Why do we need Hausdorff-ness in definition of topological manifold? 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?

I think the main reason behind the hausdorffness condition in the definition of topological manifold is that one needs to be able to do calculus on manifolds. More to the point, one needs to deal with convergent sequences on manifolds and here comes the necessity of including the hausdorffness condition.
