Empty Set Satisfies Topology Axioms

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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 $\tau = \set \O$.


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

By Union of Empty Set:

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

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

$\Box$


Open Set Axiom $\paren {\text O 2 }$: Pairwise Intersection of Open Sets

From Intersection with Empty Set:

$\O \cap \O = \O \in \tau$

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

$\Box$


Open Set Axiom $\paren {\text O 3 }$: Complete Set is Element of Topology

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

$\blacksquare$