open sets 2 (the intesection is open)

We will give a general proof by which, given open sets $O_1,O_2$  in a metric space $\Omega$ ,
the intersection of $O_1$  and $O_2$  is open.

A intersection of a finite number of open sets is open. Namely,
If $O_1,O_2,\cdots , O_k\in\Omega$ , $O_1\cap O_2,\cap \cdots  \cap O_k\in\Omega$  is open.

If a element $e$  is in $\cap_{i=1}^k O_i$ , as $e\in O_i$ for all $i=1,\cdots ,k$ ,
there is a $\delta_i>0$  such that open ball $B(e, \delta_i)$  is in $O_i$ .
We choose $\delta=\min(\delta_1,\cdots\delta_k)$ .
Then, since $B(e, \delta)\subset B(e, \delta_i)$ for all $i=1,\cdots ,k$ ,
$B(e, \delta)\subset \cap_{i=1}^k O_i$ .

Thus, we obtain the result which we want. We should remember that a set $O$  is open
if and only if, for any element $e\in O$ , there is a $\delta>0$  such that the open ball $B(e,\delta)\subset O$ .