Empty Set is Element of Topology

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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:

$\displaystyle \O = \bigcup \O \in \tau$

$\blacksquare$