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10 Chapter 2. Locales, frames and spaces
Proposition. For any b ∈ M we have
Pt(f)−1[Σb] = Σf∗(b).
Consequently, each Pt(f) is continuous and we have a covariant functor Pt : Loc → Top.
(2.3.1)
Proof. Pt(f)−1[Σb] = {p | f(p) ∈ Σb} = {p | b f(p)} = {p | f∗(b) p} = Σf∗(b). The consequence is obvious.
2.3.2. Proposition. The spectrum Pt(L) is a sober space.
Proof.Ifpqthenp∈/Σp ∋q.Thus,Pt(L)isT0.NowletFbeacompletely primefilterinΩ(Pt(L))={Σa |a∈L}.SetF ={a|Σa ∈F}.ByFact2.3, F isacompletelyprimefilterinLandhence,by1.6.1,F ={a|ap}fora point P. Thus,
Σa∈F iff p∈Σa;
in other words, F = U(p).
2.4. Locales associated with spaces. The functor Lc.
The functor Ω from 2.2 is
modified to a functor
by setting
Lc : Top → Loc
Lc(X) = Ω(X), Lc(f) = Ω(f)∗.
From the standard adjunction of preimage
f−1[B]⊆A iff B⊆Yf[XA] we immediately learn that
Lc(f)(U)=Y f[XU].
This does not seem to be a very transparent formula, but it becomes more acceptable after the following observation.
Represent a point x ∈ X by the point
x = X { x} of Lc(L). Then we have
Lc(f)(x) = f(x) (2.4.1) (indeed, Lc(f)(x) = Y f[X (X {x})] = Y f[{x}] = Y {f(x)} = f(x)).
