-known extension theories only work (nicely) in the presence of certain sep- aration axioms, e.g. the uniform completion for T2−uniform spaces or metric spaces, the Wallman compactification for T1−topological spaces or the C̆ech- Stone compactification for Tychonoff spaces. In this note we want to focus on some forms of separation axioms in the realm of approach theory and in case they determine categorically nice (i.e. epireflective) subconstructs of AP, to give a description for the corresponding epireflection arrows. We will use TOP0, resp. TOP1 and TOP2 for the full subconstructs of TOP consisting of all T0, resp. T1 and T2-spaces. 2. Separation Axioms. 2.1. The T0-axiom. In [10] the categorically correct definition of T0-objects in the setting of topological constructs in the sense of H. Herrlich (see e.g. [1]) was given and it was shown there that the resulting subconstruct of all T0-objects is the largest epireflective, not bireflective subconstruct of the considered topo- logical construct. We will now identify the T0-objects in AP. Definition 2.1. We call an approach space (X,δ) a T0-space if every contrac- tion f : I2 −→ (X,δ) is constant. We write AP0 for the full subconstruct of AP consisting of all T0−objects. Proposition 2.2. For every (X,δ) ∈ |AP|, the following assertions are equiv- alent: (1) (X,Tδ) ∈ |TOP0|, (2) (X,δ) ∈EAP(P), (3) ∀x,y ∈ X : x 6= y ⇒ ((∃ϕ ∈A(x) : ϕ(y) > 0) ∨ (∃ϕ ∈A(y) : ϕ(x) > 0)) (4) ∀x,y ∈ X : x 6= y ⇒ (∃γ ∈R : γ(x) 6= γ(y)), (5) ∀f ∈ AP(I2, (X,δ)) : f constant, (6) ∀x,y ∈ X : x 6= y ⇒A(x) 6= A(y). Proof. The implication (1)⇒(3) is obvious because it is proved in [8] that for ev- ery x ∈ X, {{ψ < ε}|ψ ∈A(x),ε > 0} is a base for the Tδ-neighbourhoodsystem at x. To verify the implication (3) ⇒ (2), note that it was proved in [8] that the source (δ(·,A) : (X,δ) −→ P : x 7→ δ(x,A))A∈2X is initial in AP. Therefore it suffices to show that it is point-separating in order to conclude that (X,δ) is a subspace of a power of P, so pick x,y ∈ X with x 6= y. Assume without loss of generality that ϕ(y) > 0 for some ϕ ∈ A(x). Then automatically δ(x,{y}) ≥ ϕ(y) > 0 and δ(y,{y}) = 0. The implication (2)⇒(1) is obvious since the concrete bicoreflector from AP onto TOP preserves products and subspaces, because ([0,∞],TδP) ∈ |TOP0| and because the latter is an epireflective subconstruct of TOP. The implication (1) ⇒(4) is proved using the implication (1) ⇒(3) because for all x ∈ X, δ(·,{x}) ∈ 478 R. Lowen and M. Sioen R. To prove the converse one, take x,y ∈ X, x 6= y and assume without loss of generality that γ(x) > α > γ(y) for some γ ∈ R,α ∈ R+. Because γ : (X,Tδ) −→ P is lower semicontinuous, {γ > α} is a Tδ-neighbourhoood of x not containing y. Furthermore, it is clear that (1) implies (5), whereas the converse implication follows by contraposition, because if x,y ∈ X were distinct points such that all neighbourhoods of x contain y and vice versa, f : I2 −→ (X,δ) defined by f(0) := x,f(1) := y would be a non-constant contraction. The implication (3) ⇒(6) is obvious and we finish with the implication (6) ⇒ (4). Take x,y ∈ X with x 6= y. According to (6), we can assume without loss of generality that there exists ϕ ∈ A(x) \ A(y). Therefore, it follows from the transition formula (distance −→ approach system) that there exists A ∈ 2X with infz∈A ϕ(z) > δ(y,A), whence automatically δ(x,A) > δ(y,A) and because δ(·,A) ∈R, we are done. � This shows that the T0-property in AP is in fact completely topological and it is proved in the next proposition that, again like in the topological case, the corresponding epireflection arrows are obtained as quotients. Proposition 2.3. AP0 is an epireflective subconstruct of AP. For any (X,δ) ∈ |AP|, we define an equivalence relation ∼ on X by x ∼ y ⇔ (∀ϕ ∈R : ϕ(x) = ϕ(y)) ⇔A(x) = A(y). Then the AP-quotient of (X,δ) with respect to ∼ gives us is an AP0-epireflection arrow for (X,δ). Proof. For every x ∈ X, we write x for the corresponding equivalence class w.r.t. ∼ and we denote the corresponding projection by π : X −→ X/ ∼: x 7→ x. By definition of ∼, it is obvious that for each γ ∈R, the map γ : X/ ∼−→ [0,∞] : x 7→ γ(x) is well-defined. From the descripiton of quotients in AP, it is now clear that the final regular function frame on X/ ∼ with respect to π : (X,R) −→ X/ ∼ is exactly R/ ∼:= {ϕ ∈ [0,∞]X/∼|ϕ◦π ∈R} = {γ|γ ∈R}. Then clearly (X/ ∼,R/ ∼) ∈ |AP0| and for every (X′,R′) ∈ |AP0| and f ∈ AP((X,R), (X′,R′)), it is clear that f : (X/ ∼,R/ ∼) −→ (X′,R′) : x 7→ f(x) is a well-defined contraction, being the unique one such that f = f ◦π. � The corresponding epireflector is denoted by T0 : AP −→ AP0. Remark 2.4. AP is universal, i.e. it is the bireflective hull of AP0 in AP. Moreover, every epireflector from AP onto one of its subconstructs is either a bireflector or the composition of a bireflector, followed by the AP0−epireflector. Proof. The universality follows immediately from the result, proved in [8], that for each (X,δ) ∈ |AP|, the source (δ(·,A) : (X,δ) −→ P)A∈2X A note on separation in AP 479 is initial in AP, because P ∈ |AP0|. The second part is proved in [10]. � 2.2. The T1-axiom. Remark 2.5. For every (X,δ) ∈ |AP|, the following assertions are equivalent: (1) (X,Tδ) ∈ |TOP1|, (2) ∀x,y ∈ X : x 6= y ⇒ (∃ϕ,ψ ∈R : (ϕ(x) < ϕ(y)) ∧ (ψ(y) < ψ(x)), (3) ∀x,y ∈ X : x 6= y ⇒ ((∃ϕ ∈A(x) : ϕ(y) > 0) ∧ (∃ψ ∈A(y) : ψ(x) > 0)). (4) ∀x,y ∈ X : x 6= y ⇒ ((A(x) 6⊂A(y)) ∧ (A(y) 6⊂A(x))). Proof. This is proved in the same way as 2.2 . � Comparing this remark with the way T1-objects are defined in TOP and the characterizations of T0-objects in AP, yields that the following definition is plausible: Definition 2.6. We call an approach space T1 if it satisfies the equivalent statements from 2.5. We define AP1 to be the full subconstruct of AP defined by all T1 approach spaces. Corollary 2.7. AP1 is an epireflective subconstruct of AP. Next we want to give an internal description of the corresponding epireflector from AP onto AP0. It is well-known that a topological space (X,T ) is T1 if and only if it is both T0 and symmetric in the sense of [3], meaning that ∀x,y ∈ X : x ∈ cl({y}) ⇔ y ∈ cl({x}). This, together with remark 2.4 above motivates the following line of working: Definition 2.8. An approach space (X,δ) is called R0 if it satisfies the con- dition ∀x ∈ X : A(x) = ⋂ y:δ(y,{x})=0 A(y). The full subconstruct of AP formed by all R0-objects is denoted by APR0 . Constructing the T1−epireflector will carry us outside of AP, into the su- perconstruct PRAP of pre-approach spaces and contractions, as introduced in [9]. Let us only recall that a pre-approach space is a pair (X,δ) with δ : X × 2X −→ [0,∞] satisfying (D1), (D2) and (D3) (such δ is called a pre-distance) and contractions are defined in the same way as above. Just as in the approach case, a pre-approach distance δ can be equivalently character- ized by resp. a pre-approach system A, a pre-hull h and a pre-approach limit λ. For details we refer to [9], but we note that, just as for distances, stepping from AP to PRAP comes down to dropping the triangular axiom for A and λ and the idempotency for h. It was proved in [9] that AP is a concretely 480 R. Lowen and M. Sioen bireflective subconstruct of PRAP. Take (X,h) ∈ |PRAP|. If γ ∈ [0,∞]X, define h0(γ) := γ and for each ordinal α ≥ 1, hα(γ) := { h(hα−1(γ)) α not a limit ordinal∧ β<α h β(γ) α limit ordinal. Then there exists some ordinal κ such that hκ(γ) = hκ+1(γ) for all γ ∈ [0,∞]X and we put h∗(γ) := hκ(γ) for each γ ∈ [0,∞]X. Then it can be proved that h∗ : [0,∞]X −→ [0,∞]X is a hull on X and idX : (X,h) −→ (X,h∗) is the reflection arrow. We will write D : PRAP −→ AP for the corncrete reflector and to simplify notations, we will write (X, D(δ)) instead of D((X,δ)) for (X,δ) ∈ |PRAP|, and analoguously for the associated pre-approach systems, pre-hulls and pre-approach limits. First note that obviously, |AP1| = |APR0|∩ |AP0|. We will first show that APR0 is a concretely bireflective subconstruct of AP, yielding at once a description of the concrete bireflector R0 : AP −→ APR0 . If (X,A) ∈ |AP|, we define a relation ∼R0 on X as follows: x ∼R0 y ⇔ δ(x,{y}) = 0. If we put A∗(x) := ⋂ y∼R0x A(y) for all x ∈ X, and A∗ := (A∗(x))x∈X, then (X,A∗) ∈ |PRAP|. Fix (X,A) ∈ |AP|. Put A0 := A and for every ordinal α ≥ 1, define Bα(x) := { (Aα−1)∗(x) α not a limit ordinal,⋂ β<αA β(x) α a limit ordinal , x ∈ X (note that (X,Bα := (Bα(x))x∈X) ∈ |PRAP|) and define Aα := D(Bα), whence (X,Aα) ∈ |AP|. Proposition 2.9. For every (X,A) ∈ |AP|, there exists an ordinal κ for which Aκ = Aκ+1. If we denote AR0 := Aκ, (X,AR0 ) ∈ |APR0| and idX : (X,A) −→ (X,AR0 ) is the APR0−bireflection arrow. Proof. Pick (X,A) ∈ |AP|. Since for all ordinals β < α, Aβ ⊃ Aα and hence 0 ≤ δAα ≤ δAβ , it is clear that Aκ+1 = Aκ, if we take a fixed ordinal κ > card([0,∞]X×2 X ). Then define AR0 := Aκ. By construction, it is obvious that (X,AR0 ) ∈ |APR0|. Now fix (X′,A′) ∈ |APR0| and f ∈ AP((X,A), (X′,A′)). Then by definition, f ∈ AP((X,A0), (X′,A′)). Now assume that f ∈ AP((X,Aα−1), (X′,A′)) for some non-limit ordinal α. In order to verify that f ∈ PRAP((X,Bα), (X′,A′)), pick x ∈ X and ϕ′ ∈ A′(f(x)). We should prove that ϕ′ ◦ f ∈ Bα(x), so let y ∈ X such that δAα−1 (y,{x}) = 0. Then automatically δ′(f(y),{f(x)}) = 0, so because (X′,A′) ∈ |APR0|, ϕ′ ∈ ⋂ z∈X′:δ′(z,{f(x)})=0 A′(z) ⊂A′(f(y)), A note on separation in AP 481 whence ϕ′◦f ∈Aα−1(y). Since D is a concrete bireflector, it now immediately follows that f ∈ AP((X,Aα), (X′,A′)). Next take a limit ordinal α and assume that f ∈ AP((X,Aβ), (X′,A′)) for all β < α. It then trivially follows that f ∈ AP((X,Aα), (X′,A′)). By transfinite induction, it now follows in particular that f ∈ AP((X,AR0 ), (X′,A′)) so we are done. � For (X,δ) ∈ |AP|, we will also use the notation (X, R0(δ)) for R0((X,δ)) and the same convention applies for the other equivalent axiomizations for approach spaces. Proposition 2.10. For every (X,R) ∈ |AP|, the AP1-epireflection is ob- tained by taking the AP0-epireflection of its APR0−bireflection, i.e. the cor- responding AP1-epireflection arrow is given by π : (X,R) −→ (X/ ∼, R0(R)/ ∼), (where ∼ and π are determined by R0(R).) Proof. It suffices to verify that (X/ ∼, R0(R)/ ∼) ∈ |AP1|. First note that, with the notations as in 2.3, for all x,y ∈ X δR0(R)/∼(x,{y}) = sup γ∈R0(R),γ(y)=0 γ(x) = sup γ∈R0(R),γ(y)=0 γ(x) = δR0(R)(x,{y}). Because (X, R0(R)) ∈ |APR0|, this implies that ∀x,y ∈ X : δR0(R)/∼(x,{y}) = 0 ⇒ δR0(R)/∼(y,{x}) = 0, i.e. that (X/ ∼,TR0(R)/∼) is a symmetric space in the sense of [3] (called an R0- space there) and since it also belongs to |TOP0|, it belongs to |TOP1| and we are done. � 2.3. The T2-axiom. If X is a set and F ⊂ [0,∞]X, we call 〈F〉 := {γ ∈ [0,∞]X | ∀ε > 0,∀M < ∞ : ∃γεM ∈ F : γ ∧M ≤ γεM + ε}, resp. c(F) := supγ∈F infx∈X γ(x), the saturation, resp. the level of F. If (X,A) ∈ |AP| and x,y ∈ X, then A(x) ∨ A(y) := 〈A(x) ∪ A(y)〉 is the supremum of A(x) and A(y) in the lattice of all saturated ideals in [0,∞]X. Remark 2.11. If (X,δ) ∈ |AP|, the following assertions are equivalent: (1) ∀x,y ∈ X : x 6= y ⇒ c(A(x) ∨A(y)) > 0, (2) ∀x,y ∈ X : x 6= y ⇒ (∃ϕ ∈A(x),∃ψ ∈A(y) : infs∈X(ϕ∨ψ)(s) > 0, (3) (X,Tδ) ∈ |TOP2|. Proof. Obvious since ({{ϕ < ε}|ε > 0,ϕ ∈ A(x)})x∈X is a base for the Tδ- neigbourhood system. � Again, a comparison with the classical topological situation and taking into account that the role of neighbourhood filters in topology is played by the so-called approach systems in approach theory, makes it plausible to define Hausdorff objects in AP in the following, again topological, way: 482 R. Lowen and M. Sioen Definition 2.12. We call an approach space T2 if it satisfies the equivalent statements from 2.11. We define AP2 to be the full subconstruct of AP defined by all T2 approach spaces. Corollary 2.13. AP2 is an epireflective subconstruct of AP. As an answer to a question raised by H. Herrlich, an internal description of the epireflector from TOP onto TOP2 was given by V. Kannan in [5], making use of a transfinite construction. We will now derive an explicit description for the epireflector from AP onto AP2, along the same lines as was done for the T1-case in the section above. First we define a property R for approach spaces, which is inspired by the notion of reciprocity for convergence spaces, as defined in [2]. If (X,A) ∈ |AP|, we define a relation ∼R on X by x ∼R y ⇔ (∃x1 := x,. . . ,xn := y : ∀i ∈{1, . . . ,n− 1} : c(A(xi) ∨A(xi+1)) = 0). Definition 2.14. We call (X,δ) ∈ |AP| an R- space if it fulfills the following condition ∀x ∈ X : A(x) = ⋂ y∼Rx A(y) and we denote the full subconstruct of AP formed by all R-spaces by APR. First note that |AP2| = |AP0| ∩ |APR|. To begin with, we will prove that APR is a bireflective subconstruct of AP by describing the bireflector R : AP −→ APR. Let (X,A) ∈ |AP|. If we put A†(x) := ⋂ y∼Rx A(y) for all x ∈ X, and A† := (A†(x))x∈X, then (X,A†) ∈ |PRAP|. Fix (X,A) ∈ |AP|. Put A0 := A and for every ordinal α ≥ 1, define Bα(x) := { (Aα−1)†(x) α not a limit ordinal,⋂ β<αA β(x) α a limit ordinal , x ∈ X (note that (X,Bα := (Bα(x))x∈X) ∈ |PRAP| and define Aα := D(Bα), whence (X,Aα) ∈ |AP|. Proposition 2.15. For all (X,A) ∈ |AP|, there exists an ordinal κ for which Aκ+1 = Aκ. If we denote AR := Aκ, (X,AR) ∈ |APR| and idX : (X,A) −→ (X,AR) is the APR−bireflection arrow. Proof. The proof is exactly the same as that of 2.9 except for verifying that for (X′,A′) ∈ |APR| and f ∈ AP((X,A), (X′,A′)), if we assume that f ∈ AP((X,Aα−1), (X′,A′)) for some non-limit ordinal α (*), it follows that f ∈ PRAP((X,Bα), (X′,A′)). Therefore, pick x ∈ X and ϕ′ ∈A′(f(x)). Assume that y ∈ X and x1 := y, . . . ,xn := x such that c(Aα−1(xi) ∨Aα−1(xi+1)) = 0 A note on separation in AP 483 for all i ∈ {1, . . . ,n− 1}. Then it follows from our assumption (*) that for all i ∈{1, . . . ,n− 1} c(A′(f(xi)) ∨A′(f(xi+1))) = sup γ∈A′(f(xi)) sup µ∈A′(f(xi+1)) inf z′∈X′ (γ ∨µ)(z′) ≤ sup γ∈A′(f(xi)) sup µ∈A′(f(xi+1)) inf z∈X (γ∨µ)(f(z)) ≤ c(Aα−1(xi)∨Aα−1(xi+1)) = 0, showing that f(y) ∼R f(x). Because (X′,A′) ∈ |APR|, this implies that ϕ′ ∈A′(f(y)) whence ϕ′ ◦f ∈Aα−1(y) and this completes the proof. � For (X,δ) ∈ |AP|, we will also use the notation (X, R(δ)) for R((X,δ)) and the same convention applies for the other equivalent axiomizations for approach spaces. Proposition 2.16. For every (X,R) ∈ |AP|, the AP2-epireflection is ob- tained by taking the AP0-epireflection of its APR−bireflection, i.e. the corre- sponding AP2-epireflection arrow is given by π : (X,R) −→ (X/ ∼, R(R)/ ∼), (where ∼ and π are detremined by R(R).) Proof. We only need to check that (X/ ∼, R(R)/ ∼) ∈ |AP2|. Now assume that x,y ∈ X for which every TR(R)/∼-neighbourhood of x meets every TR(R)/∼- neighbourhood of y. Note that TR(R) = {{µ > 0} | µ ∈ R(R)} and that with the notation introduced in 2.3 R(R)/ ∼= {γ|γ ∈ R(R)}. This yields that x ∼R y where the ∼R-relation is taken with respect to R(R), and because (X, R(R)) ∈ |APR|, it follows that AR(R)(x) = AR(R)(y), whence x = y. � 2.4. Regularity. In [11], three different suggestions for regularity were pro- posed, and in [5], it was motivated that the strongest one of them is the correct notion of regularity in the construct AP. We simply recall this definition for the sake of completeness. For any set X, let F(X) stand for the set of all filters on X and for each F ∈ F(X) and ε ≥ 0, let F(ε) denote the filter generated by {F (ε) | F ∈F}. Definition 2.17. [11], [5] An approach space (X,δ) is called regular if ∀ε ≥ 0,∀F ∈ F(X) : λ(F(ε)) ≤ λ(F) + ε. We write RAP for the full subconstruct of AP formed by all regular spaces, and it was proved in [11] that it moreover is concretely bireflective. For some other equivalent characterizations of regularity in terms of distances and ap- proach systems, we refer to [11]. Note that, where the lower separation axioms we discussed in AP all turn out to be topological in the sense that an approach space (X,δ) is Ti if and only if its topological coreflection in Ti in the classical sense (with i ∈{0, 1, 2}), 484 R. Lowen and M. Sioen the regularity condition stated above is of a purely quantitative nature. This was already noted in [11], where it is shown that δ(x,A) :=   ∞ A = ∅ 0 x ∈ A 1 x 6∈A,A infinite 2 other cases x ∈ R,A ⊂ R defines an approach distance on R such that (R,δ) 6 ∈|RAP| but (R,Tδ) is a regular topological space. It was also already stated in [11] that for topological spaces, this notion of regularity is equivalent to the classical one. 2.5. Complete Regularity. We recall the definition of uniform approach spaces from [8]. Definition 2.18. An approach space (X,δ) is called uniform if and only if there exists a collection D of ∞p-metrics on X which is closed with respect to taking finite suprema and such that ∀x ∈ X,∀A ∈ 2X : δ(x,A) = sup d∈D δd(x,A). The full subconstruct of AP formed by all uniform approach spaces is de- noted by UAP and it can be shown (see e.g. [8]) that UAP = EAP(pMET∞). It was also proved in [8] that for every (X,δ) ∈ |AP|, the corresponding UAP−epireflection arrow, which in fact is a concrete bireflection arrow, is given by idX : (X,δ) −→ (X,δu := sup d∈Gs(δ) δd), with Gs(δ) := {d|d ∞p− metric on X,δd ≤ δ}. The following proposition shows that ‘being uniform’ is precisely the correct quantified generalization of complete regularity to the approach setting and it was proved in [8] that for topological objects, these two notions are equivalent. Proposition 2.19. For every (X,δ) ∈ |AP|, the following assertions are equivalent: (1) (X,δ) ∈ |UAP|, (2) ∀x ∈ X,∀A ∈ 2X,∀ε > 0,∀ω < ∞ : ∃fωε ∈ AP((X,δ),R) : fωε (x) = 0 and ∀z ∈ A : f ω ε (z) + ε ≥ δ(x,A) ∧ω. (3) ∀x ∈ X,∀A ∈ 2X,∀ε > 0,∀ω < ∞ : ∃fωε ∈ AP((X,δ),R) bounded : fωε (x) = 0 and ∀z ∈ A : f ω ε (z) + ε ≥ δ(x,A) ∧ω. Proof. We first show that (1) implies (2). Therefore let D be a collection of ∞p-metrics on X which is closed w.r.t. the formation of finite suprema and such that δ = supd∈D δd and fix x ∈ X,A ∈ 2X,ε > 0 and ω < ∞. Then we can find dωε ∈D with δdωε (x,A) + ε > δ(x,A) ∧ω A note on separation in AP 485 and because fωε := d ω ε (x, ·) ∈ AP((X,δ),R), we are done. That (2) implies (3) is clear since for every f ∈ AP((X,δ),R) and ω < ∞ , obviously |f| ∧ ω ∈ AP((X,δ),R). Finally we show that (3) implies (1). In order to verify that (X,δ) ∈ |UAP|, it sufices to show that δ ≤ δu, so fix x ∈ X,A ∈ 2X,ε > 0 and ω < ∞. According to (3), we can find fωε ∈ AP((X,δ),R) bounded such that fωε (x) = 0 and such that f ω ε (a) + ε ≥ δ(x,A) ∧ω for each a ∈ A . If we denote the Euclidean metric on R by dE, it is clear that dE ◦ (fωε ×fωε ) is an ∞p-metric on X, which belongs to Gs(δ) because fωε is a contraction. It then is clear that δu(x,A) + ε ≥ inf a∈A |fωε (x) −f ω ε (a)| + ε ≥ δ(x,A) ∧ω, completing the proof. � To see that, like regularity, this is a numerical, non-topological separation axiom, note that d(x,y) := { |x−y|/2 x ≤ y |x−y| x > y x,y ∈ R defines a pseudo-quasi-metric on R, such that (R,Td) is a completely regular topological space but (X,δd) 6∈|UAP|. References [1] Adámek, J. Herrlich, H. and Strecker, G. Abstract and Concrete Categories, John Wiley, New York (1990) [2] Bentley, H. L., Herrlich H. and Lowen-Colebunders E. Convergence , J. Pure Appl. Alg. 68 (1990), pp. 27-45 [3] Herrlich, H. Topological Structures, Math. Centre Tracts 52 (1974), pp. 59-122 [4] Herrlich, H. Categorical Topology 1971-1981, Proc. 5th Prague Top. Symp. 1981, Hel- dermann Verlag Berlin (1983), pp. 279-383 [5] Brock, P and Kent, D. On Convergence Approach Spaces, Appl. Cat. Struct. 6 (1998), pp. 117-125 [6] Kannan, V. On a Problem of Herrlich, J. Madurai. Univ. 6 (1976), pp. 101-104 [7] Lowen, R. Approach Spaces: a Common Supercategory of TOP and MET, Math. Nachr. 141 (1989), pp. 183-226 [8] Lowen, R. Approach Spaces: the Missing Link in the Topology-Uniformity-Metric Triad, Oxford Mathematical Monographs, Oxford University Press (1997) [9] Lowen, E and Lowen, R. A Quasitopos Containing CONV and MET as full subcate- gories , Int. J. Math. and Math. Sci., 11(3) (1988), pp. 417-438 [10] Marny, T. On Epireflective Subcategories of Topological Categories, Gen. Top. Appl. 10 (1979), pp. 175-181 [11] Robeys, K., Extensions of Products of Metric Spaces, PhD Thesis Universiteit Antwer- pen, RUCA (1992) Received November 2001 Revised August 2002 486 R. Lowen and M. Sioen R. Lowen University of Antwerp, RUCA, Department of Mathematics and Computer Sci- ence, Middelheimlaan 1, B2020 Antwerp, Belgium E-mail address : rlow@ruca.ua.ac.be M. Sioen Free University of Brussels, Department of Mathematics, Pleinlaan 2, B1000 Brussels, Belgium E-mail address : msioen@vub.ac.be