Curves and Surfaces > Surfaces > Special Surfaces > Topology
Surfaces with special topological properties.
Orientability Some surfaces can be oriented; others cannot.
Compactness A bounded closed surface is said to be compact.
Completeness A surface which cannot be extended.
Covering Surfaces A family of surfaces which can be used to cover Rn.

Orientability

Many surfaces can be assigned an orientation by choosing a direction for the surface normals. For a few special surfaces, however, this process can break down and two normals can be assigned opposite directions although they are quite near each other. The global nature of the surface, then, makes it possible to assign an orientation. Examples of such surfaces are the Möbius strip, the Klein bottle, and Steiner's Roman surface.

Compactness

A surface which is both closed and bounded is said to be compact. The sphere is an excellent example.

Completeness

A surface is said to be complete if its geodesics can be extended to the entire real number line, remaining always on the surface. A direct consequence of completeness, which gives a better sense of what it means intuitively, is non-extendibility. That is, when a regular surface S is complete, there is no larger surface S' of which S is a subset. One example of a complete surface is one sheet of the cone, minus its vertex.

Covering Surfaces

If a family of surfaces can be used to cover Rn, it is said to be covering. For example, a family of spheres of increasing radii can cover Rn.