Let be distinct, and put . Then there are distinct with , and therefore . Since is an open subset of , then there must be open sets such that , where and ; this is because is a basis for . Notice that and must be disjoint, because otherwise we would have , a contradiction. So are disjoint neighborhoods of , respectively. It follows that is Hausdorff.

EDIT: The latex software is acting up, but you can quote my post to see the code.