There is a well known result that every one dimensional topological manifold without boundary is homeomorphic either to the circle or to the whole real line. However there is one detail hidden: manifold is understood to be second countable (or paracompact). If we drop this assumption it becomes possible to construct different example, so called _open long line_ or _Aleksandroff line_. It is defined as $\omega_1 \times [0,1) \setminus \{(0,0)\}$ with suitable order topology. What might be surprising, is that replacing $\omega_1$ by bigger ordinal does not produce manifold anymore (this would produce points with uncountable neighbourhood system). There is also a variant of long line "in both directions". So the natural question is: if we drop the assumption for (one dimensional) manifolds to be second countable, is it possible to characterise all of them?$
Edit: what about two dimensional case?