Page 78 - Textos de Matemática Vol. 41

64 Chapter 10. Separation I. Regular, fit and subfit
In spaces, the subfitness is slightly weaker than the separation property T1. It was introduced by Isbell in [30] and independently (as conjunctivity, because it is the opposite of the disjunctive property for distributive lattices) by Simmons in [62].
Fitness is a specifically point-free property. It appeared, first, in [30]; there we find our present formulation as an equivalent characterization, while the original definition is what we will have as the statement (iv) in Proposition 10.3 below).
Proposition. Each regular locale is fit, and each fit locale is subfit. Proof. The first statement is obvious since c→b ≥ c→0.
Now let L be fit and let a, b, c be as in the condition (Fit). Then c→b = (c→b) ∧ (b→b) = (c ∨ b)→b ̸= b
and hence c ∨ b ̸= 1. 10.2. Hereditary properties. We will need the following easy formula (S is an
arbitrary sublocale of L).
∀x ∈ L,∀s ∈ S, x→s = νS(x)→s (10.2.1)
(indeed, y ≤ x→s iff x ≤ y→s iff νS(x) ≤ y→s iff y ≤ νS(x)→s). Proposition. Let L be regular resp. fit. Then each sublocale S ⊆ L is regular
resp. fit.
Proof. Let a, b ∈ S, a b. Denote by x ∨S y the join in S. Consider the c ∈ L from the two definitions.
Let L be regular. We have νS(c)∨S a ≥ c∨a = 1 and νS(c)→0S = νS(c)→ νS(0) = c→νS(0) by (10.2.1), and c→νS(0) ≥ c→0 and hence νS(c) → 0S b. If L is fit it is even easier: again, trivially νS (c) ∨S a = 1, and by (10.2.1) νS(c)→b = c→b b.
10.3. Some characterizations of fitness. It will be useful to introduce the fol- lowing notation: for a sublocale S of a general L set
S′ =↓(S{1})(={x∈L|νS(x)̸=1}). Lemma. For any sublocale S and any c ∈ L, S ⊆ o(c) iff νS(c) = 1.
Proof. If νS(c) = 1 then for s ∈ S, by (10.2.1) and 1.3.1(H2), c→s = νS(c)→s = 1→s = s.
If s = νS(c) ̸= 1 then c ≤ s and c→s = 1 ̸= s, by 1.3.1(H3).