sublocale

A sublocale is a generalization of a subspace in the framework of point-free topology, encapsulated within the theory of locales. In the category of locales, a sublocale of a given locale X is a subobject of X. When X is described by a frame L (the lattice of opens of X under the duality between locales and frames), sublocales can be described concretely in terms of nuclei.

A nucleus on a frame L is a monotone, inflationary, idempotent map j: L → L that preserves finite meets. The fixed-point elements of j (those a with j(a) = a) form a subframe of L, and this subframe serves as the frame of opens of a sublocale of X. In this sense, sublocales are classified by nuclei, and each sublocale corresponds to a subframe determined by a suitable closure-like operator. Equivalently, sublocales can also be described as certain quotients of L by congruences, yielding a dual viewpoint in which sublocales and quotient locales correspond to each other.

Open and closed sublocales are common special cases. An open sublocale and a closed sublocale arise from particular choices of nuclei or congruences and play roles analogous to open and closed subspaces in classical topology. However, not every sublocale comes from a subset of points; some sublocales reflect purely point-free structure that has no direct topological subspace counterpart.

In summary, sublocales provide a flexible, categorical way to talk about substructures within a locale, capturing both subspace-like phenomena and more refined point-free distinctions. They are central to constructions in locale theory, topos theory, and constructive approaches to topology.