Intersection of any two topologies on a non empty set is always topology on that set. While the union of two topologies may not be a topology on that set.

Example:
            Let
           
           
             is a topology on X.
             is not a topology on X.

Given two (and in fact any number of) topologies, on X there is a topology  which is weaker than bothand, is contained in bothand contains every topology on X which is weaker than bothand.
            Similarly there is a topology  which contains both and is the weakest in the sense that if is a topology which contains both and then.
            We write  and call as the topology generated by and. is different from the union  which may not be a topology. Here is the set theoretic union of the collection and.