Page 80 - Textos de Matemática Vol. 41

66 Chapter 10. Separation I. Regular, fit and subfit Proof. (i)⇒(ii): Let b ∈ L and a = νS(b). If a∨c = 1 we have νS(b∨c) ≥ a∨c = 1
and hence b ∨ c = 1. Thus, a ≤ b, that is, b ∈ S. (ii)⇔(iii): (iii) is just an immediate reformulation of (ii). (iv)⇔(v) follows immediately from Corollary 6.2.6(1).
(iii)⇒(iv): Set S = {c(x) | x∨a = 1} and suppose c(y)∩(c(a)∨S) = O. Then, c(y) ∩ c(a) = O and, by complementarity, c(y) ⊆ o(a). Therefore, by Corollary 6.2.6(1), y ∨ a = 1, and we conclude that c(y) ⊆ S and finally c(y) = c(y) ∩ (c(a) ∨ S) = O. Thus by (iii), c(a) ∨ S = L and by complementarity again o(a) ⊆ S (⊆ o(a) by 6.2.6(1)).
(4)⇒(1): If a b we have c(b) c(a) (as b ∈ c(b)c(a)) and hence o(a) o(b). Thus there is a c such that c ∨ a = 1 and c(c) o(b), that is, c ∨ b ̸= 1.
10.4.1. We say that a localic map f : L → M is co-dense if f [L {1}] = f [L] \ {1}
(recall 2.2.1(a)) is cofinal in M {1}. From the equivalence (i)⇔(ii) in 10.4 we immediately obtain
Corollary. A locale M is subfit iff each co-dense localic map f : L → M is onto.
Note. Recall Note 6.4(2). While the density is characterized by the implication f∗(a)=0 ⇒ a=0,
the co-density is characterized by
f∗(a)=1 ⇒ a=1.
Indeed, if f[L{1}] is cofinal in M {1} and b < 1 then there is an a < 1 such that b ≤ f(a) and hence f∗(b) ≤ a < 1. Conversely, if the implication holds and b<1inM thenf∗(b)=a<1andb≤f(a).
10.5. Two properties of subfit locales. Recall 7.3. Open localic maps are pre- cisely the localic maps f : L → M for which the left adjoint f∗ is a complete Heyting homomorphism. When M is subfit one can say more (see also [54]):
10.5.1. Proposition. If M is subfit then a localic map f : L → M is open iff f∗ : M → L is a complete lattice homomorphism (i.e. f∗ has also a left adjoint).
Proof.Letf:L→Mbeanopenlocalicmap.By7.3,f∗ :M→Lis,in particular, a complete lattice homomorphism.