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:
 * $$\bigcup \varnothing = \varnothing$$

Hence the result.