Hausdorff condition in differential manifold definition

[Goldbeetle]

Dear all,
why is it needed in the diff manifold definition that the base set M is topologically Hausdorf ?
Since M is locally homeomorphic with Rn as metric space is Hausdorf, shouldn't this condition be automatically satisfied?
Thanks.
Goldbeetle

 

[zhentil]

The line with two origins is locally Euclidean, but not Hausdorff.

 

[Goldbeetle]

OK, true.