Closed manifold

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Closed manifold

In mathematics

Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, a closed manifold is a type of topological space

Topological space

Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connected space, and Continuous function ....
, namely a compact

Compact space

In mathematics, a topological space is called compact if each of its open covers has a finite set subcover.Note: Some authors such as Nicolas Bourbaki use the term "quasi-compact" for this instead, and reserve the term "compact" for topological spaces that are both Hausdorff spaces and "quasi-compact"....
manifold

Manifold

In mathematics, more specifically topology, a manifold is a topological space in which every point has a neighborhood which "resembles" Euclidean space....
without boundary. In contexts where no boundary is possible, any compact manifold is a closed manifold.

The simplest example is a circle

Circle

A circle is a simple shape of Euclidean geometry consisting of those point in a plane which are the same distance from a given point called the center....
, which is a compact one-dimensional manifold. As a counterexample, the real line

Real line

In mathematics, the real line is simply the set R of singleton real numbers.However, this term is usually used when R is to be treated as a space of some sort, such as a topological space or a vector space....
is not a closed manifold because it is not compact. As another counterexample, a disk

Disk (mathematics)

In geometry, a disk is the region in a plane bounded by a circle.A disk is said to be closed or open according to whether or not it contains the circle that constitutes its boundary....
is a compact two-dimensional manifold, but is not a closed manifold because it has a boundary.

The notion of closed manifold must not be confused with a closed set

Closed set

In topology and related branches of mathematics, a closed set is a Set whose complement is open set....
or a closed one-form

One-form

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Encyclopedia

In mathematics

Mathematics

Topological space

Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connected space, and Continuous function ....
, namely a compact

Compact space

Manifold

Circle

Real line

Disk (mathematics)

Closed set

In topology and related branches of mathematics, a closed set is a Set whose complement is open set....
or a closed one-form

One-form

In linear algebra, a one-form on a vector space is the same as a linear functional on the space. The usage of one-form in this context usually distinguishes the one-forms from higher-degree multilinear form on the space....
. A disk with its boundary is a closed set, but not a closed manifold. When people speak of a closed universe

Shape of the Universe

The shape of the Universe is an informal name for a subject of investigation within physical cosmology which describes the geometry of the universe including both #Local geometry and #Global geometry....
, they are almost certainly referring to a closed manifold, not a closed set.

Compact manifolds are, in an intuitive sense, finite. By the basic properties of compactness, a closed manifold is the disjoint union

Disjoint union

In set theory, a disjoint union is a modified union operation which indexes the elements according to which set they originated in.Formally, let be a family of sets indexed by I....
of a finite number of connected closed manifolds. One of the most basic objectives of geometric topology

Geometric topology

In mathematics, geometric topology is the study of manifolds and their embeddings. Low-dimensional topology, concerning questions of dimensions up to four, is a part of geometric topology....
is to understand what the supply of possible closed manifolds is.

Other examples of closed manifolds are the torus

Torus

In geometry, a torus is a surface of revolution generated by revolving a circle in three dimensional space about an axis coplanar with the circle, which does not touch the circle....
and the Klein bottle

Klein bottle

In mathematics, the Klein bottle is a certain non-orientability surface, i.e., a surface with no distinct "inner" and "outer" sides. Other related non-orientable objects include the M?bius strip and the real projective plane....
.

All compact topological manifolds can be embedded into for some n, by the Whitney embedding theorem

Whitney embedding theorem

In mathematics, particularly in differential topology,there are two Whitney embedding theorems:*The strong Whitney embedding theorem states that any connected differentiable manifold m-dimensional manifold can be smooth map embedding in Euclidean space -space, if m>0....
.

Contrasting terms

A compact manifold means a "manifold" that is compact as a topological space, but possibly has boundary. More precisely, it is a compact manifold with boundary (the boundary may be empty). By contrast, a closed manifold is compact without boundary.

An open manifold is a manifold without boundary with no compact component. 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

Disjoint union

In set theory, a disjoint union is a modified union operation which indexes the elements according to which set they originated in.Formally, let be a family of sets indexed by I....
of a circle and the line is non-compact, but is not an open manifold, since one component (the circle) is compact.