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 .