find all the Hausdorff topologies on 123SolutionIn a Hausdor

find all the Hausdorff topologies on {1,2,3}

Solution

In a Hausdorff topology, all singletons are closed.

Hence {1},{2},{3} are closed.

So {2,3}, {3,1}, {1,2} are open.

As finite intersections of open sets are open, it follows that the only Hausdorff topology is the discrete topology .That is all subsets are open.