The 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.
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 both and , is contained in both and contains every topology on X which is weaker than both and .
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 .