Page 40 - Textos de Matemática Vol. 41

26 Chapter 5. Sublocales, that is, generalized subspaces
Notes. By (S1) a sublocale is always non-empty since 1 = ∅ ∈ S. This should not come as a surprise. The void locale O = {0 = 1} represents the void space (each non-void space has at least two open sets).
Also realize that the void locale O does not have any point in the sense of 2.3.
Further, by (S1) and (S2) each sublocale S of L is closed under the Heyting operation and meets, and hence it is a complete Heyting algebra, and therefore a locale (with the same meets and the same Heyting operation as in L).
Proposition. A subset S ⊆ L is a sublocale iff the embedding jS : S ⊆ L is a localic map.
Proof. Let S be a sublocale. We have already observed that S is a complete Heyting algebra and hence a locale. From (S1) we see, furthermore, that the embeddingjS :S→LhasaleftadjointjS∗ :L→S.
For each a ∈ L and s ∈ S we have
jS∗(a)→s = a→s (5.2.1)
(indeed: by (H∗),
x≤jS∗(a)→siffjS∗(a)≤x→siffa≤jS(x→s)=x→s iffx≤a→s).
Thus,
jS(jS∗(a)→s) = jS∗(a)→s = a→s = a→jS(s)
and hence, by 2.2.1, jS∗ is a frame homomorphism (2.2.1(a) being in this case trivial), and jS is a localic map.
Conversely, if jS is a localic map it preserves all meets and hence we have (S1). Finally, if s ∈ S and a is general, we have, by 2.2.1(b),
a→s = a→jS(s) = jS(jS∗(a) → s) ∈ S.
Remarks. (1) Conditions (S1) and (S2) appeared originally in Exercise II.2.3 of [35]. Note that the concept of a sublocale isreminiscent of that of an ideal in a ring: for a moment think of the (big) meet as of an additive structure and of the Heyting operation → as of a multiplication (cf. [54]).
(2) The factorization from 4.6 can be interpreted as stating that, for every localic map h : L → M, h[L] is a sublocale of M and h is decomposed into localic maps j·ewherej:h[L]⊆M ande(a)=h(a).
Corollary. A sublocale T of a sublocale S of a locale L is a sublocale of L. (The composition of the embeddings is again a localic map.)