Page 54 - Textos de Matemática Vol. 41

40 Chapter 6. Some special sublocales
6.8.4. Corollary. For any locale L the following are equivalent: (i) Every sublocale of L is spatial.
(ii) Sl(L)op is spatial.
Proof. By Proposition 3.5.2, Sl(L)op is spatial iff each sublocale S of L is a join in Sl(L) of meet-irreducibles of Sl(L)op. The conclusion follows immediately from 6.8.3.
6.8.5. Remarks. (1) H. Simmons proved in ([63], Theorem 4.4) that, for any space X, L = Lc(X) satisfies condition (ii) of 6.8.4 iff X is weakly scattered. Thus we have
For a space X, all sublocales of L = Lc(X) are spatial iff X is weakly scattered.
This adds more examples of non-spatial locales to the list in 3.5.4: any non-weakly scattered space has sublocales which are not spatial.
(2) Statement (1) of Proposition 6.8.3 identifies the points of the locale Sl(L)op (sometime referred to as the full assembly of L); it gives an inverse pair of bijections between the points of the spectrum Pt(L) of a locale L and the points of the spectrum Pt(Sl(L)op) of its assembly:
|Pt(L)| oo // |Pt(Sl(L)op)|
(6.8.5)
p

// bb(p) (p) Soo S.
This shows that the spectrum of the assembly has essentially the same points as the spectrum of L. It has, however, a different topology. What is this induced topology on |Pt(L)| ?
Recall that the Skula space (sometime called the front space) of a space X has the same points of X but the finer topology (the Skula topology) generated by
{U∩(X\V)|U,V ∈Ω(X)}
as a base (that is, the Skula topology of a space X is the smallest topology on |X| that makes all the original closed sets clopen). The topology on |Pt(L)|