Closed manifold

A closed manifold is a compact topological manifold without a boundary . If a manifold without a boundary is given in the context, a compact manifold is automatically a closed one.

The simplest example is a circle with the induced canonical open topology of the . This is a compact one-dimensional manifold without a boundary. Other examples of closed manifolds are the sphere , the projective plane , the Klein bottle and the torus . ${\displaystyle \mathbb {R} ^{2}}$

The opposite example is the real number line, since it is not compact, and the two-dimensional circular disk. The latter is compact, but has an edge.

The concept of a closed manifold must not be confused with the concept of a closed set . (The latter is defined for subsets of a topological space, relative to the topology of this space.) So every submanifold of ${\displaystyle \mathbb {R} ^{n}}$ is automatically closed, as the above examples illustrate, but not necessarily closed as well.

literature

• 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 .