Depends on the definition of topology you are using. Typically, the definition only requires that the intersection of two elements of the topology is in the topology. It typically does not expressly state that the intersection of any finite number of elements in the topology is still in the topology.
To use induction, you know by definition that the intersection of two sets is in the set. For the induction step, suppose it is true for the intersection of n elements and prove it for n+1.
Also, if for all , then you have the discrete topology. Through unions, you get is the power set of . That is rarely the case. In the Euclidean case, we look at topologies where no single point sets are in the topology.