Genus

Remarks

The genus is finite and satisfies $\chi(S) = 2 - 2g(S)$, where $\chi$ is the Euler characteristic. By the classification of surfaces, $S$ is homeomorphic to the sphere if $g = 0$, and to the connected sum of $g$ tori if $g \ge 1$.

Example: for the torus $T^2 = S^1 \times S^1$, the curve $c_1 = S^1 \times \set{1}$ has connected complement $S^1 \times (S^1 \setminus \set{1})$, an open annulus. Any two disjoint simple closed curves in the torus disconnect it, so $g(T^2) = 1$, in accord with $\chi(T^2) = 0$. Non-example of a non-separating curve: on the sphere $S^2$ every simple closed curve separates the sphere into two pieces, by the Jordan curve theorem, so $g(S^2) = 0$.

For a closed non-orientable surface, the non-orientable genus is the number $k$ of copies of $\R P^2$ in its connected sum decomposition; then $\chi = 2 - k$.