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 the definition of topology, $T$ is open in $T$.

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

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

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