Open and Closed Sets in Topological Space

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

Then $S$ and $\O$ are both both open and closed in $T$.

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

From Underlying Set of Topological Space is Clopen, $S$ is both open and closed in $T$.

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

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