Page 53 - Textos de Matemática Vol. 41

6.8. Spatial sublocales 39
Proof. If x ≤ p then, by 1.3.1(H3) and (H2),
(x → p) → p = 1 → p = p.
On the other hand, if x p then x → p = p (indeed, by 1.3.1(H6), (x → p)∧x = x ∧ p ≤ p which implies x → p ≤ p; by 1.3.1(H4) p ≤ x → p always). Thus (x → p) → p = 1 → p = 1.
6.8.3. Proposition. For any sublocale S of L we have
(1) S is a meet-irreducible element of Sl(L)op iff S = bb(p)(p) for some meet-
-irreducible p of L.
(2) mPt(S) = bb(p)(p).
p∈Pt(S)
(3) S is spatial iff S = bb(p)(p).
p∈Pt(S)
Proof. (1) If S = bb(p)(p) = {p,1} (by Lemma 6.8.2), then S ̸= O and p ∈ S⊆S1∨S2 impliesp=a∧bforsomea∈S1 andb∈S2.Hencebythe meet-irreducibility of p, p = a ∈ S1 (that is, bb(p)(p) ⊆ S1) or p = b ∈ S2 (that is, bb(p)(p) ⊆ S2).
Conversely, if S is meet-irreducible in Sl(L)op then S ⊇ {a,1} for a = S ̸= 1 (because S ̸= O). Let b ∈ S such that a < b. Then S c(b) = ↑a. But S ⊆ c(b) ∨ o(b) thus by meet-irreducibility S ⊆ o(b). In particular, b ∈ S ⊆ o(b) so b ∈ o(b)∩c(b) = O that is b = 1. Hence S = {a,1}. By Proposition 6.1, a is necessarily meet-irreducible in L.
(2) The inclusion “⊇” is obvious since, by Lemma 6.8.2, bb(p)(p) = {p, 1} = Pt(S).
p∈Pt(S) p∈Pt(S)
The reverse inclusion is also obvious: for any Y ∈ mPt(S),
Y ⊆ Pt(S) ⊆ bb(p)(p), p∈Pt(S)
thus Y ∈ p∈Pt(S) bb(p)(p).
(3) is an immediate consequence of (2) and Proposition 3.5.2.