Manifold

The surface of the Earth requires (at least) two charts to include every point. Here the globe is decomposed into charts around the North and South Poles.

Figure 1: The four charts each map part of the circle to an open interval, and together cover the whole circle.

Figure 2: A circle manifold chart based on slope, covering all but one point of the circle.

In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, each point of an n-dimensional manifold has a neighbourhood that is homeomorphic to the Euclidean space of dimension n.

One-dimensional manifolds include lines and circles, but not figure eights (because they have crossing points which are not locally homeomorphic to Euclidean 1-space). Two-dimensional manifolds are also called surfaces. Examples include the plane, the sphere, and the torus, which can all be embedded (formed without self-intersections) in three dimensional real space, but also the Klein bottle and real projective plane which will always self-intersect when immersed in real space.

单词	Regular domain
释义	Regular domain 英语百科 Manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, each point of an n-dimensional manifold has a neighbourhood that is homeomorphic to the Euclidean space of dimension n. One-dimensional manifolds include lines and circles, but not figure eights (because they have crossing points which are not locally homeomorphic to Euclidean 1-space). Two-dimensional manifolds are also called surfaces. Examples include the plane, the sphere, and the torus, which can all be embedded (formed without self-intersections) in three dimensional real space, but also the Klein bottle and real projective plane which will always self-intersect when immersed in real space.
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Regular domain

Manifold