Empty Set is Element of Topology

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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:

$\ds \O = \bigcup \O$

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

$\O \in \tau$

$\blacksquare$