Closed manifold

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See also: Classification of manifolds#Point-set

In mathematics, a closed manifold is 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 or a closed one-form. 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.

[edit] 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.