Empty Set Satisfies Topology Axioms

Theorem
Let $T = \left({\varnothing, \left\{{\varnothing}\right\}}\right)$ where $\varnothing$ is the empty set.

Then $T$ satisfies the open set axioms for a topological space.

Proof
We proceed to verify the open set axioms for $\left\{{\varnothing}\right\}$ to be a topology on $\varnothing$.

Let $\tau = \left\{{\varnothing}\right\}$.

$\left({O1}\right):$ Union of Open Sets
By Union of Empty Set:
 * $\displaystyle \bigcup \tau = \varnothing \in \tau$

Thus open set axiom $\left({O1}\right)$ is satisfied.

$\left({O2}\right):$ Pairwise Intersection of Open Sets
From Intersection with Empty Set:
 * $\varnothing \cap \varnothing = \varnothing \in \tau$

and so open set axiom $\left({O2}\right)$ is satisfied.

$\left({O3}\right):$ Set Itself
By definition $\varnothing \in \tau$ and so open set axiom $\left({O3}\right)$ is satisfied.