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Closed manifold

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] 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 \mathbf{R}^n 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.

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