Definition:Nowhere Dense

Definition

Let $T = \left({S, \tau}\right)$ be a topological space.

Let $H \subseteq S$.

Then $H$ is nowhere dense in $T$ iff:

$\left({H^-}\right)^\circ = \varnothing$

where $H^-$ denotes the closure of $H$ and $H^\circ$ denotes its interior.

That is, $H$ is nowhere dense in $T$ iff the interior of its closure is empty, i.e. it is "all boundary".

Alternative Definition

A set $H$ is nowhere dense in $T$ iff $H^-$ contains no open set of $T$ which is not empty.

These definitions can be seen to be directly equivalent from the definition of interior as the union of all subsets of $H$ open in $T$.

Example

The set $\left\{{\dfrac 1 n: n \in \N}\right\}$ is nowhere dense in $\R$.

Also see

• Everywhere dense
• Dense-in-itself

• Results about topological denseness can be found here.

Sources

• Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (1970)... (previous)... (next): $\text{I}: \ \S 1$: Countability Properties