dense subspace



Given a topological space (or locale) XX, a subspace of XX is dense if its closure is all of XX.


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 point-set topology, a space is separable if and only if it has a dense subspace with countably many points.

Revised on April 1, 2015 05:18:30 by Urs Schreiber (