Empty Set is Nowhere Dense

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

Then the empty set $\O$ is nowhere dense in $T$.

Proof
From Empty Set is Closed in Topological Space, $\O$ is closed in $T$.

From Closed Set Equals its Closure:
 * $\O^- = \O$

where $\O^-$ is the the closure of $\O$.

From Empty Set is Element of Topology, $\O$ is open in $T$.

From the definition (trivially) we also have that:
 * $\O^\circ = \O$

where $\O^\circ$ is the interior of $\O$.

So:
 * $\struct {\O^-}^\circ = \O$

and so by definition $\O$ is nowhere dense in $T$.