|
|
| Line 1: |
Line 1: |
| {{Noref|date=December 2009}}
| | Hi there, I am Yoshiko Villareal but I never truly favored that name. Interviewing is what she does. Kansas is our birth place and my parents live close by. One of the issues I love most is greeting card collecting but I don't have the time recently.<br><br>Feel free to visit my web page; extended auto warranty ([http://www.indiatube.nl/profile.php?u=MaFre mouse click the up coming webpage]) |
| {{seealso|Classification of manifolds#Point-set}}
| |
| | |
| In [[mathematics]], a '''closed manifold''' is a type of [[topological space]], namely a [[compact space|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.
| |
| Other examples of closed manifolds are the [[torus]] and the [[Klein bottle]].
| |
| As a counterexample, the [[real line]] is not a closed manifold because it is not compact. A [[Disk (mathematics)|disk]] is a compact two-dimensional manifold, but is not a closed manifold because it has a boundary.
| |
| | |
| 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.
| |
| | |
| All compact topological manifolds can be embedded into <math>\mathbf{R}^n</math> 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]] of a circle and the line is non-compact, but is not an open manifold, since one component (the circle) is compact.
| |
| | |
| The notion of closed manifold is unrelated with that of a [[closed set]]. A disk with its boundary is a closed subset of the plane, but not a closed manifold.
| |
| | |
| ==Use in physics==
| |
| The notion of a "[[Shape of the Universe|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]].
| |
| | |
| == Literature ==
| |
| * [[Michael Spivak]]: ''A Comprehensive Introduction to Differential Geometry.'' Volume 1. 3rd edition with corrections. Publish or Perish, Houston TX 2005, ISBN 0-914098-70-5.
| |
| | |
| {{topology-stub}}
| |
| | |
| [[Category:Geometric topology]]
| |
| [[Category:Manifolds|*]]
| |
Hi there, I am Yoshiko Villareal but I never truly favored that name. Interviewing is what she does. Kansas is our birth place and my parents live close by. One of the issues I love most is greeting card collecting but I don't have the time recently.
Feel free to visit my web page; extended auto warranty (mouse click the up coming webpage)