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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?

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marked as duplicate by Dap, user99914, Claude Leibovici, Arnaud Mortier, Mostafa Ayaz Feb 15 '18 at 20:42

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    $\begingroup$ No this "scheme-like" definition does not implies Hausdorffness and second-countability because these properties are not local. In fact many authors does not require their manifold to be second-countable (for example $\mathbb{R}$ with the discrete topology may be considered as a manifold). It seems that one can even drop the Hausdorff hypothesis thought I never saw someone not using it. $\endgroup$ – Roland Feb 13 '18 at 13:58