Surface (topology)
As Surface designates one into that mathematical subsections that Differentialgeometrie and Topology a two-dimensional Diversity. One wins examples in the three-dimensional area, if one the surfaces of Solid bodies regarded. The surfaces of fluids objects how Rain drop or Seifenblasen represent an idealization.
Beyond the simple mathematical definition for example the surface is appropriate for one Flake, those is very finely structured.
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Definition
Surfaces in the topology
In the topology a surface is a topological two-dimensional Diversity. One can define it therefore as follows:
- A surface is in Hausdorffraum, that second Abzaehlbarkeitsaxiom fulfilled and in the each point one to open circular disk <math>D^{\circ2}</math> or to the half plane <math> \lbrace (x_1,x_2) \in \mathbb R^2 \mid x_1 \geq 0 \rbrace</math> homoeomorphe Environment possesses.
An alternative definition, done without the definition by cases reads:
- A surface is a Hausdorffraum, that the second Abzaehlbarkeitsaxiom fulfills and in the each point one to closed circular disk <math>D^{2}</math> homoeomorphe environment possesses.
Those points, the one environment homoeomorphe to the open circular disk possess, calls one internal points of the surface and the others than peripheral points. The quantity of the internal points forms the inside <math>F^ \circ</math> the surface, while the quantity of the peripheral points the edge <math> \partial F</math> forms for the surface.
A surface does not have peripheral points, thus 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 a Hausdorffraum, that the second Abzaehlbarkeitsaxiom fulfilled and in the each point one to the open circular disk <math>D^{\circ2}</math> homoeomorphe 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 is called closed, if it compactly is.
- openly
- A unberandete surface is called open, if it is not compact.
Examples
- Klein bottle
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See also
- Sex (surface)
- Riemann surface
- Complete-cash surface
- Euler characteristic
- Euler polyhedron set
- Sex (surface)
- Geodaete
- Fundamentalpolygon
- Classification set for 2-Mannigfaltigkeiten
Literature
- Ralph Stoecker, Heiner Zieschang: Algebraic topology. Teubner, Stuttgart 1988, ISBN 3-519-02226-5
- William S. Massey: Algebraic Topology: At Introduction. 1. Edition. Springer, Berlin 1967, ISBN 3-540-90271-6
