TITLE: A non-Hausdorff space with a Hausdorff subspace
QUESTION [8 upvotes]: Can anyone give an example of a non-Hausdorff space that contains a Hausdorff subspace?

REPLY [20 votes]: A point is Hausdorff, so any non-Hausdorff space gives examples.

REPLY [19 votes]: It is enough to consider the line with two origins, and than consider one 
of this lines as a subspace.