Page 44 - Textos de Matemática Vol. 41

30 Chapter 5. Sublocales, that is, generalized subspaces
Note. It has been pointed out to us by P. T. Johnstone that the first short proof of the distributivity in the vein of the above proposition is due to Dana Scott — see also [38].
5.5. Subspaces in this context. Let X be a space and let A be a subspace of X with jA : A ⊆ X the corresponding embedding. Then we have the frame homomorphism
Ω(jA) : Ω(X) → Ω(A),
and the resulting sublocale embedding jA : Lc(A) → Lc(X). We have
j A ( U ) = { V ∈ L c ( X ) | V ∩ A = U } . Thus, the subspace A is in Lc(X) represented by the sublocale
A = jA[Lc(A)]
(an open set U in A is represented by the maximal open extension V in X that
gives U = V ∩ A).
It should be noted that a sublocale of Lc(X) is not necessarily (induced
by) a subspace. Also, in general we have only
A ∩ B ⊆ A ∩ B,
not always an equality. In particular, complemented subspaces A, B of X are not always complemented as sublocales in Lc(X). For instance (see 6.3.4 below) the disjoint rationals and irrationals in the space R of reals have a very substantial (although non-spatial) intersection in Lc(R).
Remark. It should be noted that the representation of subspaces as sublocales is not always quite exact. It can happen (even in the T0-case) that A = B while A ̸= B.
The necessary and sufficient condition for correctness of our representation is that the space satisfies a very weak separation axiom:
(TD): for each x ∈ X there is an open U ∋ x such that also U {x} is open. One has
(T1) ⇒ (TD) ⇒ (T0) and none of these implications can be reversed [2].
5.6. The congruence lattice. Recall 5.3.2. Sometimes it is of advantage to work with the (dual) locale (frame) of congruences on L. It will be denoted by
C(L).