Conversely, a subset of a topological space is called nowhere dense if .
If is a dense subset of topological space , then for all non-empty open sets .
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.