Since is the set of all points such that every open neighborhood of intersects , this can equivalently be written as “every open neighborhood of every point intersects ”, or equivalently “every inhabited open set intersects ”, i.e. is inhabited for all inhabited open sets .
Contraposing this, we obtain another equivalent definition “the only open subset not intersecting is the empty set”, or “if for some open set , then ”. This is the definition usually given when is a locale: a nucleus is dense if (since is the union of all opens whose “intersection with ” is ).
If i: and are dense subspace inclusions, then so is the composite .
If is a dense subset of topological space and is connected, so is .
In locale theory, we have the curious property that any intersection of dense subspaces is still dense. (This of course fails rather badly for topological spaces, where the intersection of all dense topological subspaces is the space of isolated point?s.) One consequence is that every locale has a smallest dense sublocale, the double negation sublocale.
In constructive mathematics, the law of contraposition is not an equivalence, so we obtain two inequivalent notions of density:
Of course, strong density implies weak density, since emptiness is non-inhabitation (whereas inhabitation is stronger than non-emptiness). The two notions of density are related dually to the corresponding notions of closed subspace: is strongly dense iff its weak closure is all of , and weakly dense iff its strong closure is all of .
Note that the usual notion of density for sublocales is an analogue of weak density, and could be called such. There is also a notion of strong density for sublocales. Since strong density refers to inhabited sets, one might expect strong density for sublocales to refer to positive elements, and thus only be sensible for overt locales; but in fact it can be reformulated to make sense in all cases.
With classical logic, every truth value is either or , and we have (and any nucleus satisfies ) while . Thus classically strong and weak density coincide. To see that this is really a notion of strong density, we prove:
If is a sublocale such that and are both overt, then is strongly dense if and only if for any positive open , the intersection is also positive.
First suppose is strongly dense, and let be positive. Let be the truth value of the statement “ is positive”. We want to show that is true, for which it suffices to show that is positive, since then its covering would be inhabited and thus would be true. And since is positive, it suffices to show .
Now since is strongly dense, , which is to say that . By adjointness, therefore, to show it suffices to show . Now since is overt, can be covered by positive opens, so it suffices to show that for any positive we have . But if is positive, then is also positive, i.e. is true, and thus , which contains .
Now suppose conversely that for any positive , is also positive, and let be any truth value; we must show . Since is overt, can be covered by positive opens, so it suffices to show that for any positive we have . But by adjointness, is equivalent to , and by assumption is also positive. Thus, is positive, which means that is true, and hence and so .
Since spatial locales are overt, and their positivity predicate coincides with inhabitedness, we have in particular:
If is a subspace of a topological space, then is strongly dense as a topological subspace if and only if it is strongly dense as a sublocale.
Strongly dense sublocales are discussed in
Sketches of an Elephant, C1.1 and C1.2
Peter Johnstone, Fiberwise separation axioms for locales