Intersection of a finite collection of open sets is open

Prove that the intersection of a finite collection of open sets is open.

Easy... Given a point in the intersection, for every open set in the collection the point has a neighborhood from the basis of topology that lies inside. Take the intersection of these neighborhoods. It will lie inside the intersection set.

It's a part of the definition of topological space, by the way.

• What happens if we drop "finite"?