Let be a collection of open sets in If is finite, then the intersection is also an open set. Here is a proof:
Suppose For each let be a ball of some positive radius around which is contained entirely inside Then the intersection of the is a ball around which is contained entirely inside the intersection, so the intersection is open.
Here a ball around is a set ( a positive real number) consisting of all points such that In it is an open disk centered at of radius
Where does this proof go wrong when is infinite?