Empty Set is Element of Topology

Theorem
Let $\struct {X, \tau}$ be a topological space.

Then $\O$ is an open set of $\struct {X, \tau}$.

Proof
Axiom $(\text O 1)$ for a topology states that:
 * $\displaystyle \forall \AA \subseteq \tau: \bigcup \AA \in \tau$

By Empty Set is Subset of All Sets, we have that $\O \subseteq \tau$.

Hence, by Union of Empty Set, we have:
 * $\displaystyle \O = \bigcup \O \in \tau$