# nLab dense subspace

### Context

#### Topology

topology

algebraic topology

# Contents

## Idea

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

## Remark

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

## Properties

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

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

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

• If $X\subseteq A$ is a dense subset of topological space $A$ and $X$ is connected, so is $A$.

• 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)