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.
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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 
.
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© Robert FERRÉOL 2017