Empty Set is Nowhere Dense

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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 $T$.

$\blacksquare$


Sources