next surface | previous surface | 2D curves | 3D curves | surfaces | fractals | polyhedra |

MANIFOLD

Other names: variety, given by Beltrami in 1869, multiplicity. |

An *n*-dimensional topological manifold is a topological space homeomorphic to the *n*-dimensional Euclidian space or half-space (i.e. all the points of which have a neighborhood homeomorphic to ^{ } or ) .

The points that have a neighborhood homeomorphic to the half-space form the "boundary" of the surface; a compact manifold without boundary is said to be *closed*, a non-compact manifold without boundary is said to be *open*.

Examples:

- the *n*-dimension Euclidian space is an open manifold.

- the *n*-dimensional sphere is a closed manifold, compactification of the previous one.

- the *n*-dimensional torus is a closed manifold.

The 1-dimensional manifolds are the (topological) curves, the 2-dimensional ones are the (topological) surfaces, and the 3-dimensional ones are the 3-manifolds.

The Whitney embedding theorem states that all *n*-dimensional manifolds can be embedded (i.e. represented without breaks or intersections) in .

next surface | previous surface | 2D curves | 3D curves | surfaces | fractals | polyhedra |

© Robert FERRÉOL 2017