Page 87 - Textos de Matemática Vol. 41

11.6. Regularity and Hausdorff properties 73 Proposition. A space X is regular iff the locale Lc(X) is regular.
Proof. The pseudocomplement in Lc(X) is obviously given by the formula U∗ = X\UsothatV ≺UiffV ⊆U.Thenitisaveryeasyexercisetocheck that the classical regularity is equivalent to the condition that for each open U, U= {V|Vopen,V⊆U}.
11.6.3. Lemma. Let L be an arbitrary locale and let U ⊆ L × L be saturated and such that ∆(0) ⊆ U. Let (a∧b,a∧b) ∈ U. Then (x,y) ∈ U for every x ≺ a, y ≺ b.
Proof. Let x∗ ∨ a = y∗ ∨ b = 1. Then by 9.4
= (x∧(y∗ ∨b))⊕(y∧(x∗ ∨b))
= ((x∧y∗)∨(x∧b))⊕((y∧x∗)∨(y∧a))
= ((x∧y∗)⊕(y∧x∗))∨((x∧y∗)⊕(y∧a))∨
∨((x ∧ b) ⊕ (y ∧ x∗)) ∨ ((x ∧ b) ⊕ (a ∧ y)) ≤ (x⊕x∗)∨(y∗ ⊕y)∨(x⊕x∗)∨((a∧b)⊕(a∧b))
⊆U
since x⊕x∗, y∗ ⊕y and (a∧b)⊕(a∧b) are subsets of the saturated U.
11.6.4. Proposition. Each regular locale is I-Hausdorff (and hence DS-Hausdorff ).
Proof. If the locale is regular and if (a ∧ b, a ∧ b) ∈ U we have, in the notation ofthepreviousproof,first(a,y)=( {x|x≺a},y)∈U fory≺b,andthen (a,b)=(a, {y|y≺b})∈U.
(x,y)∈ x⊕y