Empty Set is Element of Topology

From ProofWiki
Jump to navigation Jump to search

Theorem

Let $\left({X, \tau}\right)$ be a topological space.

Then $\varnothing$ is an open set of $\left({X, \tau}\right)$.


Proof

Axiom $\left({O1}\right)$ for a topology states that:

$\displaystyle \forall \mathcal A \subseteq \tau: \bigcup \mathcal A \in \tau$

By Empty Set is Subset of All Sets, we have that $\varnothing \subseteq \tau$.

Hence, by Union of Empty Set, we have:

$\displaystyle \varnothing = \bigcup \varnothing \in \tau$

$\blacksquare$