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. 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.