Classification of compact surfaces without boundary

Every connected compact surface without boundary is homeomorphic to exactly one surface in the following list:

1. the sphere \(S^2\) (the orientable genus-\(0\) surface);
2. for each \(g\ge1\), the orientable surface \[ \Sigma_g=T^2\#\cdots\#T^2 \] with \(g\) torus summands;
3. for each \(k\ge1\), the nonorientable surface \[ N_k=\mathbb{RP}^2\#\cdots\#\mathbb{RP}^2 \] with \(k\) projective-plane summands.

The orientable and nonorientable families do not overlap. Within each family the genus is a homeomorphism invariant; for example \[ \chi(\Sigma_g)=2-2g, \qquad \chi(N_k)=2-k. \] A compact surface without boundary that is not connected is a finite disjoint union of such connected surfaces.