Empty Set is Closed/Topological Space

Theorem
Let $T$ be a topological space.

Then $\varnothing$ is closed in $T$.

Proof
From the definition of closed set, $U$ is open in $T$ iff $T \setminus U$ is closed in $T$.

From Topological Space is Open and Closed in Itself, $T$ is open in $T$.

From Set Difference with Self is Empty Set, we have $T \setminus T = \varnothing$, so $\varnothing$ is closed in $T$.