The intersection of topologies on a fixed set is again a topology
Let be a family of topologies on a set .
Then the union is again a topology on . #m/thm/topology
Proof
Since for all , it follows that .
Let be a family of open subsets under .
Then the union and hence for all ,
wherefore .
Similarly let be a finite family of open subsets under .
Then the intersection and hence for all ,
wherefore .
Therefore is a topology on .