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.
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.
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 except on . If is topological, then every open or closed sublocale of is also topological.
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.)