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72 Chapter 11. Separation II. Hausdorff axioms
Note. This separation axiom [20], dated from the same year as the Isbell’s one, originally required the u, v for a, b ̸= 1 such that a ∨ b = 1. The present formulation is a modification that is hereditary for all open sublocales.
11.5. Relation between the two Hausdorff axioms.
Lemma. For any a, b ∈ L, the set
U =↓(a,a∧b)∪↓(a∧b,b)∪n
is a saturated subset of L × L.
Proof.Let(xi,y)beinU.Ify=0then(xi,y)∈U trivially.If0̸=y≤a∧b thenxi ≤aand( xi,y)∈↓(a,a∧b)⊆U;ifya∧btheny≤bandxi ≤a∧b, and (xi,y) ∈↓(a ∧ b,b) ⊆ U. Symmetrically for (x,yi).
Proposition. Each I-Hausdorff locale is DS-Hausdorff.
Proof. Suppose L is not DS-Hausdorff. Then there are a, b ̸= a ∨ b such that whenever u ∧ v = 0 and (u, v) ≤ (a, b) then either u ≤ b or v ≤ a, that is, for thesetU fromtheLemma,∆(0)∩(a⊕b)⊆U.
Now let L be I-Hausdorff. Then, by Proposition 11.2, (a,b) belongs to (a∧b)⊕(a∧b)∨∆(0) and hence
(a,b)∈((a∧b)⊕(a∧b)∨∆(0))∩(a⊕b)⊆(a∧b)⊕(a∧b)∨U =U
(as (a ∧ b) ⊕ (a ∧ b) ⊆ U ) which is a contradiction.
11.6. Regularity and Hausdorff properties.
11.6.1. The relation “rather below”. Let a and b be elements of a locale L. We
say that a is rather below b in L and write a≺b
if a∗ ∨ b = 1.
11.6.2. Note. The formula (Reg) of 10.1 is easily translated to
a = {b | b ≺ a} for each a ∈ L. (11.6.2)
(Indeed (11.6.2) says that if a b there is an x such that x∗ ∨ a = 1 and x b. Set c = x∗ and use the fact that x ≤ x∗∗. Conversely, (Reg) means that if a{b|b≺a}thereisacsuchthatc∨a=1andc∗ {b|b≺a},which is an absurd since c∗ ≺ a.)
Here, the notion of regularity is an extension of the classical one. We have