Metric space/Structural properties of open subsets/Fact

Let ${}(M,d)$ be a metric space. Then the following properties hold.

1. The empty set ${}\emptyset$ and the total space ${}M$ are open.
2. Let ${}I$ be an arbitrary index set, and let ${}U_{i}$, ${}i\in I$, denote open sets. Then also the union

   $\bigcup _{i\in I}U_{i}$

   is open.

3. Let ${}I$ be a finite index set, and let ${}U_{i}$, ${}i\in I$, be open sets. Then also the intersection

   $\bigcap _{i\in I}U_{i}$

   is open.

Write a proof