Page 23 - Textos de Matemática Vol. 41

2.3. Points of a locale and the spectrum 9 (b) ⇒: x ≤ f(f∗(a)→b) iff f∗(x∧a) = f∗(x)∧f∗(a) ≤ b iff x∧a ≤ f(b) iff
x ≤ a→f(b).
⇐: f∗(a∧b) ≤ x iff a ≤ b→f(x) = f(f∗(b)→x) iff f∗(a) ≤ f∗(b)→x iff
f∗(a) ∧ f∗(b) ≤ x.
2.3. Points of a locale and the spectrum: a space associated with a locale. Recall the frames Ω(X) associated with spaces X (in 2.4 below we will slightly modify the construction to fit our discussion of locales). Now we will present a natural construction of a space associated with a locale.
A point in a locale L is any meet-irreducible element p ∈ L. Note that in Ω(X) of a weakly sober space X, these points are precisely the X \ {x} (recall 1.7.1), that are in one-one correspondence with the points x of the space if it is T0 (that is, if X is sober).
Lemma.
(a) Localic maps reflect tops, that is, if f(a) = 1 then a = 1. (b) Localic maps send points to points.
Proof. (a) Proposition 2.2.1(a).
(b) Let a be meet-irreducible in L and let x ∧ y ≤ f(a). Then f∗(x) ∧ f∗(y) = f∗(x ∧ y) ≤ a so that, say, f∗(x) ≤ a and x ≤ f(a) (which cannot be the top, by (a)).
For an element a ∈ L define
Σa ={p|ppointofL, ap}.
We have
Fact.Σa∧b =Σa∩Σb,Σai =Σai,Σ0 =∅andΣ1 ={allpoints}.
(All is trivial; in the first case realize that by the meet-irreducibility, a ∧ b p iff a p and b p.)
2.3.1. Define the spectrum of L as the topological space Pt(L)={p|ppointofL},{Σa |a∈L}.
By Lemma 2.3, a localic map f : L → M restricts to a map Pt(L) → Pt(M). Denote the restriction by
Pt(f) : Pt(L) → Pt(M).