Thread: Prove: The intersection of a finite collection of open sets is open in a metric space

I want to prove that the intersection of a finite collection of open sets is open in a metric space with metric p.

Definitions of open and closed:
A subset M of a metric space X is said to be open if it contains a ball about each of its points. A subset k of X is said to be closed if its complement (in X) is open.

I found several threads about this subject, but none of those threads offer a profound proof.

Originally Posted by Boyd

I want to prove that the intersection of a finite collection of open sets is open in a metric space with metric p.

This is a standard proof by induction.
First show that if two open sets have a point in common, say x, then there is a ball which is a subset of both open sets.
So the whole proof turns on proving that the intersection of two balls is open.