Empty Set is Nowhere Dense

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Theorem

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

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


Proof

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

From Closed Set Equals its Closure:

$\varnothing^- = \varnothing$

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

From Empty Set is Element of Topology, $\varnothing$ 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 $T$.

$\blacksquare$


Sources