Page 84 - Textos de Matemática Vol. 41

70 Chapter 11. Separation II. Hausdorff axioms Since the smallest element in ∆[L] is obviously ∆(0), this amounts to the
condition that
∆[L] =↑∆(0).
Lemma. The following statements about a locale L are equivalent: (i) L is I-Hausdorff.
(ii) For all saturated U ⊇ ∆(0), ∆∆∗(U) = U; in other words, the restric- tions of ∆ resp. ∆∗ to L →↑∆(0) resp. ↑∆(0) → L are mutually inverse isomorphisms.
(iii) There exists a mapping α : L →↑∆(0) such that α∆∗ = γ = (U → U ∨ ∆(0)).
Furthermore, the α in (iii) is necessarily the restriction of ∆ to L → ↑∆(0). Proof. (i)⇒(ii): For the U ⊇ ∆(0) choose a = β(U) such that U = ∆(a). Thus,
U = ∆β(U) and, by (11.1.1), ∆∗(U) = ∆∗∆β(U) = β(U). Thus, U = ∆∆∗(U). (ii)⇒(iii): Set
α = (a → ∆(a)) : L → ↑∆(0).
Then U ∨ ∆(0) = ∆∆∗(U ∨ ∆(0)) = ∆(∆∗(U) ∨ ∆∗∆(0)) = ∆(∆∗(U) ∨ 0) =
∆(∆∗(U)) = α∆∗(U).
(iii)⇒(i) is trivial since such an α is necessarily onto.
Finally, if α∆∗ = γ then α(a) = α∆∗(∆(a)) = ∆(a) ∨ ∆(0) = ∆(a). Proposition. A locale L is I-Hausdorff iff for every saturated U ⊇ ∆(0) one has
the implication
(a∧b,a∧b)∈U ⇒ (a,b)∈U.
Proof. ⇒: Let L be an I-Hausdorff locale. Then by the Lemma, ∆∆∗(U) = U.
If(a∧b,a∧b)∈U thena∧b≤∆∗(U)and(a,b)∈∆(∆∗(U))=U.
⇐: Let the implication hold and let xi, i ∈ J, be in L. Then for a saturated
U ⊇ ∆(0) we have the implication
(xi,xi)∈U ⇒ (xi,xj)∈U foranyj (∗)
(since if (xi , xi ) ∈ U then (xi ∧ xj , xi ∧ xj ) ∈ U ).