A topological space X, more precisely its associated frame Ω(X) of open sets, has typically more sublocales than subspaces. We will analyze some of Simmons (1980) and Niefield and Rosenthal (1987) results concerning sublocales induced by subspaces.
The first result concerning the question when every sublocale is (induced by) a subspace was presented by Simmons. More precisely, Simmons proved a necessary and sufficient condition for the lattice of sublocales being Boolean which is slightly different: if sublocales are in a one-to-one correspondence with subspaces (subsets) they do form a Boolean algebra, while the converse implication does not hold. Later, Niefield and Rosenthal treated more directly the question of every sublocale being spatial and gave a characterization of the respective frames. In both cases, however, the question of the one-to-one correspondence between subspaces and sublocales is somehow circumvented. While, as we have already pointed out, typically one has more sublocales than subspaces, there are already cases when there are less sublocales than subspaces. Namely, it turns out that unless the space in question satisfies a certain very weak separation condition TD, representation of subspaces of X by sublocales of Ω(X) is imperfect: distinct subspaces can induce the same sublocale.
Concentration on the properties of TD-spaces and technique of sublocales in this context allows us to present a simple, transparent and choice-free proof of the scatteredness theorem: for a TD-space X, the sublocales are in a one-to-one correspondence with subspaces iff the X is scattered.
(This is joint work with Ales Pultr).