Page 25 - Textos de Matemática Vol. 41

2.5. Reconstructing spaces and continuous mappings 11
Also note that for the representation of the points x by x we have the following useful formula (concerning U ∈ Lc(X) = Ω(X) and the open sets ΣU of the spectrum of Lc(X))
x∈ΣU iff x∈U (2.4.2) (of course: the left hand side says that U x = X {x}).
2.5. Reconstructing spaces and continuous mappings. For sober spaces the localic maps very well represent continuous maps. We have
Theorem. Let Y be sober. Then each localic map g : Lc(X ) → Lc(Y ) is Lc(f ) for precisely one continuous map f : X → Y . Thus, the restriction of the functor Lc to the subcategory of sober spaces is a full embedding.
Proof. Use 1.7.1. As x = X {x} is meet-irreducible, then by Lemma 2.3 h(x) is also meet-irreducible. Then, by sobriety, it is Y {y} for a uniquely determined y (the unicity follows from T0). Set y = f(x). Then we have, for any U ∈ Lc(Y ),
x∈f−1[U] iff f(x)∈U
iff UY{f(x)}=g(x)
iff g∗(U)x=X{x} iff x∈g∗(U).
Thus, Ω(f)(U) = f−1[U] = g∗(U) and hence, first, f is continuous, and, second, g = Lc(f), the (uniquely determined) right adjoint of Ω(f).
2.5.1. Recall the locale P from 2.2; it can be viewed as the locale Lc({·}) asso- ciated with a one-point space. The points of X are in a natural one-one corre- spondence with the continuous mappings {·} → X. By Theorem 2.5 they are, hence, in a one-one correspondence with the localic maps P → Lc(X). Note that, generally, localic maps g : P → L correspond (by g → g(0)) to the points of L in the sense of 2.3 (0 is a point of P, and on the other hand the g given by the adjoint g∗ : L → P sending x to 1 iff x z). Thus, the spectrum Pt(Lc(X)) re- constructs the space X if X is sober (analyze the topology as a simple exercise). For more see Notes 3.6 below.