nLab
dense subspace

Contents

Idea

Given a topological space (or locale) AA, a subspace XX of AA is dense if its closure is all of AA: cl(X)=Acl(X)=A.

Remark

Conversely, a subset XAX\subseteq A of a topological space AA is called nowhere dense if Int(cl(X))=Int(cl(X))=\emptyset.

Properties

  • If XAX\subseteq A is a dense subset of topological space AA, then XYX\cap Y\neq\emptyset for all non-empty open sets YY.

  • If XAX \subseteq A is a dense subset of a topological space AA and f:ABf: A \to B is an epimorphism, then the image f(X)f(X) is dense in BB.

  • If i: XAX \hookrightarrow A and j:ABj: A \hookrightarrow B are dense subspace inclusions, then so is the composite ji:XBj \circ i: X \to B.

  • If XAX\subseteq A is a dense subset of topological space AA and XX is connected, so is AA.

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

  • 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.

Revised on August 17, 2015 22:56:45 by Todd Trimble (67.81.95.215)