Empty Set is Element of Topology

Theorem

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

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

Proof

We have the axiom:

Any union of arbitrarily many elements of $\vartheta$ is an element of $\vartheta$

Let us take the union of no elements of $\vartheta$:

Then from Union of Empty Set:

$\displaystyle \bigcup \varnothing = \varnothing$

Hence the result.

$\blacksquare$