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 2-2
*n*(spheres with*n*handles) - Non-orientable with characteristic 1-2
*n*(projective planes with*n*handles) - Non-orientable with characteristic -2
*n*(Klein bottles with*n*handles)

Non-compact connected surfaces are just these with one or more punctures (missing points). A surface can be embedded in **R**^{3} if it is orientable or if it has at least one puncture. All can be embedded in **R**^{4}. To make some models, attach the sides of these (and remove the corners to puncture):

* * 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 B

sphere real projective plane Klein bottle torus (punctured: Möbius band) (sphere with handle)