Page 45 - Textos de Matemática Vol. 41

Chapter 6
Some special sublocales
6.1. One-point sublocales. A sublocale always contains the top 1 and the sublocale {1} models the void subspace. Asking about “one-element subspaces” is, hence, the question about sublocales of the form {a, 1} with a ̸= 1.
Proposition. A sublocale of L contains precisely one element a ̸= 1 if and only if a is a point of L (that is, a meet-irreducible element of L).
Proof. If a is meet-irreducible and x is general then either x ≤ a and x→a = 1, or x a and then since x ∧ (x → a) ≤ a we have x → a = a, by 1.3.1 (H3), (H4) and (H6).
If {a, 1} is a sublocale and a ̸= 1, and if x ∧ y ≤ a then x ≤ y → a. Since y → a ∈ {a,1}, if y a, that is, y→a ̸= 1, then x ≤ a = y→a.
Thus, the relation between points and one-point sublocales, p vs. {p, 1},
is the same as that between x and {x} in classical spaces. Note that in Lc(X) we have, in the notation of 5.5,
{ x } = { x , 1 } .
6.2. Open and closed sublocales. For an a ∈ L define the sublocales
c(a) = ↑a,
o(a) = {a→x | x ∈ L} = {x ∈ L | a→x = x}. 31