Proving the intersection and union of two open sets are open in $\mathbb{C}$

Question

I am looking for a proof online that shows that $V\cap U$ and $V\cup U$ are open sets considering $V$ and $U$ are open while $V \subseteq \mathbb{C}$ and $U \subseteq \mathbb{C}$.

I have the definition for an open set $U \subseteq \mathbb{C}$ where $r(z) > 0$, $z \in U$ such that for any $w \in \mathbb{C}$:

$|w − z| < r(z) ⇒ w ∈ U$

Can anyone start me off or give a hint? A geometric approach would also be great. Much appreciated.

Answer 1

if $z \in V\cap U$ you have $r_1 >0$ and $r_2 >0$ as you use in the definition of open. Use $r:=min(r_1,r_2)$

if $z \in V\cup U$ it is even more trivial. You already know that there is a small disk contained in U (or V) containing z and hence contained in the union.