1
$\begingroup$

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?

  • 0
    What do you mean by "second conutable" in the title. – 2017-01-23
  • 1
    Every closed 2-dimensional manifold is second countable, so the classification is identical whether or not one troubles to mention second countability explicitly. – 2017-01-24
  • 0
    @ Lee: so you mean, hausdorff + compact + locally euclidean imply second countable? – 2017-01-24

0 Answers 0