Page 83 - Textos de Matemática Vol. 41

Chapter 11
Separation II. Hausdorff axioms
11.1. The diagonal. The adjoint ∆∗ : L ⊕ L → L to ∆ (see 9.5.1) is given by ∆∗(U) = {x ∧ y | (x, y) ∈ U} = {x | (x, x) ∈ U}
(recall the formula (∗) in 9.3.3).
Obviously ∆∗ is onto (for instance, ∆∗(a⊕a) = a). Thus, as ∆∗∆∆∗ = ∆∗,
∆∗∆ = id. (11.1.1) From the adjunction we further have
U ⊆ ∆∆∗(U). (11.1.2) 11.2. Isbell-Hausdorff axiom. A topological space X is Hausdorff (or T2) if and
only if the diagonal
{(x,x) | x ∈ X}
is closed in X × X. Imitating this and following Isbell ([30]), a locale L is said to be Isbell-Hausdorff (strongly Hausdorff in [30]), briefly I-Hausdorff, if the diagonal sublocale ∆[L] of L ⊕ L is closed in L ⊕ L.
Since Lc(X × X) is not generally isomorphic to Lc(X) ⊕ Lc(X), the Isbell-Hausdorff property is only a formal analogy in Loc of the classical Hausdorff ax- iom, not an extension. If X is T0 and Lc(X) is Isbell- -Hausdorff then X is Hausdorff, but the reverse im- plication need not be true (cf. [35]).
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