Page 98 - Textos de Matemática Vol. 41

84 Chapter 12. Entourages and (quasi-)uniformities
(f⊕f)∗(E)=(f⊕f)∗ (a⊕b)= (f∗(a)⊕f∗(b)). (12.4.1) (a,b)∈E (a,b)∈E
Let (L,E),(M,F) be uniform (or just quasi-uniform) locales. A localic map f : (L,E) → (M,F) is uniform if for each E ∈ F, (f ⊕f)∗(E) ∈ E. The motivation is clear: recalling 7.5.2 this means, in terms of open sublocales, that
(f ⊕ f)−1(o(E)) = o((f ⊕ f)∗(E) ∈ E
for any o(E) ∈ F.
The resulting categories are the well-known categories of uniform and
quasi-
-uniform locales (usually described, in the literature, in terms of covers).
12.5. Bases and subbases. A uniformity or quasi-uniformity is often described by a basis, that is, an admissible system of entourages B satisfying (U3) such that
(U2′)E,F ∈B⇒∃G∈B, G⊆E∩F.
Then E = {E | ∃F ∈ B,F ⊆ E} is obviously a (quasi-)uniformity, and iE =iB (i=1,2).
One also constructs a (quasi-)uniformity as
{E | ∃F1, . . . , Fn ∈ S, F1 ∩ · · · ∩ Fn ⊆ E}
from a subbasis S for which only (U3) is required; here, however, one has to be
careful since iE in general does not coincide with iS.
12.6. Complete regularity. Given elements a, b of a locale L we say that a is
completely below b and write
a ≺≺ b
if there are ar for r dyadic rationals in the interval [0, 1] such that
a0 =a, a1 =b and ar ≺as forr<s. A relation R is said to be interpolative if
aRb ⇒ ∃c,aRcRb. Note that ≺≺ is the largest interpolative R ⊆≺.