Closure of Empty Set is Empty Set

Theorem

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

Then the closure of the empty set $\O$ in $T$ is $\O$.

Proof

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

The result follows from Closed Set equals its Closure.

$\blacksquare$