Empty Set is Nowhere Dense

Theorem
Let $T = \left({X, \vartheta}\right)$ be a topological space.

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

Proof
From Open and Closed Sets in a Topological Space, $\varnothing$ is closed in $T$.

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

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

Similarly, also from Open and Closed Sets in a Topological Space, $\varnothing$ is open in $T$.

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

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

So:
 * $\left({\varnothing^-}\right)^\circ = \varnothing$

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