Every simply ordered set is a Hausdorff space in the order topology.
Since it is simply ordered, the relation is reflexive, anti-symmetric, and transitive. How can this coupled with the Hausdorff condition help show that it is the order topology?
Let be distinct. Since is a total ordering on we may assume without loss of generality that . We have two choices, either there exists with in which case take and , else is empty and take and . Regardless, and are disjoint neighborhoods of respectively. The conclusion follows.