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?$

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?