Empty Set Satisfies Topology Axioms

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:

$\ds \bigcup \tau = \O \in \tau$

Thus Open Set Axiom $\paren {\text O 1 }$: Union of Open Sets 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 $\paren {\text O 2 }$: Pairwise Intersection of Open Sets is satisfied.

$\Box$

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

By definition $\O \in \tau$ and so Open Set Axiom $\paren {\text O 3 }$: Underlying Set is Element of Topology is satisfied.

$\Box$

All the open set axioms are fulfilled, and the result follows.

$\blacksquare$