Closed manifold

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 the (n-1)-th homology group $$H_{n-1}(M;\mathbb{Z}) $$ is 0 or $$\mathbb{Z}_2$$ depending on whether $$M$$ is orientable or not. This follows from an application of the universal coefficient theorem.

Let $$R$$ be a commutative ring. For $$R$$-orientable $$M$$ with fundamental class $$[M]\in H_{n}(M;R) $$, the map $$D: H^k(M;R) \to H_{n-k}(M;R)$$ defined by $$D(\alpha)=[M]\cap\alpha$$ is an isomorphism for all k. This is the Poincaré duality. In particular, every closed manifold is $$\mathbb{Z}_2$$-orientable. So there is always an isomorphism $$H^k(M;\mathbb{Z}_2) \cong H_{n-k}(M;\mathbb{Z}_2)$$.

Open manifolds
For a connected manifold, "open" is equivalent to "without boundary and non-compact", but for a disconnected manifold, open is stronger. For instance, the disjoint union of a circle and a line is non-compact since a line is non-compact, but this is not an open manifold since the circle (one of its components) is compact.

Abuse of language
Most books generally define a manifold as a space that is, locally, homeomorphic to Euclidean space (along with some other technical conditions), thus by this definition a manifold does not include its boundary when it is embedded in a larger space. However, this definition doesn’t cover some basic objects such as a closed disk, so authors sometimes define a manifold with boundary and abusively say manifold without reference to the boundary. But normally, a compact manifold (compact with respect to its underlying topology) can synonymously be used for closed manifold if the usual definition for manifold is used.

The notion of a closed manifold is unrelated to that of a closed set. A line is a closed subset of the plane, and a manifold, but not a closed manifold.

Use in physics
The notion of a "closed universe" can refer to the universe being a closed manifold but more likely refers to the universe being a manifold of constant positive Ricci curvature.