SOBER APPROACH SPACES 1. Introduction in This Article We Will

SOBER APPROACH SPACES

B. BANASCHEWSKI, R. LOWEN, C. VAN OLMEN

1. Introduction In this article we will introduce the notion of sobriety for approach spaces, modelled after the corresponding familiar concept in the case of topological spaces. Recall that the latter is closely linked to the relationship between the categories Top of topological spaces and Frm of frames, these being certain complete lattices which may be viewed as abstractly defined lattices of virtual open sets. More specifically, the connection between these categories is provided by a pair of contravariant functors, adjoint on the right, which assign to each topological space its frame of open subsets and to each frame its spectrum. Sober spaces are then those spaces for which the corresponding adjunction map is a homeomorphism. In the approach setting, the so-called regular function frame of an approach space takes over the role of the frame of open sets in topology and accordingly we introduce an abstractly defined version of this, called an approach frame. This leads to the contravariant functor R from the category AP of approach spaces to AFrm of approach frames and the spectrum functor Σ in the opposite direction. R and Σ are then adjoint on the right and an approach space X is called sober if the corresponding adjunction map X → ΣRX is an isomorphism. The purpose of this paper is to give an account of the basic facts concerning the functors R and Σ and to present a number of results about sobriety, including its reflectiveness and the relation between sobriety and completeness for uniform approach spaces.

2. Basic Definitions Approach spaces can be characterized by means of various defining struc- tures. One of these is the so-called regular function frame. This is actually the set of contractions from X to P := ([0, ∞], δP) where δP is defined by δP(x, A) := (x − sup A) ∨ 0 for all x ∈ X and A ⊂ X. Such a regular function frame can of course be characterized internally. The axioms which a collection R of [0, ∞]-valued functions has to fulfill in order to be the regular function frame of an approach space are: (R1) ∀S ⊂ R : W S ∈ R, (R2) ∀µ, ν ∈ R : µ ∧ ν ∈ R, (R3) ∀µ ∈ R, ∀α ∈ [0, ∞]: µ + α ∈ R, (R4) ∀µ ∈ R, ∀α ∈ [0, ∞]:(µ − α) ∨ 0 ∈ R.

The third author is an Aspirant of the Foundation for Scientiﬁc Research Flanders. 1 2 B. BANASCHEWSKI, R. LOWEN, C. VAN OLMEN

From this it clearly follows that a regular function frame is a frame, a complete lattice with the distributivity law _ _ ( A) ∧ b = (a ∧ b). a∈A However, it is also endowed with two families of unary operations, addition and bounded subtraction of an α ∈ [0, ∞]. For more information about frames, we refer to P.T. Johnstone [?]. 2.1. Deﬁnition. An approach frame L is a frame (with top > and bottom ⊥) equipped with two families of unitary operations, addition and subtraction of α ∈ [0, ∞]. We denote these operations Aα and Sα respectively. These operations satisfy all the identities valid for addition and subtraction by α and the frame operations in [0, ∞] and the implications Aα⊥ = ⊥ ⇒ α = 0 and Sα> = > ⇒ α 6= ∞. Notably the following hold:

(1) A0 = S0 = Id, Aα> = >, Sα⊥ = ⊥, (2) SαAα = Id if α 6= ∞ and AαSα = Aα⊥ ∨ Id, (3) AαAβ = Aα+β and SαSβ = Sα+β, W (4) α∈S Aαa = AW Sa, (5) Sαa ∧ Sβa = Sα∧βa, W W W W (6) Aα( S) = Aα(S) if S 6= ∅ and Sα( S) = Sα(S), (7) Aα(a ∧ b) = Aαa ∧ Aαb and Sα(a ∧ b) = Sαa ∧ Sαb As morphisms between two approach frames L and M, we will use frame homomorphisms h which commute with additions and subtractions, i.e. hAα = Aαh and hSα = Sαh for all α ∈ [0, ∞]. We refer to the morphisms as AFrm- homomorphisms and note that, with these morphisms, approach frames form a category which we will denote by AFrm.

We deﬁne J as the approach frame with the underlying frame [0, +∞] using the usual order and equipped with the evident operations Aα and Sα, α ∈ [0, ∞]. J is the initial object in AFrm. For every approach frame L, there exists exactly one homomorphism h : J → L and this morphism takes α to Aα⊥. Consequently, any approach frame L contains a copy of J and for simplicity in notation, we will write α for Aα⊥, and hence we will also write 0 for the bottom element ⊥ and ∞ for the top element >. 2.2. Theorem. The category of approach frames is complete and cocomplete.

Proof. AFrm clearly has products and equalizers, obviously formed at the level of the underlying sets. Further, coequalizers and coproducts are then easily obtained as approach subframes of appropriate products, where the case of coproducts is based on the fact that approach frames generated by sets of bounded size are themselves of bounded size. 2.3. Remark. The last statement also implies the so-called Solution Set Condition for the underlying set functor of AFrm, and hence the Adjoint Functor Theorem applies, showing there is a free approach frame over any set. Of course, the existence of coproducts can then be seen as a consequence of this. SOBER APPROACH SPACES 3

3. Categorical relationship between AFrm and AP For an approach space X the regular function frame RX is an approach frame with the obvious operations Aα and Sα. Furthermore, for any contraction ϕ : X → Y between approach spaces X and Y , the map Rϕ : RY → RX : γ 7→ γ ◦ ϕ is a homomorphism. In fact, R determines a contravariant functor from AP to AFrm. 3.1. Deﬁnition. The spectrum of an approach frame L is the approach space for which the set of points is the set of all homomorphisms ξ : L → J. We denote this set ΣL. The approach structure on ΣL is determined by the regular function frame which consists of all functions aˆ :ΣL → [0, ∞]: ξ 7→ ξ(a), where a ∈ L. We denote it Lb. It is clear from the deﬁnition that Lb is a subframe of [0, ∞]ΣL. That it is stable under addition and bounded subtraction follows at once from the fact that for any a ∈ L and α ∈ [0, ∞] we have

aˆ + α = Adαa and (â − α) ∨ 0 = Sdαa. For a homomorphism h : L → M we define Σh :ΣM → ΣL : ξ 7→ ξ ◦ h. This map is indeed a contraction, since for any a ∈ L and ξ ∈ ΣL we have (â ◦ Σh)(ξ) =a ˆ(ξ ◦ h) = ξ(h(a)) = hd(a)(ξ) and hencea ˆ ◦ Σh ∈ Mc for anya ˆ ∈ Lb. It is easy to see that Σ determines a contravariant functor from AFrm to AP. 3.2. Theorem. R and Σ are adjoint on the right.

Proof. Deﬁne ηL : L → RΣL : a 7→ aˆ and X : X → ΣRX : x 7→ x˜ with x˜ : RX → J : f 7→ f(x) the evaluation in x. That ηL is a homomorphism is implied by the arguments used to show that thea ˆ form an approach structure. To see that X is a contraction: for any γ ∈ RX we have that

(ˆγ ◦ X )(x) =γ ˆ(˜x) =x ˜(γ) = γ(x) meaningγ ˆ ◦ X = γ. To show the naturality of ηL and X : take f : X → Y a contraction, g ∈ RY and x ∈ X, then: (ΣRf)◦X (x) (g) = X (x)◦Rf (g) = X (x)(g◦f) = gf(x) = Y (f(x))◦g Take h : L → M a homomorphism and a ∈ L: (RΣh) ◦ ηL (a) = (RΣh)(ˆa) =a ˆ ◦ Σh = h(a)ˆ= ηM (h(a)) Finally we have the adjunction identities ˜ ˜ (ΣηL) ◦ ΣL(ξ) (a) = ξ ◦ ηL(a) = ξ(ˆa) =a ˆ(ξ) = ξ(a) (RX ) ◦ RX (γ) =γ ˆ ◦ X = γ 4 B. BANASCHEWSKI, R. LOWEN, C. VAN OLMEN

3.3. Deﬁnition. We call an approach space X sober whenever X : X → ΣRX is an isomorphism. An approach frame L is said to be spatial iﬀ ηL : L → RΣL is an isomorphism.

Note that this holds iff X is injective and surjective, hence we find that X is sober iff any ξ : RX → J is anx ˜ with x unique. 3.4. Definition. Sob is the category of sober approach spaces and contractions between such spaces. SpAFrm is the category of spatial approach spaces and homomorphisms between them. These subcategories of respectively AP and AFrm are clearly full. 3.5. Proposition. R and Σ induce a dual equivalence between Sob and SpAFrm. Proof. This is a formal consequence of the definitions of Sob and SpAFrm. 4. An alternative view on the spectrum 4.1. Definition. We call an element a ∈ L prime iff ∀b, c ∈ L : b ∧ c ≤ a ⇒ b ≤ a or c ≤ a 4.2. Lemma. Prime elements are translation-stable in the sense that for any prime a ∈ L, ∀α ∈ [0, ∞[ then Aαa is prime and if α ≤ a then Sαa is prime. Proof. For the first claim we have b ∧ c ≤ Aαa ⇔ Sα(b ∧ c) ≤ a ⇔ Sαb ≤ a or Sαc ≤ a ⇔ b ≤ Aαa or c ≤ Aαa. The second claim goes analogously. We can even give a stronger result: 4.3. Lemma. In any approach frame L, any prime a is a translation of a prime b with the property λ ≤ b ⇒ λ = 0. W Proof. For any prime a ∈ L, put λa = {α|α ≤ a}. Then λa ≤ a by the definition of approach frames. Now suppose µ ≤ Sλa a, then for ν = µ + λa,

ν ≤ Aλa Sλa a = a ∨ λa = a, which gives us µ + λa ≤ λa and therefore µ = 0. Hence u = Sλa a is a prime for which λ ≤ u implies λ = 0 and a = Aλa u. Using these two lemmas, we see that any prime element is accompanied by a [0, ∞[-indexed family of “translated” prime elements and there is a special role for primes a with λ ≤ a ⇒ λ = 0. 4.4. Deﬁnition. We call an element a ∈ L approach prime iﬀ a is prime and λ ≤ a implies λ = 0. W 4.5. Proposition. If ξ : L → J is a homomorphism then ξ∗(0) = {x|ξ(x) = 0} is approach prime. Conversely, if a ∈ L is approach prime, then ξa : L → V J with ξa(x) := {α|x ≤ Aαa} is a homomorphism. Moreover, ξξ∗(0) = ξ and (ξa)∗(0) = a. SOBER APPROACH SPACES 5

Proof. For the first claim, put a := W{x|ξ(x) = 0}, then obviously ξ(a) = 0. If b ∧ c ≤ a, then by monotonicity ξ(b) ∧ ξ(c) = ξ(b ∧ c) = 0, hence either ξ(b) = 0 or ξ(c) = 0. Finally, if α > 0 is such that α ≤ a, then ξ(α) ≤ ξ(a) = 0 which is a contradiction. To prove the second claim, note that ξ is clearly order-preserving and that ξa(0) = 0. Suppose that ξa(∞) 6= ∞, then there exists an α ∈ ]0, ∞[ such that ∞ = Aαa. Consequently 2α ≤ Aαa, and thus α = Sα2α ≤ a which is a contradiction. If S ⊂ L then we find that _ _ W S ≤ Aξa(s)a = A ξa(S)a s∈S W W and hence that ξa( S) ≤ ξa(S). Since ξ is monotone, we have the other inequality and so we find that ξ commutes with arbitrary joins. ξa(x ∧ y) = ξa(x) ∧ ξa(y): since ξ is order-preserving, we find ξ(x ∧ y) ≤ ξ(x) ∧ ξ(y). For the other inequality: suppose that x ∧ y ≤ Aαa. By lemma ??, we find that x ≤ Aαa or y ≤ Aαa. ξa(Aαx) = ξa(x) + α: take ξa(x) = β, then x ≤ Aβa ⇔ Aαx ≤ Aβ+αa if α < ∞. With α = ∞, the result is immediate. ξa(Sαx) = (ξa(x) − α) ∨ 0: Sαx ≤ Aβa ⇔ x ≤ Aα+βa. 4.6. Corollary. The space of points of L can be expressed as the set of approach prime elements. We denote this set by aprim(L).

4.7. Definitions and Notations. We will use pξ for the approach prime element associated with the homomorphism ξ : L → J. For the homomorphism associated with the approach prime element a, we will use ha. Using this definition we can easily express the approach structure on aprim(L). This must still be L, but the construction is slightly different now: ∀f ∈ L, a ∈ aprim(L): f(a) = ha(f) The distance δ on the spectrum can be constructed explicitly. First re- member the relation between the distance and the regular function frame in an approach space X:

δ(x, A) = sup{ρ(x)|ρ ∈ R, ρ|A = 0} 4.8. Proposition. On ΣL, the distance is deﬁned as: _ δ(ξ, A) = {ξ(a)|a ∈ L, ∀ζ ∈ A : ζ(a) = 0} If we work with aprim(L), this gives ^ δ(a, A) = ha( ai)

ai∈A Proof. The formula on ΣL is an easy translation of the relation between regular function frames and the distance. For aprim(L), note that ha (b) = 0 iﬀ b ≤ ai. It now follows that the i V largest element of L that has evaluation 0 for all hai must be ai. 6 B. BANASCHEWSKI, R. LOWEN, C. VAN OLMEN

5. Sobriety In this section we will (briefly) study the property of sobriety and this will include a very nice characterization of the spectrum of uniform spaces. Note that being sober is also equivalent to the fact that every approach W prime of RX is of the form (˜x)∗(0) = {ζ ∈ RX|ζ(x) = 0} = δ{x} for a unique x. Also note that sobriety in the classical frame sense can be defined likewise. In further analogy with frame theory we find counterparts of two of the basic properties concerning sobriety: the spectrum of a frame L is sober and the relation between morphisms between spaces and their corresponding frames. 5.1. Lemma. Every ΣL is sober.

Proof. We already know that (ΣηL) ◦ ΣL = IdΣL. On the other hand:

(ΣL ◦ (ΣηL))(ζ) = ΣL(ζ ◦ ηL) = (ζ ◦ ηL)˜ and for any a ∈ L we have

(ζ ◦ ηL)˜(ˆa) =a ˆ(ζ ◦ ηL) = (ζ ◦ ηL)(a) = ζ(ˆa)

This means that (ζ ◦ ηL)˜= ζ and so ΣL ◦ (ΣηL) = IdΣRΣL. Hence we find that ΣL is an isomorphism. 5.2. Proposition. Sobriety is reflective in AP, with the adjunction maps X : X → ΣRX as reflection maps. Proof. By the preceding lemma, if this map is an isomorphism then X is sober and therefore it is an isomorphism iff X is sober. 5.3. Proposition. Take Y a sober approach space and X general. Then for each homomorphism h : RY → RX, there exists exactly one contraction f : X → Y such that h = Rf. Proof. For any h : RY → RX, if −1 f = Y ◦ (Σh) ◦ X : X → Y then −1 Rf = RX ◦ RΣh ◦ (RY ) = RX ◦ RΣh ◦ ηRY = RX ◦ ηRX ◦ h = h because of the adjunction identities and naturalness of the adjunction maps. Further, if Rf = Rg, for any f, g : X → Y , we have

Y ◦ f = (ΣRf) ◦ X = (ΣRg) ◦ X = Y ◦ g and hence f = g. It is natural to wonder whether there is a relation between sobriety in the AFrm-context and the classical notion of sobriety. The next two proposi- tions will shed light on this relation. We will see that for topological spaces it means the same, but for general approach spaces it is a stronger construction. 5.4. Proposition. If an approach space X is sober (in the approach sense) then its topological coreﬂection is sober (in the topological sense). SOBER APPROACH SPACES 7

Proof. If we take A closed, join-irreducible and f, g ∈ RX, we ﬁnd −1 −1 −1 −1 f ∧ g ≤ δA ⇒ f (0) ∪ g (0) ⊇ A ⇒ f (0) ⊇ A or g (0) ⊇ A

⇒ f ≤ δf −1(0) ≤ δA or g ≤ δg−1(0) ≤ δA

Thus δA is approach prime. But since there is an isomorphism between X and ΣRX, we ﬁnd that there exists a unique x ∈ X such that δA = δ{x} and hence A = {x}.

5.5. Remark. The converse is not true, take for example (]0, 1], δE). The topological coreﬂection of this space is sober (even T3). The spectrum of the approach frame of this space however is the set of all evaluations in x for x ∈ [0, 1], since the approach prime functions are the functions fy(x) := |x − y|. But it is interesting to note that the space that we have obtained is the completion of the space we started with. 5.6. Proposition. A topological approach space X is sober iﬀ its associated topology is sober.

Proof. We only need to prove that if the coreflection is sober, then every approach prime element is of the form δA. This follows from proposition ??: if δA is prime, then A¯ must be join-irreducible in the set of closed sets of the coreflection. Now take an approach prime f ∈ RX. If f is the zero-function (thus f = δX ), there exists an element x ∈ X such that {x} = X. If f is not zero, then there exists x ∈ X such that f(x) > 0. Take > 0 such that f(x) > and we find that δ(x, {f ≤ }) > 0. Since δ{f≤} ∧ (f ∨ ) ≤ f we know that δ{f≤} ≤ f, hence ∞ ≤ δ(x, {f ≤ }) ≤ f(x). So for all finite we find {f ≤ } = {f = 0} which implies f = δ{f=0}. 5.7. Remark. Note that the above proof can be redone for approach spaces with Imδ ⊂ {0} ∪ [m, ∞] to prove that these spaces are sober iff the topological coreflection is sober. Taking a closer look at sober approach spaces, we come back to the fact that by taking ΣRX we sometimes obtain the completion of the space X. In fact we will prove that this is true for a significant class of approach spaces. First we will briefly recall some definitions and properties of approach spaces which we will need. Completeness is defined using Cauchy filters and studying their con- vergence. The definition of these notions follows directly from the definition of the limit operator (notation: λ) of a filter: given an approach space X and a filter F on X we define λF(x) = supA∈sec(F) δ(x, A) with sec(F) = {A ⊂ X|∀F ∈ F : A ∩ F 6= ∅}. A Cauchy filter is a filter F for which infx∈X λF(x) = 0 and a filter for which this infimum is a minimum is called convergent. We can also associate another operator with filters, the so-called adherence operator, defined as αF(x) = supF ∈F δ(x, F ). We also need some properties of the above concepts, namely for F, G filters with F ⊂ G and U an ultrafilter we have that αF ≤ αG ≤ λG ≤ λF , 8 B. BANASCHEWSKI, R. LOWEN, C. VAN OLMEN

αU = λU and λF(x) ≤ δ(x, {y}) + λF(y). The proof of all these properties can be found in [?], section 1.8. Note that in approach theory there is a very nice completion theory for uniform approach spaces, approach spaces that are subspaces of products of ∞p-metric spaces. With some notions recalled, we will start by showing that there is a connection between adherence and limit operators and approach prime elements. 5.8. Lemma. Take f an approach prime and F is the ﬁlter generated by {{f ≤ }| > 0}, then f = αF.

Proof. Take > 0, then (f ∨ ) ∧ δ{f≤} ≤ f. Since f ∨ 6≤ f, we find W δ{f≤} ≤ f. Hence >0 δ{f≤} ≤ f and we have that f(x) ≤ δ(x, {f ≤ })+ W so f = >0 δ{f≤} = αF. 5.9. Proposition. If f is approach prime then there exists an ultrafilter U ⊃ F such that f = λU (consequently U is a Cauchy filter). Proof. It is clear that f ≤ λU for all ultrafilters U ⊃ F. Suppose that for all those ultrafilters f 6= λU then

∀U ⊃ F, ∃xU ∈ X : f(xU ) < λU(xU ) = sup δ(xU ,U) U∈U Hence ∀U ⊃ F, ∃xU ∈ X, ∃UU ∈ U : f(xU ) < δ(xU ,UU ) which implies that ∀UU ∈ U : δUU 6≤ f.

Now there exist U1,..., Un ⊃ F and UU1 ∈ U1,...,UUn ∈ Un such that n ∪i=1UUi ∈ F. However then n min δU = δ∪n U 6≤ f i=1 Ui i=1 Ui since f is prime. We know however that f = αF = supF ∈F δF so this gives a contradiction. Thus ∃ U ⊃ F ultra, f = λU. U is also Cauchy since infx∈X f(x) = 0. 5.10. Remark. The converse of this proposition is not true, the limit operator of a Cauchy ultrafilter isn’t necessarily approach prime. In a topological space, the Cauchy filters are convergent and λU is approach prime iff lim U is a join-irreducible closed set. Thus take X := R t R with the topology generated by the following neighborhoods: V((x, 1)) := {V × {1}|V ∈ V(x)} V((x, 2)) := {(x, 2)} ∪ (V × {1}\{(x, 1)})|V ∈ V(x)}

This topology is T1. Choose an ultrafilter U ⊃ V((x, 1)) ∨ V((x, 2)), then lim U = {(x, 1), (x, 2)} which is not a join-irreducible closed set. Since the converse is not true in general, we will limit ourselves to approach spaces where it is true. 5.11. Definition. An approach space X is called semi-separated iff all Cauchy limit operators are approach prime functions. With this definition we find a first direct connection to completeness: SOBER APPROACH SPACES 9

5.12. Proposition. A sober, semi-separated approach space is complete. Proof. We have that ΣRX = {λF|F Cauchy} and by sobriety we have that every Cauchy limit operator is equal to δ{x} for a unique x ∈ X, hence it converges. A typical example of a sober, semi-separated approach space is the initial space P. The converse of the proposition is not true however. Take for example R with the finite complements topology. This is a complete, semi-separated space, but it is not sober. We will have to further limit ourselves to a smaller class of approach spaces to find a better relation between sobriety and completeness. As a first step to this smaller class we recall the notion of a supertight map from [?]. We call a function ϕ : X → [0, ∞] with X an approach space supertight iff

(1) ϕ : X → ([0, ∞], δP) is a contraction (2) infx∈X ϕ(x) = 0 (3) ∀x, y ∈ X : δ(x, {y}) ≤ ϕ(x) + ϕ(y) 5.13. Lemma. Take X an approach space. A supertight map is approach prime. Proof. Take a supertight function ϕ and suppose f ∧ g ≤ ϕ. Take x ∈ X, if g(x) ≤ ϕ(x), then for all y ∈ X we have

g(y) ≤ g(x) + δ(y, {x}) ≤ ϕ(y) + ϕ(x) + ϕ(x) = A2ϕ(x)ϕ(y). 1 Since infx∈X ϕ(x) = 0, we can find an xn with ϕ(xn) < n for all n ∈ N. For every n we have f(xn) ≤ ϕ(xn) or g(xn) ≤ ϕ(xn), so if we define If = {n|f(xn) ≤ ϕ(xn)} and analogously Ig then at least one of these sets is infinite. Suppose I is infinite, then f ≤ V A ϕ, so f ≤ ϕ. f n∈If 2ϕ(xn)

Again, the implication is strict, for example in P the δ{x} with x ∈]0, ∞[ are approach prime but not supertight. But for an interesting class of spaces we can prove a stronger result. 5.14. Proposition. Take X a uniform approach space. A function f ∈ RX is approach prime iff it is supertight. x Proof. For all d ∈ G, G the gauge associated with X, we have that gα = (α − d(·, x)) ∨ 0 is a contraction in (X, δd) and hence also in X. It is trivial x to see that gα ∧ Sαd(·, x) = 0. Now take f approach prime and suppose > 0. Take α = f(x) + , then x gα 6≤ f, hence Sαd(·, x) ≤ f. Then we also find _ Sf(x)d(·, x) = Sf(x)+d(·, x) ≤ f. >0 Put into other terms: d(y, x) ≤ f(y) + f(x). Since we find this result for all d ∈ G, we obtain that δ(y, {x}) ≤ f(x) + f(y). 10 B. BANASCHEWSKI, R. LOWEN, C. VAN OLMEN

5.15. Corollary. Take X a pseudometric space. A regular function f is approach prime iff it is supertight. From [?], we recall that an approach space X is called complemented iff it satisfies ∀F ∈ F(X): δ(x, {y}) ≤ λF(x) + λF(y) 5.16. Lemma. Take X a complemented approach space. f is approach prime iff f can be written as the limit operator of a Cauchy filter. Proof. By proposition ?? we have the implication. For a complemented space we trivially find that each limit operator of a Cauchy filter is supertight and hence by lemma ?? we find that such functions are approach prime. 5.17. Proposition. Take X a complemented approach space. Then X is sober iff it is complete and T0.

Proof. ⇒ We know that X : X → ΣRX = {λF|F Cauchy} is an isomorphism. Hence every ξλF can be written asx ˜ for a unique x ∈ X. Hence λF = δ{x}, so all Cauchy ﬁlters converge. X is also T0 since X is injective. ⇐ Since X is complete every λF with F Cauchy is equal to a function δ{x} for a unique (T0) x ∈ X. Hence X is a bijection. It is even an isomorphism since the approach structure of ΣRX is evidently the same as that of X. If we have an approach frame associated with an approach space, then ξf (g) = supx∈X g(x) − f(x) . We will use this evaluation as the basis for an extension of f ∈ RX to a function fˆ in RXˆ.

5.18. Theorem. Take X complemented and T0, then ΣRX is isomorphic to the completion of X.

Proof. Take f ∈ RX and construct fˆ : Xˆ → P by setting f(x) if λF(x) = 0 fˆ(F) = supx∈X f(x) − λF(x) if 6 ∃y ∈ X : λF(y) = 0 If we prove that fˆ is an element of RXˆ, we have an isomorphism between RX and RXˆ. Take > 0 and make ¯ : Xˆ → X a ﬁxed map for which ¯V(x) = x ∀x ∈ X λF¯(F) ≤ ∀F ∈ Xˆ

Take F ∈ Xˆ and Γ ∈ 2Xˆ : ˆ δP f(¯(F)), f(¯(Γ)) ≤ δ ¯(F), ¯(Γ) = δ ¯(F), ¯(Γ) ˆ ˆ ˆ 0 ≤ δ V(¯(F)), {F} + δ F, Γ + sup δ F , X (¯(Γ)) F 0∈Γ From the deﬁnition of ¯ it immediately follows that both δˆV(¯(F)), {F} ˆ 0 and supF 0∈Γ δ F , X (¯(Γ)) are smaller than or equal to . Hence we ﬁnd that ˆ δP f(¯(F)), f(¯(Γ)) ≤ δ F, Γ + 2 SOBER APPROACH SPACES 11

ˆ ˆ We can now start the calculation of δP f(F), f(Γ) . ˆ ˆ ˆ ˆ δP f(F), f(Γ) ≤ δP f(F), {f(¯(F))} + δP f(¯(F)), f(Γ) ˆ 0 ˆ ≤ δP f(F), {f(¯(F))} + δP f(¯(F)), f(¯(Γ)) + sup δP f(¯(F )), f(Γ) F 0∈Γ We also have that ˆ ˆ δP f(F), {f(¯(F))} = f(F) − f(¯(F)) ∨ 0 = sup f(x) − λF(x) − f(¯(F)) ∨ 0 x∈X = sup f(x) − f(¯(F)) − λF(x) ∨ 0 x∈X ≤ sup δ(x, {¯(F)}) − λF(x) ∨ 0 x∈X ≤ λF(¯(F)) ≤ and 0 ˆ 0 ˆ 00 sup δP f(¯(F )), f(Γ) = sup inf δP f(¯(F )), f(F ) F 0∈Γ F 0∈Γ F 00∈Γ = sup inf f(¯(F 0)) − sup f(x) − λF 00(x) ∨ 0 F 0∈Γ F 00∈Γ x∈X = sup inf inf f(¯(F 0)) − f(x) + λF 00(x) ∨ 0 F 0∈Γ F 00∈Γ x∈X ≤ sup inf inf δ¯(F 0), {x} + δx, {¯(F 0)} + λF 00(¯(F 0)) ∨ 0 F 0∈Γ F 00∈Γ x∈X = sup inf λF 00(¯(F 0)) ≤ F 0∈Γ F 00∈Γ So we finally find that ˆ ˆ ˆ δP f(F), f(Γ) ≤ + δ F, Γ + 2 + ˆ and hence f is a contraction. 5.19. Corollary. (1) Take X metric, ΣRX is the completion of X. (2) Uniform approach spaces are sober iff they are complete and T0. Proof. (1) In a metric space the supertight maps can be expressed as limn→∞ d(·, xn) with (xn)n a Cauchy sequence. (2) All uniform spaces are complemented. 5.20. Remark. There exist complemented approach spaces that are not uniform. Take for example all Hausdorff topological spaces that are not completely regular.

6. Spatiality Remark that in frame theory spatiality can be defined in the same way as we defined it for approach frames. Hence it seems logical that we can find results analogous to the classical theory. The first two results are examples of this. 12 B. BANASCHEWSKI, R. LOWEN, C. VAN OLMEN

6.1. Proposition. An approach frame is spatial iff each element is the meet of prime elements. Proof. ⇒ If the approach frame comes from an approach space, then we know that each δ(·, {x}) is prime and since f − f(x) ≤ δ(·, {x}) for all V contractions, we find that f = Af(x)δ(·, {x}) ⇐ Take S = {approach prime elements a}, as approach structure take L and see the elements as functions. This set of functions has all the properties of an regular function frame and it is clear that this approach space gives the approach frame we started with: if a 6= b ∈ L then there must exist an approach prime element e such that he(a) 6= he(b), suppose not, then ^ ^ a = Ahe(a)e = Ahe(b)e = b. e∈S e∈S Hence an approach frame is spatial iff the underlying frame is spatial.

6.2. Proposition. A T0 approach space X is a dense subspace of ΣRX.

Proof. If X is T0 then for all x 6= y we find δ(x, {y}) > 0 or δ(y, {x}) > 0. Hence δ(·, {y}) 6= δ(·, {x}) and since all δ(·, {y}) are prime elements, X : X → ΣRX : x 7→ δ(·, {x}) is injective. To show that X is a subspace embedding, we remark that the initial structure on X by X is just R since f ◦X (x) = f(x)(f first considered as element of RX, then seen as function on X). For the density, consider a ∈ L such that ∀x ∈ X : a(X (x)) = 0, then W obviously a = 0. Hence we find that δ(·, X (X)) = {a(·)|a ∈ L, ∀x ∈ X : a(X (x)) = 0} = 0. 6.3. Definition. An approach frame L is said to be topological iff L can be obtained as the regular function frame of a topological approach space.

6.4. Proposition. L topological iﬀ it it spatial and Sλa = a for all λ < ∞ and all approach primes a. Proof. ⇒ The approach prime elements a of the approach frame are: 0 y ∈ U θ (y) = U ∞ y 6∈ U for every closed and join-irreducible (with regard to the closed sets) U. This is because an approach prime a has values below every > 0 and for each n ∈ N there exists an x ∈ X such that a(x) > n: suppose not, ∀x ∈ X : a(x) < N and ∃x ∈ X : a(x) > N − > 0, then take a(y) a(y) ≤ N − a0(y)) = , ∞ a(y) > N − then a0 ∧ (N − ) ≤ a, but a0 and N − are not smaller than a. Finally a must be of the form θY : suppose a takes a value r > 0 other −1 r r than ∞, then consider T = a ([0, 2 ]), then θT ∧ (a ∨ 2 ) ≤ a, but θT 6≤ a r and a ∨ 2 6≤ a. ⇐ Using the isomorphism between aprim(L) and ΣL, we have an approach space, but it remains to prove that this space is topological. SOBER APPROACH SPACES 13

Take A ⊂ aprim(L) and a ∈ aprim(L), then δ(a, A) = h (V a ) = ∞ a ai∈A i if a 6≥ V a since b ≤ A c ⇔ S b ≤ c and S (V a ) = V a . And ai∈A i α α α ai∈A i ai∈A i if V a ≤ a, then obviously δ(ξ, A) = 0. ai∈A i 6.5. Definition. An approach frame is metric iff it can be obtained as the regular function frame of a ∞pq-metric space. 6.6. Proposition. X is metric iff arbitrary infima in RX are pointwise. Vp Proof. ⇒ If the pointwise infimum fi is a contraction, then it is the infimum of the fi in RX. We know fi(x) ≤ fi(y) + d(x, y) for all i, hence ^ ^ ^ fi(x) ≤ (fi(y) + d(x, y)) = fi(y) + d(x, y) and so the pointwise infimum is an element of RX. ⇐ We know ∀a ∈ A : δ(·,A) ≤ δ(·, {a}) and ^ ^ ( δ(·, {a}) − sup( δ(·, {a}))(b)) ≤ δ(·,A). a∈A b∈A But since the infimum of functions is the pointwise infimum we find that Vp a∈A δ(·, {a}) ≤ δ(·,A) and so we have equality. 6.7. Proposition. If every ξ ∈ ΣL is a complete lattice homomorphism, then ΣL is metric. V V V Proof. δ(ξ, A) = ξ( ζ∈A aζ ) = ξ(aζ ) = δ(ξ, {ζ}). It is clear that with X is metric and sober, we find that all ξ : RX → J are complete lattice homomorphisms, since every ξ is ax ˜ and the infimum V V of functions is pointwise, so ( fi)(x) = fi(x). References [1] Adámek J., Herrlich H. and Strecker G.E., Abstract and concrete categories. New York, John Wiley, 1990. [2] Banaschewski B., A Sobering Suggestion. Lecture Notes, Cape Town, 1999. [3] Johnstone P.T., Stone Spaces, Cambridge Studies in Advanced Math. no. 3. Cambridge, Cambridge University Press, 1982. [4] Lowen R., Approach spaces: a common supercategory of TOP and MET, Math. Nachr. 141 (1989), p. 183–226. [5] Lowen R., Approach Spaces: the Missing Link in the Topology-Uniformity-Metric Triad, Oxford Mathematical Monographs. Oxford, Oxford University Press, 1997. [6] Lowen R., Sioen M., A Unified Look at Completion in MET, UNIF and AP, Appl. Cat. Struct. 8 (2000), p. 447–461. [7] Robeys K., Extensions of Products of Metric Spaces, Ph D thesis, University Antwerp, 1992.