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

SOBER APPROACH SPACES 13Take A ⊂ aprim(L) and a ∈ aprim(L), then δ(a, A) = h a ( ∧ a i ∈A a i) = ∞if a ≱ ∧ a i ∈A a i since b ≤ A α c ⇔ S α b ≤ c and S α ( ∧ a i ∈A a i) = ∧ a i ∈A a i. Andif ∧ a i ∈A a i ≤ a, then obviously δ(ξ, A) = 0.□6.5. Definition. An approach frame is metric iff it can be obtained as theregular function frame of a ∞pq-metric space.6.6. Proposition. X is metric iff arbitrary infima in RX are pointwise.Proof. ⇒ If the pointwise infimum ∧p f i is a contraction, then it is theinfimum of the f i in RX.We know f i (x) ≤ f i (y) + d(x, y) for all i, hence∧fi (x) ≤ ∧ (f i (y) + d(x, y)) = ∧ f i (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∈Ab∈ABut since the infimum of functions is the pointwise infimum we find thatδ(·, {a}) ≤ δ(·, A) and so we have equality.□∧ pa∈A6.7. Proposition. If every ξ ∈ ΣL is a complete lattice homomorphism,then ΣL is metric.Proof. δ(ξ, A) = ξ( ∧ ζ∈A a ζ) = ∧ ξ(a ζ ) = ∧ δ(ξ, {ζ}).It is clear that with X is metric and sober, we find that all ξ : RX → Jare complete lattice homomorphisms, since every ξ is a ˜x and the infimumof functions is pointwise, so ( ∧ f i )(x) = ∧ f i (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-MetricTriad, 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.□