SOBER APPROACH SPACES 1. Introduction In this article we will ...

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6 B. BANASCHEWSKI, R. LOWEN, C. VAN OLMEN5. SobrietyIn this section we will (briefly) study the property of sobriety and thiswill include a very nice characterization of the spectrum of uniform spaces.Note that being sober is also equivalent to the fact that every approachprime of RX is of the form (˜x) ∗ (0) = ∨ {ζ ∈ RX|ζ(x) = 0} = δ {x} for aunique 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 basicproperties concerning sobriety: the spectrum of a frame L is sober and therelation between morphisms between spaces and their corresponding frames.5.1. Lemma. Every ΣL is sober.Proof. We already know that (Ση L ) ◦ ɛ ΣL = Id ΣL . On the other hand:and for any a ∈ L we have(ɛ ΣL ◦ (Ση L ))(ζ) = ɛ ΣL (ζ ◦ η L ) = (ζ ◦ η L )˜(ζ ◦ η L )˜(â) = â(ζ ◦ η L ) = (ζ ◦ η L )(a) = ζ(â)This means that (ζ ◦ η L )˜= ζ and so ɛ ΣL ◦ (Ση L ) = Id ΣRΣL . Hence we findthat ɛ ΣL is an isomorphism.□5.2. Proposition. Sobriety is reflective in AP, with the adjunction mapsɛ X : X → ΣRX as reflection maps.Proof. By the preceding lemma, if this map is an isomorphism then X issober and therefore it is an isomorphism iff X is sober.□5.3. Proposition. Take Y a sober approach space and X general. Then foreach homomorphism h : RY → RX, there exists exactly one contractionf : X → Y such that h = Rf.Proof. For any h : RY → RX, ifthenf = ɛ −1Y◦ (Σh) ◦ ɛ X : X → YRf = Rɛ X ◦ RΣh ◦ (Rɛ Y ) −1 = Rɛ X ◦ RΣh ◦ η RY = Rɛ X ◦ η RX ◦ h = hbecause of the adjunction identities and naturalness of the adjunction maps.Further, if Rf = Rg, for any f, g : X → Y , we haveand hence f = g.ɛ Y ◦ f = (ΣRf) ◦ ɛ X = (ΣRg) ◦ ɛ X = ɛ Y ◦ gIt is natural to wonder whether there is a relation between sobriety in theAFrm-context and the classical notion of sobriety. The next two propositionswill shed light on this relation. We will see that for topological spacesit 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 7Proof. If we take A closed, join-irreducible and f, g ∈ RX, we findf ∧ g ≤ δ A ⇒ f −1 (0) ∪ g −1 (0) ⊇ A ⇒ f −1 (0) ⊇ A or g −1 (0) ⊇ A⇒ f ≤ δ f −1 (0) ≤ δ A or g ≤ δ g −1 (0) ≤ δ AThus δ A is approach prime. But since there is an isomorphism between Xand ΣRX, we find that there exists a unique x ∈ X such that δ A = δ {x}and hence A = {x}.□5.5. Remark. The converse is not true, take for example (]0, 1], δ E ). Thetopological coreflection of this space is sober (even T 3 ). The spectrum ofthe approach frame of this space however is the set of all evaluations in xfor x ∈ [0, 1], since the approach prime functions are the functions f y (x) :=|x − y|. But it is interesting to note that the space that we have obtained isthe completion of the space we started with.5.6. Proposition. A topological approach space X is sober iff its associatedtopology is sober.Proof. We only need to prove that if the coreflection is sober, then everyapproach prime element is of the form δ A . This follows from proposition ??:if δ A is prime, then Ā must be join-irreducible in the set of closed sets of thecoreflection.Now take an approach prime f ∈ RX. If f is the zero-function (thusf = δ X ), 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 ɛ > 0 such that f(x) > ɛand we find that δ(x, {f ≤ ɛ}) > 0. Since δ {f≤ɛ} ∧ (f ∨ ɛ) ≤ f we knowthat δ {f≤ɛ} ≤ f, hence ∞ ≤ δ(x, {f ≤ ɛ}) ≤ f(x). So for all finite ɛ we find{f ≤ ɛ} = {f = 0} which implies f = δ {f=0} . □5.7. Remark. Note that the above proof can be redone for approach spaceswith Imδ ⊂ {0} ∪ [m, ∞] to prove that these spaces are sober iff the topologicalcoreflection is sober.Taking a closer look at sober approach spaces, we come back to the factthat by taking ΣRX we sometimes obtain the completion of the space X. Infact we will prove that this is true for a significant class of approach spaces.First we will briefly recall some definitions and properties of approachspaces which we will need.Completeness is defined using Cauchy filters and studying their convergence.The definition of these notions follows directly from the definitionof the limit operator (notation: λ) of a filter: given an approachspace X and a filter F on X we define λF(x) = sup A∈sec(F) δ(x, A) withsec(F) = {A ⊂ X|∀F ∈ F : A ∩ F ≠ ∅}.A Cauchy filter is a filter F for which inf x∈X λF(x) = 0 and a filter forwhich this infimum is a minimum is called convergent.We can also associate another operator with filters, the so-called adherenceoperator, defined as αF(x) = sup F ∈F δ(x, F ).We also need some properties of the above concepts, namely for F, Gfilters with F ⊂ G and U an ultrafilter we have that αF ≤ αG ≤ λG ≤ λF ,
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