The following definition of topological manifold is given in Lee's Introduction to topological manifolds (2000) on page 33:
A topological manifold is a second countable Hausdorff space that is locally Euclidean.
I omitted the dimension since it's not relevant to my question so it's not a faithful quote.
My question is: Doesn't Hausdorffness follow from being locally Euclidean? Pick $x,y$ in $M$. Then for each point there exists an open set homeomorphic to an open subset of $\mathbb R^n$. Let's call them $U_x, U_y \subset M$. Then either they are already disjoint or if they intersect, by Euclidenaity, we can pick smaller open sets inside them so that the smaller sets don't intersect. So we have a Hausdorff space.
Question 1: What am I missing? Why does this not work?
Question 2: I guess we have to require second countability because otherwise the disjoint union $\bigsqcup_{r \in \mathbb R \setminus \mathbb Q} (0,1)$ would be a $1$-manifold (that doesn't admit a countable basis) . Is that a correct counter example?
Thanks for your help.