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Concept
Closed manifold
Summary
In mathematics, a closed manifold is a manifold without boundary that is compact.
In comparison, an open manifold is a manifold without boundary that has only non-compact components.
Examples
The only connected one-dimensional example is a circle.
The sphere, torus, and the Klein bottle are all closed two-dimensional manifolds. The real projective space RPn is a closed n-dimensional manifold. The complex projective space CPn is a closed 2n-dimensional manifold.
A line is not closed because it is not compact.
A closed disk is a compact two-dimensional manifold, but it is not closed because it has a boundary.
Properties
Every closed manifold is a Euclidean neighborhood retract and thus has finitely generated homology groups.
If M is a closed connected n-manifold, the n-th homology group H_{n}(M;\mathbb{Z}) is \mathbb{Z} or 0 depending on whether M is orientable or not. Moreover, the torsion subgroup of t
Official source
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