Page 97 - Textos de Matemática Vol. 41

12.4. Uniformities and quasi-uniformities 83
Proof. Let L be a regular locale and take the system Ent(L) of all entourages ofL.Then,foreacha∈L,wehavea= {b∈L|b≺a}.Butb≺ameans that b∗ ∨ a = 1 which ensures that
Eb,a =(b∗ ⊕b∗)∨(a⊕a)
is an entourage of L. Moreover, Eb,a ◦ (b ⊕ b) ⊆ (a ⊕ a). Indeed, using Lemma
12.2.5, we get
Eb,a ◦(b⊕b) = j((b∗ ⊕b∗)∪(a⊕a))◦(b⊕b) ⊆ j(((b∗ ⊕b∗)∪(a⊕a))·(b⊕b))
⊆ a⊕a
since ((b∗ ⊕ b∗) ∪ (a ⊕ a)) · (b ⊕ b) ⊆ (a ⊕ a), as can be easily checked.
In conclusion, b ≺ a implies b Ent(L) a. Hence a={b∈L|b≺a}≤{b∈L|bEnt(L) a}≤a.
Conversely, by Proposition 12.3.2(3), bE implies b ≺ a, thus we have a={b∈L|bE a}≤{b∈L|b≺a}≤a.
12.4. Uniformities and quasi-uniformities. A uniformity on a locale L is an admissible system of entourages E such that
(U1) E∈E andE⊆F impliesF ∈E.
(U2) E,F ∈E ⇒E∩F ∈E.
(U3) For each E ∈ E there is an F ∈ E such that F ◦ F ⊆ E. (U4) E ∈ E ⇒ E−1 ∈ E.
If (U4) is not required we speak of a quasi-uniformity on L.
A uniform locale is a couple (L,E) where E is a uniformity on L. If E is just a quasi-uniformity we speak of a quasi-uniform locale.
12.4.1. Uniform homomorphisms. By Corollary 9.5.2, for any localic map f : L→M andanyE∈L⊕L,(f⊕f)(E)isgivenby
(f(a) ⊕ f(b)). (a,b)∈E
On the other hand, since (f ⊕ f)∗ = f∗ ⊕ f∗, we have