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In
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, a closed manifold is a
manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a N ...
without boundary that is
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact, a type of agreement used by U.S. states * Blood compact, an ancient ritual of the Philippines * Compact government, a t ...
. 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 A circle is a shape consisting of all point (geometry), points in a plane (mathematics), plane that are at a given distance from a given point, the Centre (geometry), centre. The distance between any point of the circle and the centre is cal ...
. The
sphere A sphere (from Ancient Greek, Greek , ) is a surface (mathematics), surface analogous to the circle, a curve. In solid geometry, a sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...
,
torus In geometry, a torus (: tori or toruses) is a surface of revolution generated by revolving a circle in three-dimensional space one full revolution about an axis that is coplanarity, coplanar with the circle. The main types of toruses inclu ...
, and the
Klein bottle In mathematics, the Klein bottle () is an example of a Orientability, non-orientable Surface (topology), surface; that is, informally, a one-sided surface which, if traveled upon, could be followed back to the point of origin while flipping the ...
are all closed two-dimensional manifolds. The
real projective space In mathematics, real projective space, denoted or is the topological space of lines passing through the origin 0 in the real space It is a compact, smooth manifold of dimension , and is a special case of a Grassmannian space. Basic properti ...
RP''n'' is a closed ''n''-dimensional manifold. The
complex projective space In mathematics, complex projective space is the projective space with respect to the field of complex numbers. By analogy, whereas the points of a real projective space label the lines through the origin of a real Euclidean space, the points of a ...
CP''n'' is a closed 2''n''-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_(M;\mathbb) is \mathbb or 0 depending on whether M is orientable or not. Moreover, the torsion subgroup of the (n-1)-th homology group H_(M;\mathbb) is 0 or \mathbb_2 depending on whether M is orientable or not. This follows from an application of the
universal coefficient theorem In algebraic topology, universal coefficient theorems establish relationships between homology groups (or cohomology groups) with different coefficients. For instance, for every topological space , its ''integral homology groups'': :H_i(X,\Z) ...
. Let R be a commutative ring. For R-orientable M with fundamental class in H_(M;R) , the map D: H^k(M;R) \to H_(M;R) defined by D(\alpha)= cap\alpha is an isomorphism for all k. This is the
Poincaré duality In mathematics, the Poincaré duality theorem, named after Henri Poincaré, is a basic result on the structure of the homology (mathematics), homology and cohomology group (mathematics), groups of manifolds. It states that if ''M'' is an ''n''-dim ...
.See Hatcher 2002, p.250 In particular, every closed manifold is \mathbb_2-orientable. So there is always an isomorphism H^k(M;\mathbb_2) \cong H_(M;\mathbb_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 In mathematics and more specifically in topology, a homeomorphism ( from Greek roots meaning "similar shape", named by Henri Poincaré), also called topological isomorphism, or bicontinuous function, is a bijective and continuous function betw ...
to
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...
(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 In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
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 In geometry, topology, and related branches of mathematics, a closed set is a Set (mathematics), set whose complement (set theory), complement is an open set. In a topological space, a closed set can be defined as a set which contains all its lim ...
. A line is a closed subset of the plane, and it is 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 In differential geometry, the Ricci curvature tensor, named after Gregorio Ricci-Curbastro, is a geometric object which is determined by a choice of Riemannian or pseudo-Riemannian metric on a manifold. It can be considered, broadly, as a measure ...
.


See also

*


References

* Michael Spivak: ''A Comprehensive Introduction to Differential Geometry.'' Volume 1. 3rd edition with corrections. Publish or Perish, Houston TX 2005, . *
Allen Hatcher Allen Edward Hatcher (born October 23, 1944) is an American mathematician specializing in geometric topology. Biography Hatcher was born in Indianapolis, Indiana. After obtaining his Bachelor of Arts, B.A. and Bachelor of Music, B.Mus. from Ober ...

''Algebraic Topology.''
Cambridge University Press, Cambridge, 2002. {{Manifolds Differential geometry Manifolds Geometric topology