Closed manifold
Encyclopedia
In mathematics
, a closed manifold is a type of topological space
, namely a compact
manifold
without boundary. In contexts where no boundary is possible, any compact manifold is a closed manifold.
The simplest example is a circle
, which is a compact one-dimensional manifold. As a counterexample, the real line
is not a closed manifold because it is not compact. As another counterexample, a disk
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
. A disk with its boundary is a closed set, but not a closed manifold. When people speak of a closed universe
, 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
of a finite number of connected closed manifolds. One of the most basic objectives of geometric topology
is to understand what the supply of possible closed manifolds is.
Other examples of closed manifolds are the torus
and the Klein bottle
.
All compact topological manifolds can be embedded into
for some n, by the Whitney embedding theorem.
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
of a circle and the line is non-compact, but is not an open manifold, since one component (the circle) is compact.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, 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, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
, namely a compact
Compact space
In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...
manifold
Manifold
In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....
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 points in a plane that are a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius....
, which is a compact one-dimensional manifold. As a counterexample, the real line
Real line
In mathematics, the real line, or real number line is the line whose points are the real numbers. That is, the real line is the set of all real numbers, viewed as a geometric space, namely the Euclidean space of dimension one...
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 geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points...
. 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 a matter of debate in physical cosmology over the local and global geometry of the universe which considers both curvature and topology, though, strictly speaking, it goes beyond both...
, 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 mathematics, the term disjoint union may refer to one of two different concepts:* In set theory, a disjoint union is a modified union operation that indexes the elements according to which set they originated in; disjoint sets have no element in common.* In probability theory , a disjoint union...
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 maps between them, particularly embeddings of one manifold into another.- Topics :...
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...
and the Klein bottle
Klein bottle
In mathematics, the Klein bottle is a non-orientable surface, informally, a surface in which notions of left and right cannot be consistently defined. Other related non-orientable objects include the Möbius strip and the real projective plane. Whereas a Möbius strip is a surface with boundary, a...
.
All compact topological manifolds can be embedded into
for some n, by the Whitney embedding theorem.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 mathematics, the term disjoint union may refer to one of two different concepts:* In set theory, a disjoint union is a modified union operation that indexes the elements according to which set they originated in; disjoint sets have no element in common.* In probability theory , a disjoint union...
of a circle and the line is non-compact, but is not an open manifold, since one component (the circle) is compact.