Empty Set Satisfies Topology Axioms

Theorem
Let $T = \struct {\O, \set \O}$ where $\O$ denotes the empty set.

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

Proof
We proceed to verify the open set axioms for $\set \O$ to be a topology on $\O$.

Let $\tau = \set \O$.

By Union of Empty Set:
 * $\displaystyle \bigcup \tau = \O \in \tau$

Thus open set axiom $\paren {\text O 1}$ is satisfied.

From Intersection with Empty Set:
 * $\O \cap \O = \O \in \tau$

and so open set axiom $\paren {\text O 2}$ is satisfied.

By definition $\O \in \tau$ and so open set axiom $\paren {\text O 3}$ is satisfied.