Closed manifold

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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.

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.

Examples[edit]

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.

Properties[edit]

All compact topological manifolds can be embedded into \mathbf{R}^n for some n, by the Whitney embedding theorem.

Contrasting terms[edit]

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[edit]

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.

References[edit]

  • 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.