Empty Set is Element of Topology

Theorem
Let $\left({X, \vartheta}\right)$ be a topological space.

Then $\varnothing$ is an open set of $\left({X, \vartheta}\right)$.

Proof
Axiom $\left({1}\right)$ for a topology states that:
 * $\displaystyle \forall \mathcal A \subseteq \vartheta: \bigcup \mathcal A \in \vartheta$

By Empty Set Subset of All, we have that $\varnothing \subseteq \vartheta$.

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