Closed manifold

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In mathematics, a closed manifold, or compact manifold, is a manifold that is compact as a topological space. In contexts where manifold includes manifolds with boundary, a closed manifold is defined as a compact manifold without boundary (whereas a compact manifold may have a boundary).

These manifolds are those that are, in an intuitive sense, finite. The simplest example of a closed manifold in one dimension is a circle, whereas the real line is not a closed manifold. 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.