A sublocale is a subspace of a locale.
It is important to understand that, even for a topological locale (which can be identified with a sober topological space), most sublocales of are not topological. Specifically, we have an inclusion function which, while injective, is usually far from surjective.
Let be a locale, which (as an object) is the same as a frame.
A sublocale of is a regular subobject of in Loc, the category of locales. Equivalently, it is a regular quotient of in Frm, the category of frames.
A sublocale is given precisely by a nucleus on the underlying frame. This is a function from the opens of to the opens of satisfying the following identities:
In other words, a sublocale of is given by a meet-preserving monad on its frame of opens.
The precise reasons why nuclei correspond to quotient frames (and hence to sublocales) is given at nucleus. But the interpretation of the operation is this: we identify two opens if they ‘agree on the sublocale’. Given an open , there will always be a largest open that is identified with , so we can also describe a subspace of a locale as an operation that maps each open to its largest representative open in the sublocale. This map is the nucleus .
Of course, every locale is a sublocale of itself. The corresponding nucleus is given by
so every open is preserved in this sublocale. Conversely, every locale has an empty sublocale, given by
Suppose that is an open in the locale . Then defines an open subspace of , also denoted , given by
so is the largest open which agrees with on . also defines a closed subspace of , denoted (or any other notation for a complement), given by
so is the largest open which agrees with except on . If is topological, then every open sublocale of is also topological; the same goes for closed sublocales, assuming excluded middle. Even in constructive mathematics, however, open and closed sublocales are complements in the lattice of sublocales.
The double negation sublocale of , denoted , is given by
This is always a dense subspace; in fact, it is the smallest dense sublocale of . (As such, even when is topological, is rarely topological; in fact, its only points are the isolated point?s of .)