In mathematics, a **closed manifold** is a manifold without boundary that is compact.

In comparison, an **open manifold** is a manifold without boundary that has only *non-compact* components.

## Examples

The only connected one-dimensional example is a circle. The torus and the Klein bottle are closed. A line is not closed because it is not compact. A closed disk is compact, but is not a closed manifold because it has a boundary.

## Open manifolds

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 a line is non-compact since a line is non-compact, but this is not an open manifold since the circle (one of its components) is compact.

## Abuse of language

This section possibly contains original research. (December 2019) |

Most books generally define a manifold as a space that is, locally, homeomorphic to Euclidean space (along with some other technical conditions),thus by this definition a manifold does not include its boundary when it is embedded in a larger space. However, this definition doesn’t cover some basic objects such as a closed disk, so authors sometimes define a manifold with boundary and abusively say *manifold* without reference to the boundary. But normally, a **compact manifold** (compact with respect to its underlying topology) can synonymously be used for **closed manifold** if the usual definition for manifold is used.

The notion of a closed manifold is unrelated to that of a closed set. A line is a closed subset of the plane, and a manifold, but not a closed manifold.

## Use in physics

The notion of a "closed universe" can refer to the universe being a closed manifold but more likely refers to the universe being a manifold of constant positive Ricci curvature.

## References

- 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.