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P. 27
Chapter 3
The spectrum adjunction. Spatiality
3.1. The specialization order in Pt(L). Recall the specialization order (in a general space)
x⊑y iff x∈{y}.
Proposition. In Pt(L) the specialization order coincides with the dual of the
We have
original partial order in L. Consequently we have, for any point p ∈ Pt(L),
Σp = Pt(L) {p} (= p as a representant of the point in Lc(Pt(L)) ).
Proof.Ifpqthenp∈/Σp ∋qandhenceq∈/{p}.Ifq∈/{p}thenforsome a,p∈/Σa ∋q;thus,aqanda≤pandhencepq.Theconsequenceis straightforward.
3.2. The natural transformation σ : LcPt → Id. For a locale L define σL : Lc(Pt(L)) → L
by setting
σL(Σa)= {b|Σb ⊆Σa}, foreacha∈L. Thus, Σb ⊆ Σa iff b ≤ σL(Σa) and hence
σL∗ =(a→Σa):L→LcPt(L)
is a left Galois adjoint of σL. By Fact 2.3, σL∗ is a frame homomorphism and hence
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