# Definition:Nowhere Dense/Definition 1

## Definition

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

Let $H \subseteq S$.

$H$ is nowhere dense in $T$ if and only if:

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

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

That is, $H$ is nowhere dense in $T$ if and only if the interior of its closure is empty.

Another way of putting it is that $H$ is nowhere dense in $T$ if and only if it consists entirely of boundary.