Equivalence of Definitions of Nowhere Dense

Theorem

Let $T = \struct {S, \tau}$ be a topological space.

Let $H \subseteq S$.

The following definitions of the concept of Nowhere Dense are equivalent:

Definition 1

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

$\paren {H^-}^\circ = \O$

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

Definition 2

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

$H^-$ contains no open set of $T$ which is non-empty

where $H^-$ denotes the closure of $H$.

Proof

$(1)$ implies $(2)$

Let $H$ be nowhere dense in $T$ by definition $1$.

Then by definition:

$\paren {H^-}^\circ = \O$

Hence by definition of interior:

the union of all subsets of $H$ which are open in $T$.

But this union is empty.

Hence all subsets of $H$ which are open in $T$ must themselves be empty.

Thus $H$ is nowhere dense in $T$ by definition $2$.

$\Box$

$(2)$ implies $(1)$

Let $H$ be nowhere dense in $T$ by definition $2$.

Then by definition:

$H^-$ contains no open set of $T$ which is non-empty

where $H^-$ denotes the closure of $H$.

Hence the union of all subsets of $H$ which are open in $T$ must itself be empty.

Thus $H$ is nowhere dense in $T$ by definition $1$.

$\blacksquare$