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

8 B. BANASCHEWSKI, R. LOWEN, C. VAN OLMENαU = λU and λF(x) ≤ δ(x, {y}) + λF(y). The proof of all these propertiescan be found in [?], section 1.8.Note that in approach theory there is a very nice completion theory foruniform 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 connectionbetween adherence and limit operators and approach prime elements.5.8. Lemma. Take f an approach prime and F is the filter generated by{{f ≤ ɛ}|ɛ > 0}, then f = αF.Proof. Take ɛ > 0, then (f ∨ ɛ) ∧ δ {f≤ɛ} ≤ f. Since f ∨ ɛ ≰ f, we findδ {f≤ɛ} ≤ f. Hence ∨ ɛ>0 δ {f≤ɛ} ≤ f and we have that f(x) ≤ δ(x, {f ≤ ɛ})+ɛso f = ∨ ɛ>0 δ {f≤ɛ} = αF.□5.9. Proposition. If f is approach prime then there exists an ultrafilterU ⊃ 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 ≠ λU then∀U ⊃ F, ∃x U ∈ X : f(x U ) < λU(x U ) = sup δ(x U , U)U∈UHence∀U ⊃ F, ∃x U ∈ X, ∃U U ∈ U : f(x U ) < δ(x U , U U )which implies that ∀U U ∈ U : δ UU ≰ f.Now there exist U 1 , . . . , U n ⊃ F and U U1 ∈ U 1 , . . . , U Un ∈ U n such that∪ n i=1 U U i∈ F.However thennmin δ U Ui= δ ∪ ni=1 i=1 U Ui ≰ fsince f is prime. We know however that f = αF = sup F ∈F δ F so this givesa contradiction. Thus ∃ U ⊃ F ultra, f = λU.U is also Cauchy since inf x∈X f(x) = 0.□5.10. Remark. The converse of this proposition is not true, the limit operatorof a Cauchy ultrafilter isn’t necessarily approach prime.In a topological space, the Cauchy filters are convergent and λU is approachprime iff lim U is a join-irreducible closed set. Thus take X := R ⊔ Rwith 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 T 1 . Choose an ultrafilter U ⊃ V((x, 1)) ∨ V((x, 2)), thenlim 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 approachspaces where it is true.5.11. Definition. An approach space X is called semi-separated iff allCauchy limit operators are approach prime functions.With this definition we find a first direct connection to completeness: