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 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 for some n, by the Whitney embedding theorem.
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
- 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.
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