TITLE: Why do we need Hausdorff-ness in definition of topological manifold?
QUESTION [32 upvotes]: Suppose $M^n$ is a topological manifold, then $M^n$ locally looks like $\mathbb{R}^n$. 
$M^n$ is locally Hausdorff, since $\mathbb{R}^n$ is Hausdorff and Hausdorff-ness is a topological invariant.
All I want to understand is: 
1) Does locally Hausdorff-ness imply Hausdorff-ness? (I can not imagine a locally Hausdorff topological space that is not globally Hausdorff) 
2) Why do we need Hausdorff-ness in definition of the topological manifold? Is locally Hausdorff-ness not sufficient? If not, why? 
Can anyone say anything that might be helpful?

REPLY [2 votes]: We know that the limit of sequences in a Hassdorf space is unique. This is one of the most important reasons why we assume such condition. For example Lie derivative or other derivations are defined by a limit.