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
- $\ds \bigcup \tau = \O \in \tau$
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.
$\blacksquare$