Infinite intersection of open sets.

In a question I'm working on it asked for an example of an infinite intersection of open sets that itself is open.

Just wondering – let's say that I were to take ${\mathbb R}^{2}$ with the usual metric defined on it. Then say the intersection the open balls centered on some arbitrary $x$ of radius less than $r$ such that $r>1$. Would this constitute a suitable answer? Thanks in advance!

No, your intersection would be the closed ball of radius $1$, which is not open.
You can just take the "constant family" $A_i=A$ for all $i$ in some (infinite) set of indices and $A$ an open set. If you want something non-constant, consider $A_n=(n,\infty)$ in $\mathbb{R}$.
Indeed, while your example is wrong, with a slight change it can be made into an construction of the second type: Instead of considering the open balls around $x$ with $r>1$, consider $r\ge 1$. Then all the open balls are supersets of the open ball with $r=1$, which is in the set, and therefore is the intersection of all those balls.