Compact connected surfaces with $\chi=0$ are the torus, Klein bottle, annulus, or Möbius band according to boundary and orientability.Suggest a slogan
Let \(\Sigma\) be a compact connected surface with Euler characteristic \(\chi(\Sigma) = 0\). If \(\Sigma\) is closed, then \(\Sigma\) is homeomorphic to the torus \(T^2\) or to the Klein bottle \(K\) [@Mun00]; [@Hat02]. If \(\Sigma\) has nonempty boundary, then \(\Sigma\) is homeomorphic to the annulus \(S^1\times I\) or to the Möbius band.
Which surfaces have Euler characteristic 0?