Empty Set Satisfies Topology Axioms

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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.

$\Box$


$\left({O2}\right):$ 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.

$\Box$


$\left({O3}\right):$ Set Itself

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

$\blacksquare$