Page 85 - Textos de Matemática Vol. 41

11.3. Hereditary properties 71 We will use condition (iii) of the Lemma. Set α(a) = (a ⊕ a) ∨ ∆(0).
Obviously α preserves meets, and by (∗) and 9.4 also α(ai) = ((ai)⊕(ai))∨∆(0)
i∈J i∈J i∈J
= ( (ai ⊕aj))∨∆(0)
i,j∈J
= ((ai ⊕ ai) ∨ ∆(0))
i∈J i∈J
Thus, α is a frame homomorphism and hence so is also α∆∗; we have α∆∗(a ⊕ b) = α(a ∧ b) = ((a ∧ b) ⊕ (a ∧ b)) ∨ ∆(0) = (a ⊕ b) ∨ ∆(0),
the last equality by the assumed implication again. Since the elements a ⊕ b generate L ⊕ L (recall 9.3.4), the statement follows.
11.3. Hereditary properties.
Proposition. Each sublocale of an I-Hausdorff locale is I-Hausdorff.
Proof. Let S ⊆ L be a sublocale and let U be saturated in S ×S; let U ⊇ ∆S (0S ) = {(x, y) ∈ S × S | x ∧ y = νS (0)}.
Then ↓U (with ↓ taken in L×L) is saturated in L×L: if (xi,y) ∈ ↓U there are x′i≥xi,y′≥ywith(x′i,y′)∈U;hence( Si∈Jx′i,y′)∈Uand
S
xi ≤ x′i ≤νS( x′i)= x′i
i∈J i∈J i∈J i∈J
so that (i∈J xi,y) ∈ ↓U. Further, ↓U ⊇ ∆(0): in fact, if x∧y = 0 then
νS(x) ∧ νS(y) = νS(0) and hence
(x,y) ≤ (νS(x),νS(y)) ∈ ∆S(0S) ⊆ U.
Nowlet(a∧b,a∧b)∈U with(a,b)∈S×S.Then(a,b)∈↓U andsince U isadown-setinS×S,(a,b)∈U.
11.4.Dowker-Strauss-Hausdorffaxiom. AlocaleLisDowker-Strauss-Hausdorff, briefly DS-Hausdorff, if for any a, b ∈ L such that a ∨ b ̸= a, b there are
u≤aandv≤bsuchthatu∧v=0, ubandva.
=
α(ai ).