Closed manifold
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 non-compact components.
Examples
The only 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
Most books generally define a manifold as a space that is, locally, diffeomorphic to Euclidean space, thus by this definition, every manifold does not have a boundary. However, this definition is too specific as it doesn’t cover even basic objects such as a closed disk, so authors usually define a manifold with boundary and abusively say manifold without reference to the boundary. Due to this, a compact manifold (compact with respect to its underlying topology) can synonymously be used for closed manifolds if the definition is taken to be original definition.
The notion of a closed manifold is unrelated with that of a closed set. A disk with its boundary is a closed subset of the plane, 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.