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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)

Non-compact connected surfaces are just these with one or more punctures (missing points). A surface can be embedded in R3 if it is orientable or if it has at least one puncture. All can be embedded in R4. 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)

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