Page 47 - Textos de Matemática Vol. 41

6.2. Open and closed sublocales 33 6.2.3. Corollary. a ≤ b iff c(a) ⊇ c(b) iff o(a) ⊆ o(b).
(The first equivalence is obvious and the second one follows from the comple- mentarity.)
6.2.4. Proposition. We have (1) i∈J c(ai) = c(i∈J ai), (2) c(a) ∨ c(b) = c(a ∧ b), (3) o(a)∩o(b)=o(a∧b), (4) i∈J o(ai) = o(i∈J ai).
Proof.Indeedx∈c(ai)iffforeveryi∈J,x≥ai iffx≥i∈Jai,and c(a)∨c(b)={x∧y|x≥a, y≥b}=c(a∧b).
The third formula follows from the second one by complementarity. Finally, we trivially have i∈J o(ai) ⊆ o(i∈J ai) and if
y = ( ai)→x ∈ o( ai) i∈J i∈J
then, by 1.3(-distr), y = (ai →x) ∈ o(ai).
By 6.2.3 and 6.2.4 the set cL of all closed sublocales of L is a subframe of Sl(L)op and the correspondence a → c(a) defines an isomorphism L → cL. On the other hand, the map a → o(a) establishes a dual poset embedding of L into the subframe oL of Sl(L)op generated by all o(a), a ∈ L.
6.2.5. L-topologies. Given a complete lattice L of subsets of a set X we can modify the standard notion of a topology on X by restriction to L: an L-topology is a set τ of elements of L such that
0L and 1L are in τ,
ifU,V ∈τ thenU∧V ∈τ,and
if Ui, i ∈ J are in τ then i∈I Ui ∈ τ.
If the elements of τ are complemented in L, as they are in our case, we have naturally defined closed sets and we can build topological notions in this vein further.
Remark. Recall the spectrum Pt(L). It is an easy exercise to prove that Σa = Pt(L) \ c(a) = o(a) ∩ Pt(L).
Thus, the topology of Pt(L) can be viewed as that of a subspace.