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8.5. Induced congruences 55 Proof. Let ν be the nucleus corresponding to both E and S. Thus,
ν(x) = {s ∈ S | x ≤ s}.
Consequently,
uEv iff ν(u)=ν(v) iff (∀s∈S, u≤siffv≤s).
Now by 8.5.1,
o(u)∩S =o(v)∩S
again.
iff iff iff
c(u)∩S =c(v)∩S
{s ∈ S | s ≥ u} = {s ∈ S | s ≥ v} (∀s∈S, u≤siffv≤s)
8.5.3. Note. We already know that a space can have sublocales that are not subspaces. In that case we have frame congruences on Ω(X) = Lc(X) that are not of the form (8.5.1). But as any frame congruence, they do have the “intersection description” of (8.5.2).
