Page 30 - Textos de Matemática Vol. 41

16 Chapter 3. The spectrum adjunction. Spatiality Conversely, if a = {p points of L | a ≤ p} then for b a there is a point
p such that a ≤ p and b p, hence p ∈ Σb Σa.
3.5.3. The Boolean case.
Proposition. Every meet-irreducible element in a Boolean algebra is a co-atom. Consequently, a Boolean locale L is spatial only if it is atomic, that is, if each a∈L is a join of atoms.
Proof. Let p be meet-irreducible in a Boolean algebra L. Let p < x. If y is the complement of x we have 0 = x ∧ y ≤ p and by meet-irreducibility y ≤ p but then y ≤ x and 1 = x ∨ y = x.
Now by 3.5.2, every element of a spatial Boolean algebra is a meet of co-atoms. By way of complements we conclude that every element is a join of atoms.
3.5.4. Notes. (1) Thus, any non-atomic complete Boolean algebra — for instance the Boolean algebra of regular open subsets of the real line (regular ≡ interior of its closure) — is an example of a non-spatial locale.
(2) On the other hand, complete atomic Boolean algebras are spatial. But those are, up to isomorphism, precisely the Boolean algebras of all subsets of sets. Thus the classes of Boolean locales and spatial locales intersect only in the trivial case of discrete spaces.
3.6. The sobrification.
Proposition. The following statements on a space X are equivalent. (i) X is sober,
(ii) λX : X → Pt(Lc(X)) is one-one onto,
(iii) λX : X → Pt(Lc(X)) is a homeomorphism.
Proof. (i)⇒(ii) follows from the definition of sobriety and (iii)⇒(1) from 2.3.2. (ii)⇒(iii): We have to prove that images of open sets U under λX are open. We
have
λX [U ] = {X {x} | x ∈ U } = {X {x} | U X {x}} = ΣU .
Notes. (1) The space Pt(Lc(X)) is usually called the sobrification of X. Let us see what happens: the map λX sends X homeomorphically to the subspace of Pt(Lc(X)) representing points by their neighbourhood systems; this is then