Wikipedia has that:
Scheme-theoretically, a manifold is a locally ringed space, whose structure sheaf is locally isomorphic to the sheaf of continuous (or differentiable, or complex-analytic, etc.) functions on Euclidean space.
I wonder if the locality assumption on the structure sheaf (every stalk is a local ring) implies Hausdorffness. If we let the line with two origins have the structure sheaf inherited from the quotient, won't the stalks at each of the origins be $\Bbb R(y)[x]_{(x)}$ and $\Bbb R(x)[y]_{(x)}$, which are local rings?
What about second countability?