Empty Set is Element of Topology

Theorem

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

Then the empty set $\O$ is an open set of $T$.

Proof

By Empty Set is Subset of All Sets:

$\O \subseteq \tau$

By Union of Empty Set:

$\O = \bigcup \O$

By Open Set Axiom $\paren {\text O 1 }$: Union of Open Sets:

$\O \in \tau$

$\blacksquare$