In mathematics, a surface is a 2-manifold. In what follows, all surfaces are considered to be second-countable (see the Topology Glossary) and without boundary.
Connected, compact surfaces can be divided into three infinite sequences:
- Orientable with characteristic 2-2n (spheres with n handles)
- Non-orientable with characteristic 1-2n (projective planes with n handles)
- Non-orientable with characteristic -2n (Klein bottles with n handles)
* * B B v v v ^ *>>>>>* *>>>>>* v v v ^ v v v v A v v A A v ^ A A v v A A v v A v v v ^ v v v v v v v ^ *<<<<<* *>>>>>* * * B Bsphere real projective plane Klein bottle torus (punctured: Möbius band) (sphere with handle)