Surface

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 (possibly infinitely many) punctures. The precise statement is that one removes a closed, totally disconnected set from a connected, compact surface. 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)