Page 19 - Textos de Matemática Vol. 41

1.7. Sobriety 5
1.6.1. Lemma. The formulas
F→{x|x∈/F}, a→{x|xa}
constitute a one-one correspondence between the set of all completely prime fil- ters and the set of all meet-irreducible elements.
Proof. The element a = {x | x ∈/ F } is ̸= 1 by complete primeness and is meet- -irreducible since by completeness a ∈/ F and hence if x ∧ y ≤ a we must have eitherx∈/F ory∈/F.ThesetF ={x|xa}isafilterbymeet-irreducibility: x ∧ y ≤ a would make eitherx ∈/ F or y ∈/ F . It is obviously completely prime.
Now by completeness, {x | x ∈/ F} is the largest element that is not in F and hence
Finally,
1.7. Sobriety.
y ∈/ { x | x { x | x ∈/ F } } i ff y ∈/ F .
{ x | x ∈/ { y | y a } } = { x | x ≤ a } = a .
Consider a point x of a topological space X, and the system U (x)
of all the open neighbourhoods of x. This system U(x) is a filter in the lattice (frame)
Ω(X )
of open sets of the space X. Moreover, it is obviously a completely prime filter (indeedif i∈J Ui ∋xtherehastobeani∈J suchthatx∈Ui).Thequestion naturally arises whether, on the other hand, every completely prime filter is a U(x) for an x ∈ X.
Note that this is analogous with the completeness of metric spaces. There one considers the Cauchy filters (filters F of open sets such that for each ε > 0 there areU,V ∈FsuchthatU⊆V anddiamV ≤ε)and the space is complete if each Cauchy filter F has a non-void intersection.
If this is the case we say that the space X is weakly sober. Weakly sober spaces are characterized by the following equivalent property that was in fact used as the original definition by Grothendieck:
each meet-irreducible U ∈ Ω(X) is of the form X \ {x} for some x ∈ X.
(Needless to say, all the X \ {x} are obviously meet-irreducible.) That is, one has