Page 49 - Textos de Matemática Vol. 41

6.5. Every sublocale is constructed from closed and open ones 35
Notes. (1) This is a surprising fact. The reader may see now that there are sublocales of classical spaces which are not necessarily subspaces. For instance, in the real line R (more precisely, Lc(R)), BL is a sublocale of both the subspace of the rationals and that of the irrationals.
In fact, it is very seldom the case that all sublocales of a space are spaces ([45], [63]). This happens iff the space X in question is weakly scattered [45] (corrupt in [63]), that is, if in X every non-void closed set contains a weakly isolated point (x is a weakly isolated point of a closed F ⊆ X if there is an open set U such that ∅ ̸= F ∩ U ⊆ {x}) (see 6.8 below).
(2) More generally, one speaks of a dense localic map f : L → M if f[L] is dense in M . In the language of the adjoint frame homomorphisms this is characterized by the implication
f∗(a)=0 ⇒ a=0. (∗)
(Indeed, f is dense iff 0 ∈ f[L] iff f(0) = 0; therefore if f is dense and f∗(a) = 0 then ff∗(a) = 0, which implies a = 0; if (∗) holds then from f∗f(0) ≤ 0 it follows that f(0) = 0.)
(3) The BL is a Boolean algebra (the well-known Booleanization of L). For Boolean sublocales see more information in 6.6 below.
6.5. Every sublocale is constructed from closed and open ones. Proposition. Let S be a sublocale. Then
S = {c(x) ∨ o(y) | νS(x) = νS(y)}.
Proof. Let a ∈ S and let νS(x) = νS(y). Then, by Proposition 5.2, x→a = νS(x)→a = νS(y)→a = y→a
and hence, by 1.3.1(H9),
a = (a ∨ x) ∧ (x→a) = (a ∨ x) ∧ (y→a) ∈ c(x) ∨ o(y).
Conversely, let a be in
{c(x) ∨ o(y) | νS (x) = νS (y)}.
Then in particular a ∈ c(νS(a))∨o(a). Hence, a = y∧(a→z) for some y ≥ νS(a) (≥ a) and z ∈ L. Then
1 = a→a = (a→y) ∧ (a→(a→z)) = 1 ∧ (a→z) = a→z
by 1.3.1(H1), (H8) and (H3), and hence a = y ∧ (a → z) = y ≥ νS(a). Thus, a = νS(a) ∈ S.