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)

In music surface is the character of the salient immediate detail, it may be thought of as content plus the more general definition of texture, and is often opposed to form.