Translation and analysis of words by artificial intelligence
On this page you can get a detailed analysis of a word or phrase, produced by the best artificial intelligence technology to date:
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- etymology
punctured surface - translation to russian
![disk surface]], square surface, and hemisphere surface. The boundaries are shown in red. All three of these are topologically equivalent to each other.](https://commons.wikimedia.org/wiki/Special:FilePath/SurfacesWithAndWithoutBoundary.svg?width=200 "disk surface]], square surface, and hemisphere surface. The boundaries are shown in red. All three of these are topologically equivalent to each other.")
![[[sphere]]](https://commons.wikimedia.org/wiki/Special:FilePath/SphereAsSquare.svg?width=200 "[[sphere]]")
![[[real projective plane]]](https://commons.wikimedia.org/wiki/Special:FilePath/ProjectivePlaneAsSquare.svg?width=200 "[[real projective plane]]")
![[[torus]]](https://commons.wikimedia.org/wiki/Special:FilePath/TorusAsSquare.svg?width=200 "[[torus]]")
![[[Klein bottle]]](https://commons.wikimedia.org/wiki/Special:FilePath/KleinBottleAsSquare.svg?width=200 "[[Klein bottle]]")
математика
поверхность с вынутой точкой
математика
открытая поверхность
математика
топологическая поверхность
Definition
Wikipedia
In the part of mathematics referred to as topology, a surface is a two-dimensional manifold. Some surfaces arise as the boundaries of three-dimensional solids; for example, the sphere is the boundary of the solid ball. Other surfaces arise as graphs of functions of two variables; see the figure at right. However, surfaces can also be defined abstractly, without reference to any ambient space. For example, the Klein bottle is a surface that cannot be embedded in three-dimensional Euclidean space.
Topological surfaces are sometimes equipped with additional information, such as a Riemannian metric or a complex structure, that connects them to other disciplines within mathematics, such as differential geometry and complex analysis. The various mathematical notions of surface can be used to model surfaces in the physical world.
