Topological space

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 . 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 .

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