Open and Closed Sets in Topological Space

Theorem
Let $T$ be a topological space.

Then $T$ and $\varnothing$ are both open and 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 both open and closed in $T$.

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

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