I quote from wiki: "The classification theorem of closed surfaces states that any connected closed surface is homeomorphic to some member of one of these three families:
the sphere;
the connected sum of g tori, for $g\geq 1$;
the connected sum of k real projective planes, for $k\geq 1$."
What if we drop the "second countable" requirement? E.g. using the long line $L$, we can construct surfaces like $L^2$ and $S^1\times L$, though these are not closed.
How to classify all Hausdorff, connected, closed, 2-dimensional topological manifold?