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

4 B. BANASCHEWSKI, R. LOWEN, C. VAN OLMEN3.3. Definition. We call an approach space X sober whenever ɛ X : X →ΣRX is an isomorphism.An approach frame L is said to be spatial iff η L : L → RΣL is an isomorphism.Note that this holds iff ɛ X is injective and surjective, hence we find thatX is sober iff any ξ : RX → J is an ˜x with x unique.3.4. Definition. Sob is the category of sober approach spaces and contractionsbetween such spaces.SpAFrm is the category of spatial approach spaces and homomorphismsbetween them.These subcategories of respectively AP and AFrm are clearly full.3.5. Proposition. R and Σ induce a dual equivalence between Sob andSpAFrm.Proof. This is a formal consequence of the definitions of Sob and SpAFrm.□4. An alternative view on the spectrum4.1. Definition. We call an element a ∈ L prime iff∀b, c ∈ L : b ∧ c ≤ a ⇒ b ≤ a or c ≤ a4.2. Lemma. Prime elements are translation-stable in the sense that forany prime a ∈ L, ∀α ∈ [0, ∞[ then A α a is prime and if α ≤ a then S α a isprime.Proof. For the first claim we haveb ∧ c ≤ A α a ⇔ S α (b ∧ c) ≤ a ⇔ S α b ≤ a or S α c ≤ a ⇔ b ≤ A α a or c ≤ A α a.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 aprime b with the property λ ≤ b ⇒ λ = 0.Proof. For any prime a ∈ L, put λ a = ∨ {α|α ≤ a}. Then λ a ≤ a by thedefinition of approach frames. Now suppose µ ≤ S λa a, then for ν = µ + λ a ,ν ≤ A λa S λa a = a ∨ λ a = a, which gives us µ + λ a ≤ λ a and therefore µ = 0.Hence u = S λa a is a prime for which λ ≤ u implies λ = 0 and a = A λa u. □Using these two lemmas, we see that any prime element is accompaniedby a [0, ∞[-indexed family of “translated” prime elements and there is aspecial role for primes a with λ ≤ a ⇒ λ = 0.4.4. Definition. We call an element a ∈ L approach prime iff a is primeand λ ≤ a implies λ = 0.4.5. 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 ξ a : L →J with ξ a (x) := ∧ {α|x ≤ A α a} is a homomorphism. Moreover, ξ ξ∗(0) = ξand (ξ a ) ∗ (0) = a.□