Sublocale lattices

On Point-finiteness in Pointfree Topology

“…However, if L is scattered, that is, if every congruence of L is complemented [15] or, equivalently, [16] the conditions in 3.1(4) and 3.1(1) are equivalent, as for spaces: …”

“…Indeed each family of interior-preserving open covers of a space (X, T ) generates, in a canonical way (usually referred to as the Fletcher construction [8]) a transitive quasi-uniformity on X The partially ordered set S(X) is a complete lattice, much more complicated than its topological counterpart. Indeed, in the latter every element has a complement (which makes it a complete Boolean algebra) whilst in the former most elements are not complemented: in general, for each locale X, S(X) is a co-frame (that is, it satisfies the dual law of (1.1)) but it is not a frame unless X is scattered [16]; indeed, S ∧ i∈I S i = i∈I (S ∧ S i ) for all {S i | i ∈ I} ⊆ S(X) if and only if S is complemented [10]. However, there are sufficient complemented elements in order for S(X) to be generated by them (i.e.…”

“…and it is a cover of X if i∈I S i = X [16]. Similarly, we say that S is interior-preserving if for all J ⊆ I,…”

“…Further, S is locally finite [16] if there exists an open cover C of X such that for all C ∈ C there exists a finite set I C for which C ∧ j∈J S j = C ∧ j∈J∩I C S j for all J ⊆ I. The cover C is said to witness the local finiteness of S.…”

Remainders in pointfree topology

“…Thus, in the category of locales one should take the co-Heyting operator in SpLq as the natural substitute for the set-theoretic difference. This idea goes back to Isbell and Plewe [23,32,33] (cf. [29,VI.5]) and provides the right definition for the remainder of a locale and the corresponding concept of mapping remainder preservation.…”