Page 29 - Textos de Matemática Vol. 41

3.5. Spatial locales 15 The other equality is easier in the adjoint form:
Lc(λX)∗σ∗ (U)=λ−1[ΣU]={x|U X{x}}=U. Lc(X) X
Remark. From the equation σLc(X)Lc(λX ) = id we see that σLc(X) is onto. As it is one-one anyway (recall (3.2.1)) we have that
σLc(X) : LcPtLc(X) → Lc(X) is an isomorphism. (3.4.1) 3.5. Spatial locales. A locale (frame) L is said to be spatial if it is isomorphic
to a topology of a space.
3.5.1. Proposition. The following statements on a locale L are equivalent: (i) L is spatial,
(ii) σL : Lc(Pt(L)) → L is a complete lattice isomorphism, (iii) σL∗ : L → Lc(Pt(L)) is a complete lattice isomorphism, (iv) σL is onto,
(v) σL∗ is one-one.
Proof. The implications (ii)⇒(i), (ii)⇒(iv) and (iii)⇒(v) are obvious. By (3.2.1) any of (iv), (v) implies any of (ii), (iii), (iv), (v) (σL and σL∗ are monotone maps and any of (iv), (v) makes them inverse to each other, and hence complete lattice isomorphisms).
It remains to prove that (i)⇒(ii). Recall (3.4.1). If h : L → Lc(X) is an isomorphism we have the isomorphism σL = h−1 · σLc(X) · LcPt(h).
3.5.2. The equivalence (i) ⇔ (v) can be modified to another spatiality criterion. Proposition. A locale L is spatial if and only if each element a ∈ L is a meet of
meet-irreducible ones. Proof. σL∗ is one-one iff
a̸=b ⇒ Σa̸=Σb which can be reformulated as
ba ⇒ ΣbΣa.
T h u s, i f b a t h e r e i s a p o i n t p ∈ Σ b Σ a , t h a t i s , a ≤ p a n d b p . C o n s e q u e n t l y , a= {ppointsofL|a≤p}.