Page 100 - Textos de Matemática Vol. 41

86 Chapter 12. Entourages and (quasi-)uniformities (3) If L admits a uniformity then it is completely regular by (2). Now let L be
completely regular and, for each sequence
a1 ≺≺a2 ≺≺···≺≺an define an entourage Ea1,a2,...,an as
(a2 ⊕a2)∨(a∗1 ∧a3 ⊕a∗1 ∧a3)∨(a∗2 ∧a4 ⊕a∗2 ∧a4)∨···∨
∨(a∗n−2 ∧ an ⊕ a∗n−2 ∧ an) ∨ (a∗n−1 ⊕ a∗n−1).
Ifweinterpolatea1 ≺≺b1 ≺≺c1 ≺≺a2 ≺≺b2 ≺≺···≺≺cn−1 ≺≺an andset F = Ea1,b1,c1,a2,b2,...,cn−1,an we easily check that
F ◦ F ⊆ Ea1 ,a2 ,...,an .
Thus the system S of all the Ea1,a2,...,an is a subbasis of a uniformity E: S is admissible since if b ≺≺ a then, as we saw in 12.3.4, Eb,a ◦(b⊕b) ⊆ (a⊕a) and since S ⊆ E, E is admissible as well.
12.7. The biframe induced by a quasi-uniform locale. Given a quasi-uniform locale (L,E) let
L1(E):=a∈L|a={b∈L|b1E a} and 2
L2(E):= a∈L|a= {b∈L|bE a} .
Using 12.3.2 it is straightforward to check that L1(E) and L2(E) are subframes
of L. Further the admissibility of E implies that the triple (L, L1(E), L2(E))
is a biframe [8] (i.e. L is generated by the union L1(E) ∪ L2(E)).
The condition that (L, L1 (E ), L2 (E )) is a biframe char-
acterizes precisely the admissibility of E (cf. [49]).
12.7.1. The biframe (L, L1 (E ), L2 (E )) is the induced biframe of (L, E ). This is the point-free counterpart of the classical fact that any quasi-uniform space (X,E) induces a bitopological structure on X.
Moreover, a biframe (L, L1, L2) is quasi-uniformizable (in the sense that L admits a quasi-uniformity E whose induced biframe is precisely (L, L1, L2)) iff it is com- pletely regular (cf. [49]).