Page 31 - Textos de Matemática Vol. 41
P. 31
3.6. The sobrification 17 extended by the system of all the completely prime filters. Thus, we have here a
sort of completion of X with respect to the property in the definition of sobriety. (2) For the categorically-minded: what we have learned here is that the subcat-
egory Sob of sober spaces is reflective in Top, with the reflection λ : Id → PtLc.
(3) In Proposition 3.5.1, we have seen that if a locale L is spatial then it is isomorphic to the open set lattice of its spectrum Pt(L). If it is not, then there is no use to search for another space X to obtain L ∼= Ω(X).
On the other hand, the X such that L ∼= Ω(X) is not uniquely deter- mined: by (3.4.1) the lattice of open sets of a space and that of its sobrifica- tion are isomorphic. But nothing worse can happen: if there is an isomorphism h : Lc(X) → Lc(Y ) we have an isomorphism Pt(h) : Pt(Lc(X)) → Pt(Lc(Y )). Thus, the sobrifications are isomorphic. This is, of course, no surprise. We know since 2.5 that a sober space can be reconstructed from its lattice of open sets.
