nLab
dense subspace

Contents

Idea

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

Properties

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 March 31, 2013 04:20:55 by Urs Schreiber (89.204.155.146)