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36 Chapter 6. Some special sublocales
6.6. Boolean sublocales. As observed in Notes 5.2, a sublocale S ⊆ L is a locale itself, with thesame Heyting operation. The (intrinsic) pseudocomplement in S, that is, a→ S, will be denoted by
a∗S .
Generalizing the definition of BL from 6.4, set for an a ∈ L
b(a) = {x→a | x ∈ L}.
From 1.3 (-distr), 1.3.1(H8) and (S2) we immediately obtain
6.6.1. Proposition. b(a) is the least sublocale in L containing a.
6.6.2. Lemma. b(a) has the following properties: (1) b(a)=a.
(2) x ∈ b(a) if and only if (x→a)→a = x.
Proof. (1) a ≤ x→a by 1.3.1(H4), and a = 1→a by 1.3.1(H2).
(2) The implication ⇐ is trivial.
⇒ : ((x→a)→a)→a = x∗S∗S∗S = x∗S = x→a by Proposition 1.4.1(3).
6.6.3. Proposition. A sublocale S ⊆ L is a Boolean algebra iff S = b(a) for some a ∈ L.
Proof. ⇐ : b(a) is Boolean. Indeed, recall 1.4.3. By 6.6.2 we have, for x in S = b(a), x∗S∗S = x.
⇒ : Let S ⊆ L be a Boolean sublocale. Set a = S. Let x be in S. The pseudocomplement in S is a complement and hence we have x = x∗S∗S = (x→ a)→a∈b(a).Ontheotherhand,ifx∈b(a)thenx∈Ssincea= S∈S. Thus, S = b(a).
6.6.4. Corollary. BL is the only dense Boolean sublocale of L. 6.6.5. From 6.6.1 we immediately obtain
Corollary. Each sublocale S ⊆ L is a join in Sl(L) (indeed a union)
S = {b(a) | a ∈ S} (= {b(a) | a ∈ S})
of Boolean sublocales.