Locally Hausdorff not implies Hausdorff

Statement
A locally Hausdorff space need not be Hausdorff.

Related facts

 * Hausdorffness is closure-local
 * Hausdorffness is hereditary

Proof
An example is the line with two origins, which illustrates the more general fact that a locally Euclidean space need not be Hausdorff, and hence need not be a manifold.