# Surface one calls

surface ( topology) in the mathematical subsections of the Differentialgeometrie and topology a two-dimensional diversity. One wins examples in the three-dimensional area, if one regards the surfaces of solid bodies. The surfaces of fluids objects such as rain drops or Seifenblasen represent an idealization.

Beyond the simple mathematical definition lies for example the surface of a flake, which is very finely structured.

## definition

### surfaces in the topology

in the topology is a surface a topological two-dimensional diversity. One can define it therefore as follows:

A surface is fulfilled to that a Hausdorffraum, the second Abzählbarkeitsaxiom and in each point one to the open circular disk [itex] D^ {\ circ2}< /math> or to the half plane [itex] \ lbrace (x_1, x_2) \ in \ mathbb R^2 \ avoided x_1 \ geq 0 \ rbrace< /math> homöomorphe environment possesses.

An alternative definition, which reads done without the definition by cases:

A surface is fulfilled to that a Hausdorffraum, the second Abzählbarkeitsaxiom and in each point one to the closed circular disk [itex] D^ {2}< /math> homöomorphe environment possesses.

One calls those points, which possess an environment homöomorphe to the open circular disk, internal points of the surface and the others than peripheral points. The quantity of the internal points forms the inside [itex] for F^ \ circ< /math> the surface, while the quantity of the peripheral points the edge [itex] \ partial F< /math> forms for the surface.

If a surface does not have peripheral points, then one of a unberandeten surface or surface without edge speaks. Otherwise one calls the surface bound or surface with edge.

For unberandete surfaces the above definition shortens:

A unberandete surface is fulfilled to that a Hausdorffraum, the second Abzählbarkeitsaxiom and in each point one to the open circular disk [itex] D^ {\ circ2}< /math> homöomorphe environment possesses.

### surfaces in the Differenzialgeometrie

in the Differenzialgeometrie is a surface a differentiable two-dimensional diversity. For the exact definition see differentiable diversity.

## mathematical attributes for unberandete surfaces

closed
a surface without edge means closed, if it is compact.
openly
a unberandete surface means open, if it is not compact.

## examples

 Commons: Surface - pictures, videos and/or audio files