Page 20 - Textos de Matemática Vol. 41

6 Chapter 1. Preliminaries
1.7.1. Proposition. The following statements are equivalent: (i) X is weakly sober.
(ii) The meet-irreducible elements of Ω(X) are precisely the open sets X \ {x}.
Proof. Use 1.6.1. Let (ii) hold and let F be a completely prime filter in Ω(X). The set
V= {U|U∈/F}
is then meet-irreducible (the complete primeness is needed for V ̸= X) and henceV =X\{x}forsomex∈XandwehaveU∈/FiffUX\{x},thatis, U ∈ F iff x ∈ U.
Conversely, if X is weakly sober and V is meet-irreducible then {U | U V }
is a completely prime filter, and hence one of the U(x). Then U V iff x ∈ U iffUX\{x},andV =X\{x}.
Sober spaces are weakly sober spaces that are, moreover, T0. This amounts to stating that a completely prime filter is the neighbourhood system of precisely one point, or that the x in X \ {x} in the second statement of 1.7.1 is uniquely determined.
It is an easy exercise to show that (e.g.) each Haus- dorff space is sober. But a sober space does not have to be even T1: for instance every finite T0 space is sober (on the other hand, T1 does not imply sobriety either).
1.8. What we will need about categories. Not much, only the basics, to be found in any standard monograph, for instance [1] or [42].
The reader should know that an adjoint situation L ⊣ R (L and R are adjoint functors, L on the left and R on the right) between categories A and B,
given by
εAB :B(L(A),B)∼=A(A,R(B)), can be equivalently described by natural transformations
σ:LR→Id, ρ:Id→RL
such that all the σL(A) · L(ρA) and R(σB ) · ρR(B) are identities.